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Policy for Continuous +Control Tasks +Chao Li* +Institute of Automation, Chinese +Academy of Sciences, China +lichao2021@ia.ac.cn +Chen Gong∗ +Institute of Automation, Chinese +Academy of Sciences, China +gongchen2020@ia.ac.cn +Qiang He +University of Tubingen, Germany +qianghe97@gmail.com +Xinwen Hou† +Institute of Automation, Chinese +Academy of Sciences, Beijing, China +xinwen.hou@ia.ac.cn +Yu Liu +Institute of Automation, Chinese +Academy of Sciences, Beijing, China +yu.liu@ia.ac.cn +ABSTRACT +The deep reinforcement learning (DRL) algorithm works brilliantly +on solving various complex control tasks. This phenomenal success +can be partly attributed to DRL encouraging intelligent agents to +sufficiently explore the environment and collect diverse experiences +during the agent training process. Therefore, exploration plays a +significant role in accessing an optimal policy for DRL. Despite +recent works making great progress in continuous control tasks, +exploration in these tasks has remained insufficiently investigated. +To explicitly encourage exploration in continuous control tasks, +we propose CCEP (Centralized Cooperative Exploration Policy), +which utilizes underestimation and overestimation of value func- +tions to maintain the capacity of exploration. CCEP first keeps two +value functions initialized with different parameters, and generates +diverse policies with multiple exploration styles from a pair of value +functions. In addition, a centralized policy framework ensures that +CCEP achieves message delivery between multiple policies, fur- +thermore contributing to exploring the environment cooperatively. +Extensive experimental results demonstrate that CCEP achieves +higher exploration capacity. Empirical analysis shows diverse ex- +ploration styles in the learned policies by CCEP, reaping benefits in +more exploration regions. And this exploration capacity of CCEP +ensures it outperforms the current state-of-the-art methods across +multiple continuous control tasks shown in experiments. +KEYWORDS +Deep Reinforcement Learning, Cooperative Exploration, Continu- +ous Control Tasks +1 +INTRODUCTION +Deep reinforcement learning (DRL) [46], which utilizes deep neu- +ral networks to learn an optimal policy, works brilliantly and has +demonstrated beyond human-level performance in solving various +challenging sequential decision-making tasks e.g., video games [17, +34, 35, 52, 54], autonomous driving [18, 53], robotic control tasks [2, +31], etc. In DRL settings, an agent needs to sufficiently explore the +environment and collect a set of experiences to obtain an optimal +policy. The agent aims to learn an optimal policy to maximize its ex- +pected cumulative rewards through trial and error. Therefore, DRL +∗These two authors contributed equally. +†The corresponding author. +can be regarded as learning from reward feedback from environ- +ments. It is essential that during the training phase the agent should +be encouraged to explore the environments and gather sufficient +reward signals for well-training. +In DRL, exploration has obsessed with a critical problem: submit- +ting solutions too quickly without sufficient exploration, leading +to getting stuck at local minima or even complete failure. DRL +researchers adopt the neural network to yield the policy with sig- +nificant feature extraction and expression capabilities in a range of +continuous control tasks. Whereas this phenomenal practice has +achieved great performance, it is still obsessed with the notorious +insufficient exploration problem in continuous control tasks. Good +exploration becomes extremely difficult when the environment +is distracting or provides little feedback. Whereas existing explo- +ration methods remain a problematic drawback – lacking diversity +to explore. The classic exploration methods such as 𝜖-Greedy strate- +gies [35] or Gaussian noise [31] indirectly and implicitly change +the style of the exploration. However, in massive situations, diverse +styles of exploration are necessary. For instance, in chess games, +players should perform different styles of policies (e.g., radical, con- +servative, etc.) to keep competitive when facing various situations; +humanoid robots attempt diverse control styles and eventually learn +to walk efficiently. +Recent studies enrich diverse styles of policies by construct- +ing relationships between the distribution of policy and trajecto- +ries [1, 13, 26, 27, 43]. In Diversity is All You Need (DIAYN) [11], +authors highlight the diversity of policies plays a significant role in +well-training agents. It trains the policy that maximizes the mutual +information between the latent variable and the states, then altering +the latent variable of the policy network creates multiple policies +performing disparately. Although this interesting viewpoint has +attracted a spectrum of following works [1, 11, 13, 20, 43], the afore- +mentioned methods achieve diverse policies depending on the task +in an unsupervised way, resulting in the algorithm performing in- +sufficient generality in different tasks. In fact, the ideal algorithm +to implement the diverse policies should be general across a range +of tasks, which motivates us to design a task-oriented algorithm +working towards developing various policies. Eysenbach et al. pro- +posed [12] that the learned skills could not construct all the state +marginal distributions in the downstream tasks. For task relevance, +the mutual information is added as an intrinsic reward for em- +powerment [7, 36], but these methods change the original reward +arXiv:2301.02375v1 [cs.LG] 6 Jan 2023 + +function resulting in the performance being extremely sensitive to +the trade-off between original and intrinsic rewards. +Our method insights from an interesting phenomenon during +the exploration. The critic aims to approximate the accumulated +reward by bootstrapping in the actor-critic framework. However, +the different critic functions may have great differences even if they +approximate the same target due to the function approximation er- +ror. For instance, Twin Delayed Deep Deterministic policy gradient +(TD3) [16] presents that two value functions with different initial +parameters perform quite differently with identical targets. It is +knotty to measure whether a value function is exact or not, and +the gap between these two value functions is termed as controversy, +which sees a decreasing trend along with the value function updat- +ing process. Our intuition can be ascribed that controversy in the +value estimation will lead to sub-optimal policies, and these policies +have a bias toward message acquisition known as the style. +This paper highlights that controversy can be utilized to encour- +age policies to yield multiple styles, and encourages exploration +for a continuous control task by applying multi-styled policies. +Our paper contributes three aspects. (1) We first describe that the +estimation bias in double value functions can lead to various ex- +ploration styles. (2) This paper proposes the CCEP algorithm, +encouraging diverse exploration for environments by cooperation +from multi-styled policies. (3) Finally, in CCEP, we design a novel +framework, termed as the centralized value function framework, +which is updated by experience collected from all the policies and +accomplishes the message delivery mechanism between different +policies. Extensive experiments are conducted on the MuJoCo plat- +form to evaluate the effectiveness of our method. The results reveal +that the proposed CCEP approach attains substantial improvements +in both average return and sample efficiency on the baseline across +selected environments, and the average return of agents trained +using CCEP is 6.7% higher than that of the baseline. Besides, CCEP +also allows agents to explore more states during the same train- +ing time steps as the baseline. Additional analysis indicates that +message delivery leading to the cooperative multi-styled policies +further enhanced the exploration efficiency by 8.6% compared with +that of without cooperation. +We organize the rest of this paper as follows. Section 2 briefly +explains the concepts of RL. We elaborate on how to perform the +CCEP algorithm in Section 3. We introduce our experimental set- +tings in Section 4. Section 5 presents and analyzes results to validate +the effectiveness of CCEP, after which section 6 discusses related +works. Finally, we conclude our paper in Section 7. The code and +documentation are released in the link for validating reproducibil- +ity.1 +2 +PRELIMINARIES +Reinforcement learning (RL) aims at training an agent to tackle the +sequential decision problems that can be formalized as a Markov +Decision Process (MDP). This process can be defined as a tuple +(S, A, 𝑃,𝑟,𝛾), where S is the state space, A is the action space, +𝑃 : S × A × S ↦→ [0, 1] denotes the transition probability, 𝑟 (𝑠,𝑎) is +the reward function 𝑟 : S ×A ↦→ R, determining the reward agents +will receive in the state 𝑠 while executing the action 𝑎. The 𝛾 ∈ +1https://github.com/Jincate/CCEP +(0, 1) is the discount factor. The return is defined as the discounted +accumulated reward. +𝑅 = +∞ +∑︁ +𝑡=0 +𝛾𝑡𝑟 (𝑠𝑡,𝑎𝑡) +(1) +In the DRL community, developers usually use the neural network +parameterized with 𝜙 to indicate the policy 𝜋(𝑎|𝑠), which inputs +an observation and outputs an action. The goal of DRL is to solve +this MDP process and find the optimal policy 𝜋𝜙∗ : S ↦→ A with +parameter 𝜙∗ that maximizes the expected accumulated return. +𝜙∗ = arg max +𝜙 +E𝑎𝑡 ∼𝜋𝜙 (·|𝑠𝑡 ),𝑠𝑡+1∼𝑃 (·|𝑠𝑡,𝑎𝑡 ) +� ∞ +∑︁ +𝑡=0 +𝛾𝑡𝑟 (𝑠𝑡,𝑎𝑡) +� +(2) +David Silver, et al. [44] propose that solving Eq. (2) with determin- +istic policy gradient strategy, +∇𝜙 𝐽 (𝜙) = E𝑠𝑡+1∼𝑃 (·|𝑠𝑡,𝑎𝑡 ) +� +∇𝑎𝑄𝜋 (𝑠,𝑎)|𝑎=𝜋 (𝑠)∇𝜙𝜋𝜙 (𝑠) +� +(3) +where 𝑄𝜋 (𝑠,𝑎) = E𝑎𝑡 ∼𝜋𝜙 (·|𝑠𝑡 ),𝑠𝑡+1∼𝑃 (·|𝑠𝑡,𝑎𝑡 ) [𝑅|𝑠,𝑎] is known as the +value function, indicating how good it is for an agent to pick action +𝑎 while being in state 𝑠. To use the gradient-based approach (e.g., +Stochastic Gradient Descent [41]) to solve this equation, deep Q- +learning uses the neural network to approximate the value function. +The value function parameterized with 𝜃 is updated by minimizing +the temporal difference (TD) error [45] between the estimated value +of the subsequent state 𝑠′ and the current state 𝑠. +𝜃∗ = arg min +𝜃 +E +� +𝑟 (𝑠,𝑎) + 𝛾𝑄𝜋 +𝜃 (𝑠′,𝑎′) − 𝑄𝜋 +𝜃 (𝑠,𝑎) +�2 +(4) +We store the trajectories of the agent exploring the environment +in a replay buffer [32] from which sample a random mini-batch of +samples, updating the parameters mentioned above. +3 +CENTRALIZED COOPERATIVE +EXPLORATION POLICY +This section details technologies of CCEP (Centralized Cooper- +ative Exploration Policy). We first analyze value estimation bias +from function approximation errors and generate multi-styled value +functions by encouraging overestimation bias and underestimation +bias for the value functions, respectively. To achieve multi-styled +exploration, we propose a multi-objective update method for train- +ing policy and randomly select one policy to explore at each time +step. These historical trajectories during exploration are stored for +training a single policy function to achieve cooperative message +delivery. We denote our policy as 𝜋(𝑠,𝑧), where 𝑧 is a one-hot label +and represents different policies. In this work, we focus on gener- +ating multiple policies with different styles to encourage diverse +exploration. We implement our method based on TD3 [16] which +maintains double critics and uses the minimum of the critics as the +target estimate. +3.1 +Function Approximation Error +This section shows that there exist approximation errors in the value +function optimization and can accumulate to substantial scales. The +accumulated approximation error will lead to value estimation bias, +which plays a significant role in policy improvement. + +Sample one style per step +𝒛~𝒑(𝒛) +𝒂𝒕~𝝅𝝓(𝒂𝒕|𝒔𝒕, 𝒛𝒕) +STYLE +𝒔𝒕�𝟏~𝒑 (𝒔𝒕�𝟏|𝒔𝒕, 𝒂𝒕) +ENVIRONMENT +𝒂𝒕 +REPLAY BUFFER +𝒔𝒕�𝟏 +𝒛 +𝒛 +𝒔𝒕�𝟏 +Sample a batch of N transitions +(𝒔𝒕, 𝒂𝒕, 𝒛𝒕, 𝒔𝒕�𝟏, 𝒛𝒕�𝟏, 𝒓) +UPDATE CRITICS +𝜽𝒊 +𝒕�𝟏 ←𝐚𝐫𝐠 𝐦𝐢𝐧 +𝜽𝒊 𝑵�𝟏∑�𝒚 − 𝑸𝜽𝒊(𝒔, 𝒂)� +𝟐 +UPDATE POLICY +Generate critics by Eq. (9) +𝝓𝒕�𝟏 ←𝐚𝐫𝐠 𝐦𝐚𝐱 +𝝓 +𝟏 +𝟒 � +𝒌=𝟏 +𝟒 +𝑸𝒌�𝒔, 𝝅(𝒔, 𝒛𝒌)� +𝝓𝒕�𝟏 +POLICY + No.1 +POLICY + No.2 +POLICY + No.3 +POLICY + No.4 +Cooperative exploration +Centralization +Figure 1: The workflow of CCEP Algorithm. The agent 𝜋 interacts with the environment with diverse style cooperatively and +produce the transition 𝑠𝑡 → 𝑠𝑡+1. The actor and critic are updated over a mini-batch of the transition samples. A centralized +policy with four different styles is learned from the multi-styled critics. +In value-based deep reinforcement, deep neural networks ap- +proximate the value functions, and the function approximation +error exists correspondingly. One major source of the function +approximation error comes from the optimization procedure. In +this procedure, stochastic gradient descent, which uses a batch of +random samples for gradient update each time, is the mainstream +method due to the consideration of computational resources and +training efficiency. However, as [40] has indicated, a mini-batch +gradient update may have unpredictable effects on samples outside +the training batch, which leads to the function approximation error. +For explanation, we use 𝑒𝑡 to represent the approximation error of +the value function with the state-action pair(𝑠𝑡,𝑎𝑡) as input and +approximation error 𝑒𝑡 can be modeled as follows: +𝑄𝜃 (𝑠𝑡,𝑎𝑡) = 𝑟 (𝑠𝑡,𝑎𝑡) + 𝛾E[𝑄𝜃 (𝑠𝑡+1,𝑎𝑡+1)] − 𝑒𝑡 +(5) +Approximation errors influence the value estimation when using +the value function as an estimator. The estimation may be skewed +towards an overestimation, causing a wrong estimate for a given +state. This leads to a problem of an optimal action being chosen but +replaced by a sub-optimal action, owing to the overestimation of +a sub-optimal action. Thus, the overestimation bias is a common +problem in Q-Learning with discrete actions, as we choose the +seemly best action 𝑎𝑡+1 in the target value. Still, there is little chance +for the optimal state-action pair to be updated. +E[ max +𝑎𝑡+1∈A 𝑄(𝑠𝑡+1,𝑎𝑡+1)] ≥ +max +𝑎𝑡+1∈A E[𝑄(𝑠𝑡+1,𝑎𝑡+1)] +(6) +Mentioned overestimated bias can also occur in continuous con- +trol tasks[16], since the policy approximator always provides the +optimal action at the current state based on the value function. +While this bias can be quite small in an individual update, the bias +can be accumulated to a substantial overestimation. Eq.(5) can be +expanded as follows: +𝑄𝜃 (𝑠𝑡,𝑎𝑡) = E𝑎𝑡 ∼𝜋𝜙 (·|𝑠𝑡 ),𝑠𝑡+1∼𝑃 +� ∞ +∑︁ +𝑡=0 +𝛾𝑡 (𝑟 (𝑠𝑡,𝑎𝑡) − 𝑒𝑡) +� +(7) +Previous works such as Double Q-learning [23] and Double DQN [24] +are proposed to alleviate value functions of underestimating. The +idea is to maintain two independent estimators in which one is +used for estimation while the other is for selecting maximal action. +Similarly, as an extension in dealing with continuous control tasks, +TD3 [16] reduce the overestimation bias by using double value func- +tions and taking the minimum between the two value functions +for an estimation which suffers from underestimation problems as +well [50, 51]. +Does the estimation error influence the performance? Given +a continuous control task, we use 𝑓 to approximate the true under- +lying value function 𝑄∗, which indicates the accumulated reward +obtained by acting 𝑎 before taking optimal policy 𝜋∗ at state 𝑠. 𝑉 +represents the true underlying value function(which is not known +during training). 𝑉 ∗ and 𝑉 𝜋𝑓 represent the accumulated return +obtained by adopting the optimal policy 𝜋∗ and 𝜋𝑓 in state 𝑠 re- +spectively in which 𝜋𝑓 is a learned policy by maximizing the value +function approximate 𝑓 . +Lemma 1. (Performance Gap). The performance gap of the policy +between the optimal policy 𝜋∗ and the learned policy 𝜋𝑓 is defined +by an infinity norm ∥𝑉 ∗ − 𝑉 𝜋𝑓 ∥∞ and we have +∥𝑉 ∗ − 𝑉 𝜋𝑓 ∥∞ ≤ 2∥𝑓 − 𝑄∗∥∞ +1 − 𝛾 +We provide proof detail of Lemma 1 in Supplementary A. This +inequality indicates that the performance gap of the policy can be +bounded by the estimation error of the value function and accurate +value estimate can reduce the upper bound of the performance gap + +and enhance the performance. +Do overestimation bias and underestimation bias affect per- +formance in the same way? An empirical study shows that esti- +mation bias may not always be a detrimental problem while both +underestimation bias and overestimation bias may improve learn- +ing performance which depends on the environment [30]. As an +example, for an unknown area with high stochasticity, overesti- +mation bias may help to explore the overestimated area but un- +derestimation bias prevents this. However, if these areas of high +stochasticity are given low values, the overestimation bias may lead +to excess exploration in low-value regions. The fact is that we can +not choose the environment and these different circumstances can +always occur during exploration. Our method is designed to utilize +the difference in exploration behavior brought by estimation bias +to encourage multi-styled exploration. +3.2 +Multi-Style Critics: Radical, Conservative +As mentioned above, function approximation error exists in value +functions and can accumulate to substantial scales which have a +great influence on the value estimation resulting in overestimation +or underestimation bias. Estimation bias has been researched in +recent works [16, 25, 50, 51]. While these works focus on an accurate +value estimation and discussed the method to control the estimation +bias with the use of multiple value functions for auxiliary, they +just choose one of the value functions, which seems to be the +most accurate, for policy update neglecting other value functions. +However, there is no accurate value function without trial and +error. In this section, we show how to utilize the estimation bias +and introduce our method for the generalization of multi-styled +critics. +Our intuition is that there are different degrees of estimation +bias in double randomly initialized value functions when perform- +ing function approximation. However, the estimation bias can be +controlled by applying a maximum operator and minimum op- +erator, namely the maximum of the two estimates is relatively +overestimated and the minimum of the two estimates is relatively +underestimated. Two different estimates raise a controversy about +which critic gives the accurate estimate. The best way to resolve +the controversy is to follow one of the critics to explore and collect +reward messages. While controversy does not always exist because +there is only one accurate value, the critics reach an agreement +when the state value has been exactly estimated. And the existence +of controversy means more exploration is needed. +We start by maintaining double randomly initialized value func- +tions 𝑄𝜃1 and 𝑄𝜃2 with parameters 𝜃1 and 𝜃2 respectively and up- +date the value function with TD3 [16] which takes the minimum +between the two value functions as the target value estimate: +𝑦 = 𝑟 + min +𝑖=1,2𝑄𝜃𝑖 (𝑠′,𝑎′),𝑎′ ∼ 𝜋𝜙 +(8) +But the two randomly initialized value functions potentially have +different value estimations for a given state-action pair due to the +accumulated function approximation error. This difference leads to +the result that the two critics may give two different suggestions +for the best action choice. While these estimates are relatively +overestimated or underestimated, these different criteria for a given +state-action pair may lead to a different style of action choice. It +Algorithm 1 Centralized Cooperative Exploration Policy (CCEP) +Initialize critic networks 𝑄𝜃1,𝑄𝜃2 +Initialize actor network 𝜋𝜙 with random parameters 𝜃1,𝜃2,𝜙 +Initialize target networks 𝜃 ′ +1 ← 𝜃1,𝜃 ′ +2 ← 𝜃2,𝜙′ ← 𝜙 +Initialize replay buffer B +Initialize number of skills K +1: for 𝑡 = 1 to 𝑇 do +2: +Sample a skill 𝑧 from 𝑝(𝑧) +3: +Select action with noise 𝑎 ∼ 𝜋𝜙 (𝑠,𝑎) + 𝜖,𝜖 ∼ N (0, 𝜎) +4: +Observe a reward 𝑟 and a new state 𝑠′ +5: +Store transition tuple (𝑠,𝑧,𝑎,𝑟,𝑠′,𝑧′) in B +6: +Sample mini-batch of 𝑁 transitions (𝑠,𝑧,𝑎,𝑟,𝑠′,𝑧′) from 𝐵 +7: +𝑎′ ← 𝜋𝜙′(𝑠′,𝑧′) + 𝜖,𝜖 ∼ 𝑐𝑙𝑖𝑝(N (0, 𝜎), −𝑐,𝑐) +8: +𝑦 = 𝑟 + 𝛾 min𝑖=1,2 𝑄𝜃′ +𝑖 (𝑠′,𝑎′) +9: +Update critics: 𝜃𝑖 ← arg min𝜃𝑖 𝑁 −1 �(𝑦 − 𝑄𝜃𝑖 (𝑠,𝑎))2 +10: +if t mod d then +11: +Update policy: +12: +∇𝜙 𝐽 (𝜙) = 𝑁 −1K−1 � ∇𝑎𝑄 𝑗 (𝑠,𝑎)|𝑎=𝜋𝜙 (𝑠,𝑧)∇𝜙𝜋𝜙 (𝑠,𝑧) +13: +Update target networks +14: +𝜃 ′ +𝑖 ← 𝜏𝜃𝑖 + (1 − 𝜏)𝜃 ′ +𝑖 +15: +𝜙′ +𝑖 ← 𝜏𝜙𝑖 + (1 − 𝜏)𝜙′ +𝑖 +is reasonable the estimation is radical if we choose the maximum +value of the two to estimate and the estimation is conservative if +we choose the minimum value of the two. Thus, we consider four +critics: +𝑄 𝑗 (𝑠,𝑎) = + + +𝑄𝜃1 (𝑠,𝑎) +𝑗 = 0 +𝑄𝜃2 (𝑠,𝑎) +𝑗 = 1 +max(𝑄𝜃1 (𝑠,𝑎),𝑄𝜃2 (𝑠,𝑎)) +𝑗 = 2 +min(𝑄𝜃1 (𝑠,𝑎),𝑄𝜃2 (𝑠,𝑎)) +𝑗 = 3 +(9) +There exists controversy among these critics, and the controversy +can further influence the performance of the policy learned. +3.3 +Opposite Value Functions +This approach for generating diverse styles raises the problem that +the value functions may not provide sufficient difference in style +when the controversy disappear. This phenomenon is very com- +mon when the value function converges. But we don’t want this to +happen too soon, because we want the value functions to provide +more exploration for the policy. The controversy exists due to the +randomly initialized parameters of the neural networks and the +error accumulation. But actually, there is a small probability that +the two networks have great similarities, which will lead to double +consistent critics. This is not what we want, because consistent +critics mean monotonous policy. To avoid this, we try to enlarge the +controversy. The solution in this paper is to learn two opposite tar- +gets respectively for the two networks, where one of the networks +approximates the positive value and another approximates the neg- +ative one. This approach is equivalent to adding a factor -1 to the +final layer of either network. We find that the controversy is guar- +anteed with this simple network structure change. To provide some +intuition, we compared the controversy changes after the double +value functions learn the opposite target. We express the amount + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Time Steps (1e6) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Average Error +(a) HalfCheetah-v3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Time Steps (1e6) +0 +1 +2 +3 +Average Error +Target +same target +opposite target +(b) Walker2d-v3 +Figure 2: Measuring the error between double critics given +same/opposite targets in TD3 on MuJoCo environments over +1 million time steps +of controversy between the two value functions by the errors of +state values in a batch of samples. Figure 2 shows the controversy +measuring over MuJoCo [48] environments in HalfCheetah-v3 and +Walker2d-v3. The results show that with the simple network struc- +ture change, the controversy is enlarged. But this approach will +not influence the value estimation because we just fine-tune the +structure. +3.4 +Centralized Cooperation +With four critics, we train a centralized cooperative policy to encour- +age multi-styled explorations through diverse value estimations. +We model this problem as a multi-objective optimization problem. +The target is to train multiple policies, with each policy targeting an +individual value function. We express the policy function as 𝜋(𝑠,𝑧), +with state 𝑠 and latent variable 𝑧 as input. The latent variable 𝑧, +which is a one-hot label in our method, represents different policies. +The architecture of our centralized cooperative policy is shown +in Figure 3. This idea comes from skill discovery method [11, 43], +which use the latent variable 𝑧 to express different skills. And in +skill discovery, the target is to maximize the mutual information be- +tween latent variable 𝑧 and some aspects of the trajectories, which +is a different target for a different latent variable 𝑧. Our method +encourages diverse styles of policies by different targets as well. +Particularly, we sample latent variable 𝑧 from set {0, 1, 2, 3} and +encode it in a one-hot label. For a given latent variable 𝑧, the policy +targets 𝑧-th value functions in Eq.(9). With different latent variable +𝑧, the policy shows diverse styles due to the multi-styled targets. +We make an experiment showing that there exists different explo- +ration preferences for these policies (Section 5.2) In the exploration +procedure, we randomly sample latent variable 𝑧 and make deci- +sions by policy 𝜋(𝑠,𝑧). This approach enables diverse styles to be +applied at each time step. Broadly speaking, our exploration policy +has the following characteristics: Multi-styled, Centralized, and +Cooperative. +Multi-styled. We train four policies to accomplish the explo- +ration. These policies learn from the corresponding value function +𝑄 𝑗: +𝜋∗ +𝑗 = arg max +𝜋 +𝑄 𝑗 +(10) +There are four value estimators, in which two of them (𝑗 = 0, 1) +are normal but different estimators, one (𝑗 = 2) is an overestimated +Policy 1 +Policy 2 +Policy 3 +Policy 4 +Centralized +Cooperative +Policy +Environment +Label variance +Input +Output +Figure 3: The architecture of our centralized cooperative pol- +icy. The agent cooperatively explores the environments by +selecting one of the styles at each time step. The style se- +lection process is implemented by sampling latent variable +𝑧. Policies with diverse styles exchange messages through a +centralized network. +estimator compared to the other (𝑗 = 3) and helps encourage ex- +plorations in overestimated actions, the last remaining one (𝑗 = 3) +is a conservative estimator and brings more exploitation as illus- +trated in the previous section. It is appropriate for the policies to +perform in a variety of ways given the varied estimators they use +(e.g., conservative, radical). +Centralized. Our policy is a centralized policy because we make +use of all the policies learned in each episode. At each time step𝑡, we +sample one of these policies for exploration. It allows us to generate +a variety of trajectories adopting this exploration approach as this +centralized policy can generate 4𝑛 types of trajectories theoretically +for 𝑛-step exploration, compared to using a single policy that can +only generate one. These trajectories are stored as experience and +maintain the update for a pair of centralized value functions. +Cooperative. We update the policy cooperatively. With multi- +ple policies learning their respective value functions, “knowledge” +learned by each policy cannot be shared. Our method is to learn a +single network for policies and learn cooperatively [4]. To repre- +sent different policies, we feed latent variables 𝑧 which are one-hot +labels as extra input to the network. The policy which inputs latent +variable 𝑧 and state𝑠 and outputs action𝑎 can be defined as 𝜋(𝑠,𝑧). +We sample 𝑧 to represent the sampling of different policies. Thus, +the policy can be updated by taking deterministic policy gradient. +∇𝜙 𝐽 (𝜙) = 𝑁 −1K−1 ∑︁ +∇𝑎𝑄 𝑗 (𝑠,𝑎)|𝑎=𝜋𝜙 (𝑠,𝑧)∇𝜙𝜋𝜙 (𝑠,𝑧) +(11) +Where 𝑄 𝑗 (𝑠,𝑎) refer to the multi-styled critics in Eq.(9), K is the +number of styles which is 4 in this algorithm. The specific algorithm +is shown in Algorithm 1. +4 +EXPERIMENTAL SETTINGS +To evaluate our method, we test our algorithm on the suit of +MujoCo [48] continuous control tasks, including HalfCheetah-v3, + +G(a) +(b) +(c) +(d) +Figure 4: Screenshots of MuJoCo environments. (a) Ant-v3, +(b) HalfCheetah-v3, (c) Walker2d-v3, (d) Hopper-v3 +Hopper-v3, Walker2d-v3, Ant-v3, Pusher-v2 and Humanoid-v3 (the +screenshots are presented in Figure 4). +For implementation, our method builds on TD3 [16], and for com- +parison, we also establish three-layer feedforward neural networks +with 256 hidden nodes per hidden layer for both critics and actors. +Particularly, the actor takes state 𝑠 and latent variable 𝑧 concate- +nated as input, where the latent variable 𝑧 is encoded as one-hot +label. At each time step, both networks are trained with a mini- +batch of 256 samples. We apply soft updates for target networks as +well. +We compared our algorithm against some classic algorithms +such as DDPG [30], which is an efficient off-policy reinforcement +learning method for continuous tasks; PPO [42], the state-of-the- +art policy gradient algorithms; TD3 [16], which is an extension +to DDPG; SAC [22], which is an entropy-based method with high +sample efficiency. Further, we compared our algorithm with the +latest algorithm in solving the exploration problems in continuous +control tasks such as OAC [8], which makes improvements on SAC +for better exploration. We implement DDPG and PPO by OpenAI’s +baselines repository and SAC, TD3, and OAC by the github the +author provided. And we use the parameter the author recommend +for implementation. The details of the implementation are shown +in Supplementary B. +5 +EXPERIMENTAL RESULTS AND ANALYSIS +5.1 +Evaluation +To validate the performance of the CCEP algorithm, we evaluate +our algorithm in MuJoCo continuous control suites. We perform +interactions for 1 million steps in 10 different seeds and evaluate +the algorithm over 10 episodes every 5k steps. Our results report +the mean scores and standard deviations in the 10 seeds. We show +learning curves in Figure 5 and the max average return over 10 trials +of 1 million time steps in Table 1. The learning curves in 1 million +time steps show that our algorithm achieves a higher sample effi- +ciency compared with the latest algorithm. Furthermore, the results +in the Table 1 indicates that our algorithm shows superior perfor- +mance. And in HalfCheetah-v3, Walker2d-v3, Hopper-v3, Ant-v3, +Table 1: The highest average return over 10 trials of 1 million +time steps. The maximum value for each task is bolded. +Environment +Ours +OAC +SAC +TD3 +DDPG +PPO +HalfCheetah +11945 +9921 +11129 +9758 +8469 +3681 +Hopper +3636 +3364 +3357 +3479 +2709 +3365 +Walker2d +4706 +4458 +4349 +4229 +3669 +3668 +Ant +5630 +4519 +5084 +5142 +1808 +909 +Pusher +-21 +-25 +-20 +-25 +-29 +-21 +Humanoid +5325 +5747 +5523 +5356 +1728 +586 +our algorithm outperforms all the other baselines and achieve sig- +nificant improvements. While in the Pusher-v2 task, our algorithm +show higher stability than that of TD3. For further evaluation, we +evaluate our algorithm in the state-based suite PyBullet [9] which +is considered to be harder than the suite MuJoCo. Our algorithm +still shows better performance compared to the baseline algorithms. +The corresponding results are shown in Supplementary C.1. +5.2 +Policy Style +To ensure that our proposed CCEP algorithm learns diverse styles, +we compared the distribution of explored trajectories when ex- +ploring with a single style only. We test the algorithm in Ant-v3 +environment over 1𝑒6 time steps and use the states sampled to rep- +resent the trajectories. Figure 6 shows the states explored by each +style at 1𝑒5, 2𝑒5 and 3𝑒5 learning steps, and a more detailed results +are shown in Supplementary C.3. We collect the states sampled over +10 episodes with different seeds and apply t-SNE [49] for better +visualization. The results show that while part of the states can be +gathered by all styles which implies a compromise in controversy, +there is a considerably large region of states that can only be ex- +plored by a unique style of policy. Though different styles, diverse +styles come to be in compromise as training process goes on. This +phenomenon suggests that CCEP behaves in multi-styled explo- +ration which leads to an exploration preference, and styles come +to an agreement with sufficient exploration. Another phenomenal +conclusion is that although the style tends to be consistent, new +styles are emerging which brings enduring exploration capabilities. +5.3 +Measuring Exploration Ability +The critical problem of our proposed method is whether we achieve +higher sample efficiency. Although the learning curves (Figure 5) +gives considerably convincing results, a more intuitive result has +been given in Figure 7. We compared the exploration of CCEP +with that of TD3 and SAC (which achieve the trade-off between +exploration and exploitation by entropy regularization.) over 10 +episodes with different seeds (Figure 7). For a fair comparison, these +methods are trained in Ant-v3 with the same seed at half of the +training process. In order to get reliable results, the states explored +are gathered in 10 episodes with different seeds. We still apply the +same t-SNE [49] transformation to the states explored by all of the +algorithms for better visualization. While there are great differences +between the states explored by TD3 (green) and SAC (blue), the +result shows that our algorithm (red) explores a wider range of +states which even covers that TD3 and SAC explored. + +0.00 +0.25 +0.50 +0.75 +1.00 +0 +5000 +10000 +Average Return +HalfCheetah-v3 +0.00 +0.25 +0.50 +0.75 +1.00 +0 +2000 +4000 +Walker2d-v3 +0.00 +0.25 +0.50 +0.75 +1.00 +0 +1000 +2000 +3000 +4000 +Hopper-v3 +0.00 +0.25 +0.50 +0.75 +1.00 +Time Steps (1e6) +0 +2000 +4000 +Average Return +Humanoid-v3 +0.00 +0.25 +0.50 +0.75 +1.00 +Time Steps (1e6) +2000 +0 +2000 +4000 +6000 +Ant-v3 +0.00 +0.25 +0.50 +0.75 +1.00 +Time Steps (1e6) +80 +60 +40 +20 +Pusher-v2 +Ours +OAC +PPO +DDPG +TD3 +SAC +Figure 5: Learning curves for 6 MuJoCo continuous control tasks.For better visualization,the curves are smoothed uniformly. +The bolded line represents the average evaluation over 10 seeds. The shaded region represents a standard deviation of the +average evaluation over 10 seeds. +(1) Learning Steps = 100000 +(2) Learning Steps = 200000 +(3) Learning Steps = 300000 +POLICY No.1 +POLICY No.2 +POLICY No.3 +POLICY No.4 +Figure 6: The states visited by each style. For better visualization, the states get dimension reduction by t-SNE. The points with +different color represents the states visited by the policy with the style. The distance between points represents the difference +between states. +5.4 +Ablation Study +We perform an ablation study to understand the contribution of the +cooperation between policies for message delivery. The results are +shown in Table 2 where we compare the performance of training +policies by removing policy cooperation and training them sepa- +rately. We perform interactions for 1 million time steps for each +method. The results show that without cooperation, the policy net- +work not only trains 4 times more network parameters but also +fails to reduce performance. And this performance degradation is +even more pronounced on Walker2d. Additional learning curves +can be found in Supplementary C.2. +6 +RELATED WORK +This section discusses several methods proposed recently for im- +proving the exploration of deep reinforcement learning. +A range of works take an effort in encouraging explorations with + +(1) Ours +(2) TD3 +(3) SAC +Figure 7: Measuring the exploration region. Comparison of exploration capabilities of TD3 (green), SAC (blue) and Ours (red). +The points represent region explored by each method in 10 episodes. All the states get dimension reduction by the same t-SNE +transformation for better visualization. +Table 2: Max Average Return over 5 trials of 1 million time +steps, comparing ablation over cooperation for message de- +livery. The maximum value for each task is bolded. +Method +HCheetah +Hopper +Walker2d +Ant +CCEP +11969 +3672 +4789 +5488 +CCEP-Cooperation +11384 +3583 +4087 +4907 +TD3 +9792 +3531 +4190 +4810 +the use of randomness over model parameters [6]. Another preva- +lent series of works propose to enhance exploration by simulta- +neously maximizing the expected return and entropy of the pol- +icy [15, 21, 22, 39, 55]. Whereas, these methods do not provide +heuristic knowledge to guide the exploration, which can be consid- +ered to be insufficient and time-consuming. +To achieve effective exploration, the curiosity mechanism [19, 38] +has been proposed in recent works, e.g., the counted-based ap- +proaches [33] which quantify the “novelty” of a state by the times +visited. However, these methods maintain the state-action visitation +counts which make it challenging in solving high-dimensional or +continuous tasks. Other works rely on errors in predicting dynam- +ics, which have been used to address the difficulties in complex +environments [5, 37, 38]. Though the Intrinsic Curiosity Module +(ICM) [37] maintains a predictor on state transitions and considers +the prediction error as an intrinsic reward, Random Network Distil- +lation (RND) [5] utilizes the prediction errors of networks trained +on historical trajectories to quantify the novelty of states, which is +effective and easy to implement in real applications. +Another direction in previous work is to study exploration in +hierarchical reinforcement learning (HRL) [3, 47]. These methods +are insight from the fact that developers prefer to divide the com- +prehensive and knotty problems into several solvable sub-problems. +There are some further studies on hierarchy in terms of tasks, rep- +resentative of which are goal-based reinforcement learning and +skill discovery. The similarity of these approaches is that they both +identify different policies by utilizing latent variables. In goal-based +RL, the latent variables are defined by the policy’s goal, which +aims to complete several sub-goals and accomplish the whole task. +These methods introduce prior human knowledge, causing them +to work brilliantly on some tasks but fail when unaware of human +knowledge. Despite our method also introducing latent variables to +represent different styles of policies, all the policies share the same +objective, nevertheless differing in the road to reach the destination. +Skill discovery methods, which adopt the latent variable to repre- +sent the skill learned from the policy, introduce mutual information +to organize relationships between the latent variable 𝑧 and some +aspects of the trajectories to acquire diverse skills (also known +as style) [1, 11, 13, 20, 43]. Nevertheless, these methods train the +policy in an unsupervised way [11, 13, 43], suggesting that the +skills trained are unaware of task-driven, and they cannot rep- +resent the optimal policies when adapted to downstream tasks +illustrated in [12]. Our method avoids this issue because we train +the policy task-oriented and demonstrate the benefit brought by +the attention of these policies to the state value making them differ +considerably in exploration style. For task relevance, some related +works that learn skills by jointly learning a set of skills and a meta- +controller [3, 10, 13, 14, 28, 29]. The options of the meta-controller +control different attentions of each policy. However, these meth- +ods usually choose the best option to explore and rarely execute +sub-optimal options, leading to the drawback – the algorithm tends +to ignore sub-optimal actions that maybe fail in most states but +are effective in a few critical scenarios. Our proposed approach +randomly selects different styles of policies for directed coopera- +tive exploration, which are improved accordingly with the value +function and produce different styles due to differences in attention. +7 +CONCLUSION +In the value-based method, value estimation bias has been a com- +mon problem. While different estimation bias in double value func- +tions lead to value function controversy, the controversy can be +utilized to encourage policies to yield multiple styles. In this paper, +we aim at encouraging explorations by multi-styled policies. We + +start by analysis on estimation bias during the value function train- +ing process and its effect on the exploration. We then encourage +this controversy between the value functions and generate four +critics for producing multi-styled policies. Finally, we apply these +policies with diverse styles for centralized cooperative exploration +which perform superior sample efficiency in the test environment. +Though there are a lot of works focusing on reducing the estimation +bias for an accurate value estimation, few works try to utilize these +inevitable errors to make improvements. Our results show that it +is also an option to use the errors to encourage explorations. For +future work, it is an exciting avenue for focusing on more expres- +sive policy styles. A style that can be represented as a continuous +distribution may be more efficient and more expressive. +REFERENCES +[1] Joshua Achiam, Harrison Edwards, Dario Amodei, and Pieter Abbeel. 2018. +Variational option discovery algorithms. CoRR abs/1807.10299 (2018). +http: +//arxiv.org/abs/1807.10299 +[2] Marcin Andrychowicz, Filip Wolski, Alex Ray, Jonas Schneider, Rachel Fong, +Peter Welinder, Bob McGrew, Josh Tobin, OpenAI Pieter Abbeel, and Wojciech +Zaremba. 2017. Hindsight experience replay. 30 (2017), 5048–5058. +[3] Pierre-Luc Bacon, Jean Harb, and Doina Precup. 2017. The option-critic archi- +tecture. In Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 31. +1726–1734. +[4] Yunpeng Bai, Chen Gong, Bin Zhang, Guoliang Fan, Xinwen Hou, and Yu Lu. 2022. +Cooperative Multi-Agent Reinforcement Learning with Hypergraph Convolution. +In 2022 International Joint Conference on Neural Networks (IJCNN). 1–8. https: +//doi.org/10.1109/IJCNN55064.2022.9891942 +[5] Yuri Burda, Harrison Edwards, Amos Storkey, et al. 2019. Exploration by random +network distillation. In 7th International Conference on Learning Representations, +ICLR 2019, New Orleans, LA, USA, May 6-9, 2019. OpenReview.net. +[6] Olivier Chapelle and Lihong Li. 2011. An Empirical Evaluation of Thompson +Sampling. In Advances in Neural Information Processing Systems, J. Shawe-Taylor, +R. Zemel, P. Bartlett, F. Pereira, and K.Q. Weinberger (Eds.), Vol. 24. Curran +Associates, Inc., 2249–2257. +https://proceedings.neurips.cc/paper/2011/file/ +e53a0a2978c28872a4505bdb51db06dc-Paper.pdf +[7] Jongwook Choi, Archit Sharma, Honglak Lee, Sergey Levine, and Shixiang Shane +Gu. 2021. +Variational empowerment as representation learning for goal- +conditioned reinforcement learning. In International Conference on Machine +Learning. PMLR, 1953–1963. +[8] Kamil Ciosek, Quan Vuong, Robert Loftin, and Katja Hofmann. 2019. Better +Exploration with Optimistic Actor Critic. In Advances in Neural Information Pro- +cessing Systems 32: Annual Conference on Neural Information Processing Systems +2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, Hanna M. Wal- +lach, Hugo Larochelle, Alina Beygelzimer, Florence d’Alché-Buc, Emily B. Fox, +and Roman Garnett (Eds.). 1785–1796. https://proceedings.neurips.cc/paper/ +2019/hash/a34bacf839b923770b2c360eefa26748-Abstract.html +[9] Erwin Coumans and Yunfei Bai. 2016. Pybullet, a python module for physics +simulation for games, robotics and machine learning. (2016). +[10] Peter Dayan and Geoffrey E Hinton. 1992. Feudal reinforcement learning. Ad- +vances in neural information processing systems 5 (1992), 271–278. +[11] Benjamin Eysenbach, Abhishek Gupta, Julian Ibarz, and Sergey Levine. 2019. +Diversity is all you need: Learning skills without a reward function, In 7th +International Conference on Learning Representations, ICLR 2019. arXiv preprint +arXiv:1802.06070. +[12] Benjamin Eysenbach, Ruslan Salakhutdinov, and Sergey Levine. 2022. The infor- +mation geometry of unsupervised reinforcement learning. https://openreview. +net/forum?id=3wU2UX0voE +[13] Carlos Florensa, Yan Duan, and Pieter Abbeel. 2017. Stochastic neural networks +for hierarchical reinforcement learning. In 5th International Conference on Learn- +ing Representations, ICLR 2017. OpenReview.net, Toulon, France. +[14] Kevin Frans, Jonathan Ho, Xi Chen, Pieter Abbeel, and John Schulman. 2018. +Meta learning shared hierarchies. In 6th International Conference on Learning +Representations, ICLR 2018. OpenReview.net. https://openreview.net/forum?id= +SyX0IeWAW +[15] Justin Fu, John D. Co-Reyes, and Sergey Levine. 2017. EX2: Exploration with +Exemplar Models for Deep Reinforcement Learning. In Proceedings of the 31st In- +ternational Conference on Neural Information Processing Systems (Long Beach, Cal- +ifornia, USA) (NIPS’17). Curran Associates Inc., Red Hook, NY, USA, 2574–2584. +[16] Scott Fujimoto, Herke Hoof, and David Meger. 2018. Addressing function ap- +proximation error in actor-critic methods. In Proceedings of the 35th International +Conference on Machine Learning, ICML 2018 (Proceedings of Machine Learning Re- +search, Vol. 80). PMLR, PMLR, Stockholmsmässan, Stockholm, Sweden, 1582–1591. +http://proceedings.mlr.press/v80/fujimoto18a.html +[17] Chen Gong, Qiang He, Yunpeng Bai, Xinwen Hou, et al. 2021. Wide-Sense +Stationary Policy Optimization with Bellman Residual on Video Games. In 2021 +IEEE International Conference on Multimedia and Expo, ICME. IEEE, 1–6. +[18] Chen Gong, Zhou Yang, Yunpeng Bai, Junda He, Jieke Shi, Arunesh Sinha, Bowen +Xu, Xinwen Hou, Guoliang Fan, and David Lo. 2022. Mind Your Data! Hiding +Backdoors in Offline Reinforcement Learning Datasets. CoRR abs/2210.04688 +(2022). https://doi.org/10.48550/arXiv.2210.04688 arXiv:2210.04688 +[19] Chen Gong, Zhou Yang, Yunpeng Bai, Jieke Shi, Arunesh Sinha, Bowen Xu, +David Lo, Xinwen Hou, and Guoliang Fan. 2022. Curiosity-Driven and Victim- +Aware Adversarial Policies. In Proceedings of the 38th Annual Computer Security +Applications Conference (Austin, TX, USA) (ACSAC ’22). 186–200. https://doi.org/ +10.1145/3564625.3564636 +[20] Karol Gregor, Danilo Jimenez Rezende, and Daan Wierstra. 2017. Variational +intrinsic control. In 5th International Conference on Learning Representations, ICLR +2017. OpenReview.net. https://openreview.net/forum?id=Skc-Fo4Yg +[21] Tuomas Haarnoja, Haoran Tang, Pieter Abbeel, and Sergey Levine. 2017. Rein- +forcement Learning with Deep Energy-Based Policies. In Proceedings of the 34th +International Conference on Machine Learning (Sydney, NSW, Australia) (ICML’17, +Vol. 70). JMLR.org, 1352–1361. +[22] Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and Sergey Levine. 2018. Soft +actor-critic: Off-policy maximum entropy deep reinforcement learning with a +stochastic actor. In Proceedings of the 35th International Conference on Machine +Learning, ICML 2018 (Proceedings of Machine Learning Research, Vol. 80). PMLR, +PMLR, Stockholmsmässan, Stockholm, Sweden, 1856–1865. +[23] Hado Hasselt. 2010. Double Q-learning. In Advances in Neural Information Pro- +cessing Systems, J. Lafferty, C. Williams, J. Shawe-Taylor, R. Zemel, and A. Culotta +(Eds.), Vol. 23. Curran Associates, Inc. +https://proceedings.neurips.cc/paper/ +2010/file/091d584fced301b442654dd8c23b3fc9-Paper.pdf +[24] Hado van Hasselt, Arthur Guez, and David Silver. 2016. Deep Reinforcement +Learning with Double Q-Learning. In Proceedings of the Thirtieth AAAI Conference +on Artificial Intelligence (Phoenix, Arizona) (AAAI’16). AAAI Press, 2094–2100. +[25] Qiang He and Xinwen Hou. 2020. WD3: Taming the Estimation Bias in Deep +Reinforcement Learning. In 32nd IEEE International Conference on Tools with +Artificial Intelligence, ICTAI 2020, Baltimore, MD, USA, November 9-11, 2020. IEEE, +391–398. https://doi.org/10.1109/ICTAI50040.2020.00068 +[26] Qiang He, Xinwen Hou, and Yu Liu. 2022. POPO: Pessimistic Offline Policy +Optimization. In IEEE International Conference on Acoustics, Speech and Signal +Processing, ICASSP 2022, Virtual and Singapore, 23-27 May 2022. IEEE, 4008–4012. +https://doi.org/10.1109/ICASSP43922.2022.9747886 +[27] Qiang He, Huangyuan Su, Jieyu Zhang, and Xinwen Hou. 2022. Representation +Gap in Deep Reinforcement Learning. CoRR abs/2205.14557 (2022). +https: +//doi.org/10.48550/arXiv.2205.14557 arXiv:2205.14557 +[28] Nicolas Heess, Greg Wayne, Yuval Tassa, Timothy Lillicrap, Martin Riedmiller, +and David Silver. 2016. Learning and transfer of modulated locomotor controllers. +CoRR abs/1610.05182 (2016). http://arxiv.org/abs/1610.05182 +[29] Sanjay Krishnan, Roy Fox, Ion Stoica, and Ken Goldberg. 2017. Ddco: Discovery +of deep continuous options for robot learning from demonstrations. In Conference +on robot learning, Vol. 78. PMLR, PMLR, 418–437. http://proceedings.mlr.press/ +v78/krishnan17a.html +[30] Qingfeng Lan, Yangchen Pan, Alona Fyshe, and Martha White. 2020. Maxmin +Q-learning: Controlling the Estimation Bias of Q-learning. In 8th International +Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April +26-30, 2020. OpenReview.net. https://openreview.net/forum?id=Bkg0u3Etwr +[31] Timothy P Lillicrap, Jonathan J Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, +Yuval Tassa, David Silver, and Daan Wierstra. 2016. Continuous control with +deep reinforcement learning. (2016). +[32] Long-Ji Lin. 1992. Self-improving reactive agents based on reinforcement learning, +planning and teaching. Machine Learning 8, 3 (1992), 293–321. https://doi.org/ +10.1007/BF00992699 +[33] Marlos C. Machado, Marc G. Bellemare, and Michael Bowling. 2020. Count- +Based Exploration with the Successor Representation. Proceedings of the AAAI +Conference on Artificial Intelligence 34, 04, 5125–5133. +[34] Volodymyr Mnih, Adria Puigdomenech Badia, Mehdi Mirza, Alex Graves, Tim- +othy Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu. 2016. Asyn- +chronous methods for deep reinforcement learning. In International conference +on machine learning (JMLR Workshop and Conference Proceedings, Vol. 48), Maria- +Florina Balcan and Kilian Q. Weinberger (Eds.). PMLR, 1928–1937. +[35] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A. Rusu, Joel Veness, +Marc G. Bellemare, Alex Graves, Martin A. Riedmiller, Andreas Fidjeland, Georg +Ostrovski, Stig Petersen, Charles Beattie, Amir Sadik, Ioannis Antonoglou, Helen +King, Dharshan Kumaran, Daan Wierstra, Shane Legg, and Demis Hassabis. 2015. +Human-level control through deep reinforcement learning. nature 518, 7540 +(2015), 529–533. +[36] Shakir Mohamed and Danilo Jimenez Rezende. 2015. Variational information +maximisation for intrinsically motivated reinforcement learning. Advances in +neural information processing systems 28 (2015), 2125–2133. + +[37] Deepak Pathak, Pulkit Agrawal, Alexei A. Efros, and Trevor Darrell. 2017. +Curiosity-driven Exploration by Self-supervised Prediction. In Proceedings of +the 34th International Conference on Machine Learning (Proceedings of Machine +Learning Research, Vol. 70). PMLR, 2778–2787. +[38] Roberta Raileanu and Tim Rocktäschel. 2020. RIDE: Rewarding Impact-Driven +Exploration for Procedurally-Generated Environments. In 8th International Con- +ference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, +2020. OpenReview.net. +[39] Konrad Rawlik, Marc Toussaint, and Sethu Vijayakumar. 2012. On Stochastic +Optimal Control and Reinforcement Learning by Approximate Inference. In +Proceedings of Robotics: Science and Systems. Sydney, Australia. https://doi.org/ +10.15607/RSS.2012.VIII.045 +[40] Martin Riedmiller. 2005. Neural Fitted Q Iteration – First Experiences with a Data +Efficient Neural Reinforcement Learning Method. In Machine Learning: ECML +2005, João Gama, Rui Camacho, Pavel B. Brazdil, Alípio Mário Jorge, and Luís +Torgo (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg, 317–328. +[41] Herbert Robbins and Sutton Monro. 1951. A stochastic approximation method. +The annals of mathematical statistics (1951), 400–407. +[42] John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. +2017. Proximal Policy Optimization Algorithms. CoRR abs/1707.06347 (2017). +arXiv:1707.06347 http://arxiv.org/abs/1707.06347 +[43] Archit Sharma, Shixiang Gu, Sergey Levine, Vikash Kumar, and Karol Hausman. +2020. Dynamics-aware unsupervised discovery of skills. In 8th International +Conference on Learning Representations, ICLR 2020. OpenReview.net, Addis Ababa, +Ethiopia. https://openreview.net/forum?id=HJgLZR4KvH +[44] David Silver, Guy Lever, Nicolas Heess, Thomas Degris, Daan Wierstra, and Mar- +tin Riedmiller. 2014. Deterministic Policy Gradient Algorithms. In Proceedings of +the 31st International Conference on International Conference on Machine Learning +- Volume 32 (Beijing, China) (ICML’14). JMLR.org, I–387–I–395. +[45] Richard S. Sutton. 1988. Learning to predict by the methods of temporal differ- +ences. Machine Learning 3, 1 (1988), 9–44. https://doi.org/10.1007/BF00115009 +[46] Richard S. Sutton and Andrew G. Barto. 2018. Reinforcement learning: An intro- +duction. MIT press. +[47] Richard S Sutton, Doina Precup, and Satinder Singh. 1999. Between MDPs and +semi-MDPs: A framework for temporal abstraction in reinforcement learning. +Artificial intelligence 112, 1-2 (1999), 181–211. +[48] Emanuel Todorov, Tom Erez, and Yuval Tassa. 2012. MuJoCo: A physics engine +for model-based control. In 2012 IEEE/RSJ International Conference on Intelligent +Robots and Systems. 5026–5033. https://doi.org/10.1109/IROS.2012.6386109 +[49] Laurens van der Maaten and Geoffrey Hinton. 2008. Visualizing Data using +t-SNE. Journal of Machine Learning Research 9, 86 (2008), 2579–2605. +http: +//jmlr.org/papers/v9/vandermaaten08a.html +[50] Wei Wei, Yujia Zhang, Jiye Liang, Lin Li, and Yyuze Li. 2022. Controlling Under- +estimation Bias in Reinforcement Learning via Quasi-median Operation. Proceed- +ings of the AAAI Conference on Artificial Intelligence 36, 8 (Jun. 2022), 8621–8628. +https://doi.org/10.1609/aaai.v36i8.20840 +[51] Dongming Wu, Xingping Dong, Jianbing Shen, and Steven C. H. Hoi. 2020. +Reducing Estimation Bias via Triplet-Average Deep Deterministic Policy Gradient. +IEEE Transactions on Neural Networks and Learning Systems 31, 11 (2020), 4933– +4945. https://doi.org/10.1109/TNNLS.2019.2959129 +[52] Zhiwei Xu, Yunpeng Bai, Dapeng Li, Bin Zhang, and Guoliang Fan. 2022. SIDE: +State Inference for Partially Observable Cooperative Multi-Agent Reinforcement +Learning. In Proceedings of the 21st International Conference on Autonomous Agents +and Multiagent Systems. International Foundation for Autonomous Agents and +Multiagent Systems (IFAAMAS), 1400–1408. +[53] Yingjun Ye, Xiaohui Zhang, and Jian Sun. 2019. Automated vehicle’s behavior +decision making using deep reinforcement learning and high-fidelity simulation +environment. Transportation Research Part C: Emerging Technologies 107 (2019), +155–170. +[54] Xianjie Zhang, Yu Liu, Hangyu Mao, and Chao Yu. 2022. Common belief multi- +agent reinforcement learning based on variational recurrent models. Neurocom- +puting 513 (2022), 341–350. https://doi.org/10.1016/j.neucom.2022.09.144 +[55] Brian D Ziebart, Andrew L Maas, J Andrew Bagnell, Anind K Dey, et al. 2008. +Maximum entropy inverse reinforcement learning.. In Aaai, Vol. 8. Chicago, IL, +USA, 1433–1438. + +Supplementary Materials +A +PROOF OF LEMMA 1 +(Lemma 1)(Performance Gap). Let 𝑉 ∗ be the ground truth state +value in Bellman value iterations, 𝑄∗ be the ground truth state +action value, 𝑉 𝜋𝑓 be the state value when applying learned policy +𝜋𝑓 , 𝑓 be the value function approximator. The performance gap of +the policy between the optimal policy 𝜋∗ and the learned policy 𝜋𝑓 +is defined by an infinity norm ∥𝑉 ∗ − 𝑉 𝜋𝑓 ∥∞ and we have +∥𝑉 ∗ − 𝑉 𝜋𝑓 ∥∞ ≤ 2∥𝑓 −𝑄∗ ∥∞ +1−𝛾 +Proof. For any 𝑠 ∈ S +𝑉 ∗(𝑠) − 𝑉 𝜋𝑓 (𝑠) =𝑄∗(𝑠, 𝜋∗(𝑠)) − 𝑄∗(𝑠, 𝜋𝑓 (𝑠)) ++ 𝑄∗(𝑠, 𝜋𝑓 (𝑠)) − 𝑄∗(𝑠, 𝜋𝑓 (𝑠)) +≤𝑄∗(𝑠, 𝜋∗(𝑠)) − 𝑓 (𝑠, 𝜋∗(𝑠)) ++ 𝑓 (𝑠, 𝜋𝑓 (𝑠)) − 𝑄∗(𝑠, 𝜋𝑓 (𝑠)) ++ 𝛾E𝑠′∼𝑃 (𝑠,𝜋𝑓 (𝑠)) [𝑉 ∗(𝑠′) − 𝑉 𝜋𝑓 (𝑠′)] +≤2∥𝑓 − 𝑄∗∥∞ + 𝛾∥𝑉 ∗ − 𝑉 𝜋𝑓 ∥∞ +B +EXPERIMENTAL DETAILS +B.1 +Environments +We evaluate the performance of CCEP on environments from Mu- +joCo Control Suite [48]which can be listed as HalfCheetah-v3, Ant- +v3, Walker2d-v3, Humanoid-v3, Hopper-v3, and Pusher-v2, and the +specific parameters of these environments are listed in Table 3. We +use the publicly available environments without any modification. +B.2 +Implementation and Hyper-parameters +Here, we describe certain implementation details of CCEP. For +our implementation of CCEP, we follows a standard actor-critic +framework. +B.3 +Soft Actor-Critic Implementation Details +For implementation of SAC, we use the code the author provided +and use the parameters the author recommended. We use a single +Gaussian distribution and use the environment-dependent reward +scaling as described by the authors. For a fair comparison, we +apply the version of soft target update and train one iteration per +time step. We use the reward scales as the author recommended +(except for Pusher-v2 which is not mentioned by the author in +the article). Considering that there are similar action dimensions +between Pusher-v2 and HalfCheetah-v3, we set the same reward +scale for Pusher-v2. The specific reward scales for each environment +is shown in Table 4. +Table 3: Environment Specific Parameters +Environment +State Dimensions +Action Dimensions +Ant-v3 +111 +8 +HalfCheetah-v3 +17 +6 +Hopper-v3 +11 +3 +Humanoid-v3 +376 +17 +Pusher-v2 +23 +7 +Walker2d-v3 +17 +6 +Table 4: SAC Environment Specific Parameters +Environment +Reward Scale +Ant-v3 +5 +HalfCheetah-v3 +5 +Hopper-v3 +5 +Humanoid-v3 +20 +Pusher-v2 +5 +Walker2d +5 +B.4 +Optimistic Actor-Critic Implementation +Details +The implementation of OAC is mainly based on the open source +code. We set the hyper-parameters the same as OAC used in MuJoCo +which is listed in Table 5. And for fair comparison, we train with 1 +training gradient per environment step. We use the same reward +scales as SAC, listed in Table 4. +Table 5: SAC Environment Specific Parameters +Parameter +Value +shift multiplier +√ +2𝛿 +6.86 +𝛽𝑈 𝐵 +4.66 +𝛽𝐿𝐵 +-3.65 +B.5 +Reproducing Other Baselines +For reproduction of TD3, we use the official implementation ( +https://github.com/sfujim/TD3). For reproduction of DDPG and +PPO we use OpenAI’s baselines repository and apply default hyper- +parameters. +Table 6: CCEP Parameters settings +Parameter +Value +Exploration policy +N (0, 0.1),𝑧 ∼ 𝑝(𝑧) +Number of policy +4 +Variance of exploration noise +0.2 +Random starting exploration time steps +2.5 × 104 +Optimizer +Adam[30] +Learning rate for actor +3 × 10−4 +Learning rate for critic +3 × 10−4 +Replay buffer size +1 × 106 +Batch size +256 +Discount (𝛾) +0.99 +Number of hidden layers +2 +Number of hidden units per layer +256 +Activation function +ReLU +Iterations per time step +1 +Target smoothing coefficient (𝜂) +5 × 10−3 +Variance of target policy smoothing +0.2 +Noise clip range +[−0.5, 0.5] +Target critic update interval +2 + +0.00 +0.25 +0.50 +0.75 +1.00 +Time Steps (1e6) +0 +2000 +4000 +6000 +8000 +10000 +12000 +Average Return +(a) HalfCheetah-v3 +0.00 +0.25 +0.50 +0.75 +1.00 +Time Steps (1e6) +0 +1000 +2000 +3000 +4000 +5000 +(b) Walker2d-v3 +0.00 +0.25 +0.50 +0.75 +1.00 +Time Steps (1e6) +0 +1000 +2000 +3000 +(c) Hopper-v3 +0.00 +0.25 +0.50 +0.75 +1.00 +Time Steps (1e6) +0 +1000 +2000 +3000 +4000 +5000 +6000 +Average Return +(d) Ant-v3 +CCEP +CCEP-Cooperation +TD3 +Figure 8: Ablation over the use of cooperation. Comparison of CCEP, TD3 and the subtraction of cooperation (CCEP- +cooperation). +0.00 +0.25 +0.50 +0.75 +1.00 +Time Steps (1e6) +0 +500 +1000 +1500 +2000 +Average Return +(a) Walker2D +0.00 +0.25 +0.50 +0.75 +1.00 +Time Steps (1e6) +0 +1000 +2000 +3000 +(b) Ant +0.00 +0.25 +0.50 +0.75 +1.00 +Time Steps (1e6) +0 +500 +1000 +1500 +2000 +2500 +(c) Hopper +0.00 +0.25 +0.50 +0.75 +1.00 +Time Steps (1e6) +1000 +0 +1000 +2000 +3000 +Average Return +(d) HalfCheetah +TD3 +Ours +SAC +DDPG +PPO +Figure 9: Learning curves for 4 PyBullet continuous control tasks. For better visualization, the curves are smoothed uniformly. +The bolded line represents the average evaluation over 10 seeds. The shaded region represents the standard deviation of the +average evaluation over 10 seeds. +Table 7: Evaluation in PyBullet control suite. The highest average return over 10 trials of 1 million time steps. The maximum +value for each task is bolded. +Pybullet Environment +Ours +SAC +TD3 +DDPG +PPO +HalfCheetah +2670 ± 275 +2494 ± 266 +2415 ± 236 +1120 ± 373 +465 ± 30 +Hopper +2254 ± 186 +2167 ± 323 +1860 ± 288 +1762 ± 368 +623 ± 131 +Walker2d +1829 ± 418 +1369 ± 408 +1676 ± 342 +929 ± 345 +509 ± 106 +Ant +3175 ± 184 +2423 ± 680 +2711 ± 253 +483 ± 70 +578 ± 19 +C +ADDITIONAL EXPERIMENTS +C.1 +Additional Evaluation +For an additional Evaluation, We conduct experiments on the state- +based PyBullet [9] suite which is based on the well-known open- +source physics engine bullet and is packaged as a Python module for +robot simulation and learning. The suite of Pybullet is considered +to be a harder environment than MuJoCo [48]. We choose TD3 +[16], SAC [22], PPO [42], DDPG [31] as our baselines due to their +superior performance. We perform interactions for 1 million steps +in 10 different seeds and evaluate the algorithm over 10 episodes +every 5k steps. We evaluate our algorithm in HalfCheetah, Hopper, +Walker2d and ant in the suite of pybullet. Our results report the +mean scores and standard deviations in the 10 seeds. We show the +learning curves in Figure 9 and the max average return over 10 +trials in Table 7. +C.2 +Additional Ablation Results +We compare the learning curves of CCEP, TD3 and the subtraction +of cooperation (CCEP-cooperation) for better understanding the +contribution of policy cooperation (Section 5.4). We perform inter- +actions for 1 million steps in 10 different seeds and evaluate over +10 episodes every 5k steps. Our results report the mean scores and +standard deviations in the 10 seeds. We show the learning curves +in Figure 8 +C.3 +Supplementary Results +We provide supplementary results for Section 5.2. Figure 10 shows +the states visited by each style over 1M time steps with intervals +of 100k. The results show that different styles get consistent but + +new styles emerges as well, which brings enduring exploration +capabilities. + +(1) Learning Steps = 100000 +(2) Learning Steps = 200000 +(3) Learning Steps = 300000 +(4) Learning Steps = 400000 +(5) Learning Steps = 500000 +(6) Learning Steps = 600000 +(7) Learning Steps = 700000 +(8) Learning Steps = 800000 +(9) Learning Steps = 900000 +POLICY No.1 +POLICY No.2 +POLICY No.3 +POLICY No.4 +Figure 10: The states visited by each style. For better visualization, the states get dimension reduction by t-SNE. The points with +different color represents the states visited by the policy with the style. The distance between points represents the difference +between states. + diff --git a/09E0T4oBgHgl3EQfdQDB/content/tmp_files/load_file.txt b/09E0T4oBgHgl3EQfdQDB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..82368276fab96ef6d130c6a67a22435f7d392469 --- /dev/null +++ b/09E0T4oBgHgl3EQfdQDB/content/tmp_files/load_file.txt @@ -0,0 +1,909 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf,len=908 +page_content='Centralized Cooperative Exploration Policy for Continuous Control Tasks Chao Li* Institute of Automation, Chinese Academy of Sciences, China lichao2021@ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='cn Chen Gong∗ Institute of Automation, Chinese Academy of Sciences, China gongchen2020@ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='cn Qiang He University of Tubingen, Germany qianghe97@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='com Xinwen Hou† Institute of Automation, Chinese Academy of Sciences, Beijing, China xinwen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='hou@ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='cn Yu Liu Institute of Automation, Chinese Academy of Sciences, Beijing, China yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='liu@ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='cn ABSTRACT The deep reinforcement learning (DRL) algorithm works brilliantly on solving various complex control tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' This phenomenal success can be partly attributed to DRL encouraging intelligent agents to sufficiently explore the environment and collect diverse experiences during the agent training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Therefore, exploration plays a significant role in accessing an optimal policy for DRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Despite recent works making great progress in continuous control tasks, exploration in these tasks has remained insufficiently investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' To explicitly encourage exploration in continuous control tasks, we propose CCEP (Centralized Cooperative Exploration Policy), which utilizes underestimation and overestimation of value func- tions to maintain the capacity of exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' CCEP first keeps two value functions initialized with different parameters, and generates diverse policies with multiple exploration styles from a pair of value functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In addition, a centralized policy framework ensures that CCEP achieves message delivery between multiple policies, fur- thermore contributing to exploring the environment cooperatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Extensive experimental results demonstrate that CCEP achieves higher exploration capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Empirical analysis shows diverse ex- ploration styles in the learned policies by CCEP, reaping benefits in more exploration regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' And this exploration capacity of CCEP ensures it outperforms the current state-of-the-art methods across multiple continuous control tasks shown in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' KEYWORDS Deep Reinforcement Learning, Cooperative Exploration, Continu- ous Control Tasks 1 INTRODUCTION Deep reinforcement learning (DRL) [46], which utilizes deep neu- ral networks to learn an optimal policy, works brilliantly and has demonstrated beyond human-level performance in solving various challenging sequential decision-making tasks e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=', video games [17, 34, 35, 52, 54], autonomous driving [18, 53], robotic control tasks [2, 31], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In DRL settings, an agent needs to sufficiently explore the environment and collect a set of experiences to obtain an optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The agent aims to learn an optimal policy to maximize its ex- pected cumulative rewards through trial and error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Therefore, DRL ∗These two authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' †The corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' can be regarded as learning from reward feedback from environ- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' It is essential that during the training phase the agent should be encouraged to explore the environments and gather sufficient reward signals for well-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In DRL, exploration has obsessed with a critical problem: submit- ting solutions too quickly without sufficient exploration, leading to getting stuck at local minima or even complete failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' DRL researchers adopt the neural network to yield the policy with sig- nificant feature extraction and expression capabilities in a range of continuous control tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Whereas this phenomenal practice has achieved great performance, it is still obsessed with the notorious insufficient exploration problem in continuous control tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Good exploration becomes extremely difficult when the environment is distracting or provides little feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Whereas existing explo- ration methods remain a problematic drawback – lacking diversity to explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The classic exploration methods such as 𝜖-Greedy strate- gies [35] or Gaussian noise [31] indirectly and implicitly change the style of the exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' However, in massive situations, diverse styles of exploration are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' For instance, in chess games, players should perform different styles of policies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=', radical, con- servative, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=') to keep competitive when facing various situations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' humanoid robots attempt diverse control styles and eventually learn to walk efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Recent studies enrich diverse styles of policies by construct- ing relationships between the distribution of policy and trajecto- ries [1, 13, 26, 27, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In Diversity is All You Need (DIAYN) [11], authors highlight the diversity of policies plays a significant role in well-training agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' It trains the policy that maximizes the mutual information between the latent variable and the states, then altering the latent variable of the policy network creates multiple policies performing disparately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Although this interesting viewpoint has attracted a spectrum of following works [1, 11, 13, 20, 43], the afore- mentioned methods achieve diverse policies depending on the task in an unsupervised way, resulting in the algorithm performing in- sufficient generality in different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In fact, the ideal algorithm to implement the diverse policies should be general across a range of tasks, which motivates us to design a task-oriented algorithm working towards developing various policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Eysenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' pro- posed [12] that the learned skills could not construct all the state marginal distributions in the downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' For task relevance, the mutual information is added as an intrinsic reward for em- powerment [7, 36], but these methods change the original reward arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='02375v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='LG] 6 Jan 2023 function resulting in the performance being extremely sensitive to the trade-off between original and intrinsic rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Our method insights from an interesting phenomenon during the exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The critic aims to approximate the accumulated reward by bootstrapping in the actor-critic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' However, the different critic functions may have great differences even if they approximate the same target due to the function approximation er- ror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' For instance, Twin Delayed Deep Deterministic policy gradient (TD3) [16] presents that two value functions with different initial parameters perform quite differently with identical targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' It is knotty to measure whether a value function is exact or not, and the gap between these two value functions is termed as controversy, which sees a decreasing trend along with the value function updat- ing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Our intuition can be ascribed that controversy in the value estimation will lead to sub-optimal policies, and these policies have a bias toward message acquisition known as the style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' This paper highlights that controversy can be utilized to encour- age policies to yield multiple styles, and encourages exploration for a continuous control task by applying multi-styled policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Our paper contributes three aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' (1) We first describe that the estimation bias in double value functions can lead to various ex- ploration styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' (2) This paper proposes the CCEP algorithm, encouraging diverse exploration for environments by cooperation from multi-styled policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' (3) Finally, in CCEP, we design a novel framework, termed as the centralized value function framework, which is updated by experience collected from all the policies and accomplishes the message delivery mechanism between different policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Extensive experiments are conducted on the MuJoCo plat- form to evaluate the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The results reveal that the proposed CCEP approach attains substantial improvements in both average return and sample efficiency on the baseline across selected environments, and the average return of agents trained using CCEP is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='7% higher than that of the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Besides, CCEP also allows agents to explore more states during the same train- ing time steps as the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Additional analysis indicates that message delivery leading to the cooperative multi-styled policies further enhanced the exploration efficiency by 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='6% compared with that of without cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We organize the rest of this paper as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Section 2 briefly explains the concepts of RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We elaborate on how to perform the CCEP algorithm in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We introduce our experimental set- tings in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Section 5 presents and analyzes results to validate the effectiveness of CCEP, after which section 6 discusses related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Finally, we conclude our paper in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The code and documentation are released in the link for validating reproducibil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1 2 PRELIMINARIES Reinforcement learning (RL) aims at training an agent to tackle the sequential decision problems that can be formalized as a Markov Decision Process (MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' This process can be defined as a tuple (S, A, 𝑃,𝑟,𝛾), where S is the state space, A is the action space, 𝑃 : S × A × S ↦→ [0, 1] denotes the transition probability, 𝑟 (𝑠,𝑎) is the reward function 𝑟 : S ×A ↦→ R, determining the reward agents will receive in the state 𝑠 while executing the action 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The 𝛾 ∈ 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='com/Jincate/CCEP (0, 1) is the discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The return is defined as the discounted accumulated reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 𝑅 = ∞ ∑︁ 𝑡=0 𝛾𝑡𝑟 (𝑠𝑡,𝑎𝑡) (1) In the DRL community, developers usually use the neural network parameterized with 𝜙 to indicate the policy 𝜋(𝑎|𝑠), which inputs an observation and outputs an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The goal of DRL is to solve this MDP process and find the optimal policy 𝜋𝜙∗ : S ↦→ A with parameter 𝜙∗ that maximizes the expected accumulated return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 𝜙∗ = arg max 𝜙 E𝑎𝑡 ∼𝜋𝜙 (·|𝑠𝑡 ),𝑠𝑡+1∼𝑃 (·|𝑠𝑡,𝑎𝑡 ) � ∞ ∑︁ 𝑡=0 𝛾𝑡𝑟 (𝑠𝑡,𝑎𝑡) � (2) David Silver, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [44] propose that solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' (2) with determin- istic policy gradient strategy, ∇𝜙 𝐽 (𝜙) = E𝑠𝑡+1∼𝑃 (·|𝑠𝑡,𝑎𝑡 ) � ∇𝑎𝑄𝜋 (𝑠,𝑎)|𝑎=𝜋 (𝑠)∇𝜙𝜋𝜙 (𝑠) � (3) where 𝑄𝜋 (𝑠,𝑎) = E𝑎𝑡 ∼𝜋𝜙 (·|𝑠𝑡 ),𝑠𝑡+1∼𝑃 (·|𝑠𝑡,𝑎𝑡 ) [𝑅|𝑠,𝑎] is known as the value function, indicating how good it is for an agent to pick action 𝑎 while being in state 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' To use the gradient-based approach (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=', Stochastic Gradient Descent [41]) to solve this equation, deep Q- learning uses the neural network to approximate the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The value function parameterized with 𝜃 is updated by minimizing the temporal difference (TD) error [45] between the estimated value of the subsequent state 𝑠′ and the current state 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 𝜃∗ = arg min 𝜃 E � 𝑟 (𝑠,𝑎) + 𝛾𝑄𝜋 𝜃 (𝑠′,𝑎′) − 𝑄𝜋 𝜃 (𝑠,𝑎) �2 (4) We store the trajectories of the agent exploring the environment in a replay buffer [32] from which sample a random mini-batch of samples, updating the parameters mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 3 CENTRALIZED COOPERATIVE EXPLORATION POLICY This section details technologies of CCEP (Centralized Cooper- ative Exploration Policy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We first analyze value estimation bias from function approximation errors and generate multi-styled value functions by encouraging overestimation bias and underestimation bias for the value functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' To achieve multi-styled exploration, we propose a multi-objective update method for train- ing policy and randomly select one policy to explore at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' These historical trajectories during exploration are stored for training a single policy function to achieve cooperative message delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We denote our policy as 𝜋(𝑠,𝑧), where 𝑧 is a one-hot label and represents different policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In this work, we focus on gener- ating multiple policies with different styles to encourage diverse exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We implement our method based on TD3 [16] which maintains double critics and uses the minimum of the critics as the target estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1 Function Approximation Error This section shows that there exist approximation errors in the value function optimization and can accumulate to substantial scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The accumulated approximation error will lead to value estimation bias, which plays a significant role in policy improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Sample one style per step 𝒛~𝒑(𝒛) 𝒂𝒕~𝝅𝝓(𝒂𝒕|𝒔𝒕, 𝒛𝒕) STYLE 𝒔𝒕�𝟏~𝒑 (𝒔𝒕�𝟏|𝒔𝒕, 𝒂𝒕) ENVIRONMENT 𝒂𝒕 REPLAY BUFFER 𝒔𝒕�𝟏 𝒛 𝒛 𝒔𝒕�𝟏 Sample a batch of N transitions (𝒔𝒕, 𝒂𝒕, 𝒛𝒕, 𝒔𝒕�𝟏, 𝒛𝒕�𝟏, 𝒓) UPDATE CRITICS 𝜽𝒊 𝒕�𝟏 ←𝐚𝐫𝐠 𝐦𝐢𝐧 𝜽𝒊 𝑵�𝟏∑�𝒚 − 𝑸𝜽𝒊(𝒔, 𝒂)� 𝟐 UPDATE POLICY Generate critics by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' (9) 𝝓𝒕�𝟏 ←𝐚𝐫𝐠 𝐦𝐚𝐱 𝝓 𝟏 𝟒 � 𝒌=𝟏 𝟒 𝑸𝒌�𝒔, 𝝅(𝒔, 𝒛𝒌)� 𝝓𝒕�𝟏 POLICY No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1 POLICY No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2 POLICY No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='3 POLICY No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='4 Cooperative exploration Centralization Figure 1: The workflow of CCEP Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The agent 𝜋 interacts with the environment with diverse style cooperatively and produce the transition 𝑠𝑡 → 𝑠𝑡+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The actor and critic are updated over a mini-batch of the transition samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' A centralized policy with four different styles is learned from the multi-styled critics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In value-based deep reinforcement, deep neural networks ap- proximate the value functions, and the function approximation error exists correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' One major source of the function approximation error comes from the optimization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In this procedure, stochastic gradient descent, which uses a batch of random samples for gradient update each time, is the mainstream method due to the consideration of computational resources and training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' However, as [40] has indicated, a mini-batch gradient update may have unpredictable effects on samples outside the training batch, which leads to the function approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' For explanation, we use 𝑒𝑡 to represent the approximation error of the value function with the state-action pair(𝑠𝑡,𝑎𝑡) as input and approximation error 𝑒𝑡 can be modeled as follows: 𝑄𝜃 (𝑠𝑡,𝑎𝑡) = 𝑟 (𝑠𝑡,𝑎𝑡) + 𝛾E[𝑄𝜃 (𝑠𝑡+1,𝑎𝑡+1)] − 𝑒𝑡 (5) Approximation errors influence the value estimation when using the value function as an estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The estimation may be skewed towards an overestimation, causing a wrong estimate for a given state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' This leads to a problem of an optimal action being chosen but replaced by a sub-optimal action, owing to the overestimation of a sub-optimal action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Thus, the overestimation bias is a common problem in Q-Learning with discrete actions, as we choose the seemly best action 𝑎𝑡+1 in the target value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Still, there is little chance for the optimal state-action pair to be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' E[ max 𝑎𝑡+1∈A 𝑄(𝑠𝑡+1,𝑎𝑡+1)] ≥ max 𝑎𝑡+1∈A E[𝑄(𝑠𝑡+1,𝑎𝑡+1)] (6) Mentioned overestimated bias can also occur in continuous con- trol tasks[16], since the policy approximator always provides the optimal action at the current state based on the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' While this bias can be quite small in an individual update, the bias can be accumulated to a substantial overestimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' (5) can be expanded as follows: 𝑄𝜃 (𝑠𝑡,𝑎𝑡) = E𝑎𝑡 ∼𝜋𝜙 (·|𝑠𝑡 ),𝑠𝑡+1∼𝑃 � ∞ ∑︁ 𝑡=0 𝛾𝑡 (𝑟 (𝑠𝑡,𝑎𝑡) − 𝑒𝑡) � (7) Previous works such as Double Q-learning [23] and Double DQN [24] are proposed to alleviate value functions of underestimating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The idea is to maintain two independent estimators in which one is used for estimation while the other is for selecting maximal action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Similarly, as an extension in dealing with continuous control tasks, TD3 [16] reduce the overestimation bias by using double value func- tions and taking the minimum between the two value functions for an estimation which suffers from underestimation problems as well [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Does the estimation error influence the performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Given a continuous control task, we use 𝑓 to approximate the true under- lying value function 𝑄∗, which indicates the accumulated reward obtained by acting 𝑎 before taking optimal policy 𝜋∗ at state 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 𝑉 represents the true underlying value function(which is not known during training).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 𝑉 ∗ and 𝑉 𝜋𝑓 represent the accumulated return obtained by adopting the optimal policy 𝜋∗ and 𝜋𝑓 in state 𝑠 re- spectively in which 𝜋𝑓 is a learned policy by maximizing the value function approximate 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' (Performance Gap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The performance gap of the policy between the optimal policy 𝜋∗ and the learned policy 𝜋𝑓 is defined by an infinity norm ∥𝑉 ∗ − 𝑉 𝜋𝑓 ∥∞ and we have ∥𝑉 ∗ − 𝑉 𝜋𝑓 ∥∞ ≤ 2∥𝑓 − 𝑄∗∥∞ 1 − 𝛾 We provide proof detail of Lemma 1 in Supplementary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' This inequality indicates that the performance gap of the policy can be bounded by the estimation error of the value function and accurate value estimate can reduce the upper bound of the performance gap and enhance the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Do overestimation bias and underestimation bias affect per- formance in the same way?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' An empirical study shows that esti- mation bias may not always be a detrimental problem while both underestimation bias and overestimation bias may improve learn- ing performance which depends on the environment [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' As an example, for an unknown area with high stochasticity, overesti- mation bias may help to explore the overestimated area but un- derestimation bias prevents this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' However, if these areas of high stochasticity are given low values, the overestimation bias may lead to excess exploration in low-value regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The fact is that we can not choose the environment and these different circumstances can always occur during exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Our method is designed to utilize the difference in exploration behavior brought by estimation bias to encourage multi-styled exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2 Multi-Style Critics: Radical, Conservative As mentioned above, function approximation error exists in value functions and can accumulate to substantial scales which have a great influence on the value estimation resulting in overestimation or underestimation bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Estimation bias has been researched in recent works [16, 25, 50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' While these works focus on an accurate value estimation and discussed the method to control the estimation bias with the use of multiple value functions for auxiliary, they just choose one of the value functions, which seems to be the most accurate, for policy update neglecting other value functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' However, there is no accurate value function without trial and error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In this section, we show how to utilize the estimation bias and introduce our method for the generalization of multi-styled critics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Our intuition is that there are different degrees of estimation bias in double randomly initialized value functions when perform- ing function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' However, the estimation bias can be controlled by applying a maximum operator and minimum op- erator, namely the maximum of the two estimates is relatively overestimated and the minimum of the two estimates is relatively underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Two different estimates raise a controversy about which critic gives the accurate estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The best way to resolve the controversy is to follow one of the critics to explore and collect reward messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' While controversy does not always exist because there is only one accurate value, the critics reach an agreement when the state value has been exactly estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' And the existence of controversy means more exploration is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We start by maintaining double randomly initialized value func- tions 𝑄𝜃1 and 𝑄𝜃2 with parameters 𝜃1 and 𝜃2 respectively and up- date the value function with TD3 [16] which takes the minimum between the two value functions as the target value estimate: 𝑦 = 𝑟 + min 𝑖=1,2𝑄𝜃𝑖 (𝑠′,𝑎′),𝑎′ ∼ 𝜋𝜙 (8) But the two randomly initialized value functions potentially have different value estimations for a given state-action pair due to the accumulated function approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' This difference leads to the result that the two critics may give two different suggestions for the best action choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' While these estimates are relatively overestimated or underestimated, these different criteria for a given state-action pair may lead to a different style of action choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' It Algorithm 1 Centralized Cooperative Exploration Policy (CCEP) Initialize critic networks 𝑄𝜃1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑄𝜃2 Initialize actor network 𝜋𝜙 with random parameters 𝜃1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝜃2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝜙 Initialize target networks 𝜃 ′ 1 ← 𝜃1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝜃 ′ 2 ← 𝜃2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝜙′ ← 𝜙 Initialize replay buffer B Initialize number of skills K 1: for 𝑡 = 1 to 𝑇 do 2: Sample a skill 𝑧 from 𝑝(𝑧) 3: Select action with noise 𝑎 ∼ 𝜋𝜙 (𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑎) + 𝜖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝜖 ∼ N (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 𝜎) 4: Observe a reward 𝑟 and a new state 𝑠′ 5: Store transition tuple (𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑠′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑧′) in B 6: Sample mini-batch of 𝑁 transitions (𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑠′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑧′) from 𝐵 7: 𝑎′ ← 𝜋𝜙′(𝑠′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑧′) + 𝜖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝜖 ∼ 𝑐𝑙𝑖𝑝(N (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 𝜎),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' −𝑐,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑐) 8: 𝑦 = 𝑟 + 𝛾 min𝑖=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2 𝑄𝜃′ 𝑖 (𝑠′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑎′) 9: Update critics: 𝜃𝑖 ← arg min𝜃𝑖 𝑁 −1 �(𝑦 − 𝑄𝜃𝑖 (𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑎))2 10: if t mod d then 11: Update policy: 12: ∇𝜙 𝐽 (𝜙) = 𝑁 −1K−1 � ∇𝑎𝑄 𝑗 (𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑎)|𝑎=𝜋𝜙 (𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑧)∇𝜙𝜋𝜙 (𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='𝑧) 13: Update target networks 14: 𝜃 ′ 𝑖 ← 𝜏𝜃𝑖 + (1 − 𝜏)𝜃 ′ 𝑖 15: 𝜙′ 𝑖 ← 𝜏𝜙𝑖 + (1 − 𝜏)𝜙′ 𝑖 is reasonable the estimation is radical if we choose the maximum value of the two to estimate and the estimation is conservative if we choose the minimum value of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Thus, we consider four critics: 𝑄 𝑗 (𝑠,𝑎) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 𝑄𝜃1 (𝑠,𝑎) 𝑗 = 0 𝑄𝜃2 (𝑠,𝑎) 𝑗 = 1 max(𝑄𝜃1 (𝑠,𝑎),𝑄𝜃2 (𝑠,𝑎)) 𝑗 = 2 min(𝑄𝜃1 (𝑠,𝑎),𝑄𝜃2 (𝑠,𝑎)) 𝑗 = 3 (9) There exists controversy among these critics, and the controversy can further influence the performance of the policy learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='3 Opposite Value Functions This approach for generating diverse styles raises the problem that the value functions may not provide sufficient difference in style when the controversy disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' This phenomenon is very com- mon when the value function converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' But we don’t want this to happen too soon, because we want the value functions to provide more exploration for the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The controversy exists due to the randomly initialized parameters of the neural networks and the error accumulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' But actually, there is a small probability that the two networks have great similarities, which will lead to double consistent critics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' This is not what we want, because consistent critics mean monotonous policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' To avoid this, we try to enlarge the controversy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The solution in this paper is to learn two opposite tar- gets respectively for the two networks, where one of the networks approximates the positive value and another approximates the neg- ative one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' This approach is equivalent to adding a factor -1 to the final layer of either network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We find that the controversy is guar- anteed with this simple network structure change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' To provide some intuition, we compared the controversy changes after the double value functions learn the opposite target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We express the amount 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='0 Time Steps (1e6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='5 Average Error (a) HalfCheetah-v3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='0 Time Steps (1e6) 0 1 2 3 Average Error Target same target opposite target (b) Walker2d-v3 Figure 2: Measuring the error between double critics given same/opposite targets in TD3 on MuJoCo environments over 1 million time steps of controversy between the two value functions by the errors of state values in a batch of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Figure 2 shows the controversy measuring over MuJoCo [48] environments in HalfCheetah-v3 and Walker2d-v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The results show that with the simple network struc- ture change, the controversy is enlarged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' But this approach will not influence the value estimation because we just fine-tune the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='4 Centralized Cooperation With four critics, we train a centralized cooperative policy to encour- age multi-styled explorations through diverse value estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We model this problem as a multi-objective optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The target is to train multiple policies, with each policy targeting an individual value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We express the policy function as 𝜋(𝑠,𝑧), with state 𝑠 and latent variable 𝑧 as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The latent variable 𝑧, which is a one-hot label in our method, represents different policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The architecture of our centralized cooperative policy is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' This idea comes from skill discovery method [11, 43], which use the latent variable 𝑧 to express different skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' And in skill discovery, the target is to maximize the mutual information be- tween latent variable 𝑧 and some aspects of the trajectories, which is a different target for a different latent variable 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Our method encourages diverse styles of policies by different targets as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Particularly, we sample latent variable 𝑧 from set {0, 1, 2, 3} and encode it in a one-hot label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' For a given latent variable 𝑧, the policy targets 𝑧-th value functions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='(9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' With different latent variable 𝑧, the policy shows diverse styles due to the multi-styled targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We make an experiment showing that there exists different explo- ration preferences for these policies (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2) In the exploration procedure, we randomly sample latent variable 𝑧 and make deci- sions by policy 𝜋(𝑠,𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' This approach enables diverse styles to be applied at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Broadly speaking, our exploration policy has the following characteristics: Multi-styled, Centralized, and Cooperative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Multi-styled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We train four policies to accomplish the explo- ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' These policies learn from the corresponding value function 𝑄 𝑗: 𝜋∗ 𝑗 = arg max 𝜋 𝑄 𝑗 (10) There are four value estimators, in which two of them (𝑗 = 0, 1) are normal but different estimators, one (𝑗 = 2) is an overestimated Policy 1 Policy 2 Policy 3 Policy 4 Centralized Cooperative Policy Environment Label variance Input Output Figure 3: The architecture of our centralized cooperative pol- icy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The agent cooperatively explores the environments by selecting one of the styles at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The style se- lection process is implemented by sampling latent variable 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Policies with diverse styles exchange messages through a centralized network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' estimator compared to the other (𝑗 = 3) and helps encourage ex- plorations in overestimated actions, the last remaining one (𝑗 = 3) is a conservative estimator and brings more exploitation as illus- trated in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' It is appropriate for the policies to perform in a variety of ways given the varied estimators they use (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=', conservative, radical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Centralized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Our policy is a centralized policy because we make use of all the policies learned in each episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' At each time step𝑡, we sample one of these policies for exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' It allows us to generate a variety of trajectories adopting this exploration approach as this centralized policy can generate 4𝑛 types of trajectories theoretically for 𝑛-step exploration, compared to using a single policy that can only generate one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' These trajectories are stored as experience and maintain the update for a pair of centralized value functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Cooperative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We update the policy cooperatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' With multi- ple policies learning their respective value functions, “knowledge” learned by each policy cannot be shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Our method is to learn a single network for policies and learn cooperatively [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' To repre- sent different policies, we feed latent variables 𝑧 which are one-hot labels as extra input to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The policy which inputs latent variable 𝑧 and state𝑠 and outputs action𝑎 can be defined as 𝜋(𝑠,𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We sample 𝑧 to represent the sampling of different policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Thus, the policy can be updated by taking deterministic policy gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' ∇𝜙 𝐽 (𝜙) = 𝑁 −1K−1 ∑︁ ∇𝑎𝑄 𝑗 (𝑠,𝑎)|𝑎=𝜋𝜙 (𝑠,𝑧)∇𝜙𝜋𝜙 (𝑠,𝑧) (11) Where 𝑄 𝑗 (𝑠,𝑎) refer to the multi-styled critics in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' (9), K is the number of styles which is 4 in this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The specific algorithm is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 4 EXPERIMENTAL SETTINGS To evaluate our method, we test our algorithm on the suit of MujoCo [48] continuous control tasks, including HalfCheetah-v3, G(a) (b) (c) (d) Figure 4: Screenshots of MuJoCo environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' (a) Ant-v3, (b) HalfCheetah-v3, (c) Walker2d-v3, (d) Hopper-v3 Hopper-v3, Walker2d-v3, Ant-v3, Pusher-v2 and Humanoid-v3 (the screenshots are presented in Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' For implementation, our method builds on TD3 [16], and for com- parison, we also establish three-layer feedforward neural networks with 256 hidden nodes per hidden layer for both critics and actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Particularly, the actor takes state 𝑠 and latent variable 𝑧 concate- nated as input, where the latent variable 𝑧 is encoded as one-hot label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' At each time step, both networks are trained with a mini- batch of 256 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We apply soft updates for target networks as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We compared our algorithm against some classic algorithms such as DDPG [30], which is an efficient off-policy reinforcement learning method for continuous tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' PPO [42], the state-of-the- art policy gradient algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' TD3 [16], which is an extension to DDPG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' SAC [22], which is an entropy-based method with high sample efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Further, we compared our algorithm with the latest algorithm in solving the exploration problems in continuous control tasks such as OAC [8], which makes improvements on SAC for better exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We implement DDPG and PPO by OpenAI’s baselines repository and SAC, TD3, and OAC by the github the author provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' And we use the parameter the author recommend for implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The details of the implementation are shown in Supplementary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 5 EXPERIMENTAL RESULTS AND ANALYSIS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1 Evaluation To validate the performance of the CCEP algorithm, we evaluate our algorithm in MuJoCo continuous control suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We perform interactions for 1 million steps in 10 different seeds and evaluate the algorithm over 10 episodes every 5k steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Our results report the mean scores and standard deviations in the 10 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We show learning curves in Figure 5 and the max average return over 10 trials of 1 million time steps in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The learning curves in 1 million time steps show that our algorithm achieves a higher sample effi- ciency compared with the latest algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Furthermore, the results in the Table 1 indicates that our algorithm shows superior perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' And in HalfCheetah-v3, Walker2d-v3, Hopper-v3, Ant-v3, Table 1: The highest average return over 10 trials of 1 million time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The maximum value for each task is bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Environment Ours OAC SAC TD3 DDPG PPO HalfCheetah 11945 9921 11129 9758 8469 3681 Hopper 3636 3364 3357 3479 2709 3365 Walker2d 4706 4458 4349 4229 3669 3668 Ant 5630 4519 5084 5142 1808 909 Pusher 21 25 20 25 29 21 Humanoid 5325 5747 5523 5356 1728 586 our algorithm outperforms all the other baselines and achieve sig- nificant improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' While in the Pusher-v2 task, our algorithm show higher stability than that of TD3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' For further evaluation, we evaluate our algorithm in the state-based suite PyBullet [9] which is considered to be harder than the suite MuJoCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Our algorithm still shows better performance compared to the baseline algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The corresponding results are shown in Supplementary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2 Policy Style To ensure that our proposed CCEP algorithm learns diverse styles, we compared the distribution of explored trajectories when ex- ploring with a single style only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We test the algorithm in Ant-v3 environment over 1𝑒6 time steps and use the states sampled to rep- resent the trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Figure 6 shows the states explored by each style at 1𝑒5, 2𝑒5 and 3𝑒5 learning steps, and a more detailed results are shown in Supplementary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We collect the states sampled over 10 episodes with different seeds and apply t-SNE [49] for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The results show that while part of the states can be gathered by all styles which implies a compromise in controversy, there is a considerably large region of states that can only be ex- plored by a unique style of policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Though different styles, diverse styles come to be in compromise as training process goes on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' This phenomenon suggests that CCEP behaves in multi-styled explo- ration which leads to an exploration preference, and styles come to an agreement with sufficient exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Another phenomenal conclusion is that although the style tends to be consistent, new styles are emerging which brings enduring exploration capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='3 Measuring Exploration Ability The critical problem of our proposed method is whether we achieve higher sample efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Although the learning curves (Figure 5) gives considerably convincing results, a more intuitive result has been given in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We compared the exploration of CCEP with that of TD3 and SAC (which achieve the trade-off between exploration and exploitation by entropy regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=') over 10 episodes with different seeds (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' For a fair comparison, these methods are trained in Ant-v3 with the same seed at half of the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In order to get reliable results, the states explored are gathered in 10 episodes with different seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We still apply the same t-SNE [49] transformation to the states explored by all of the algorithms for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' While there are great differences between the states explored by TD3 (green) and SAC (blue), the result shows that our algorithm (red) explores a wider range of states which even covers that TD3 and SAC explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 0 5000 10000 Average Return HalfCheetah-v3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 0 2000 4000 Walker2d-v3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 0 1000 2000 3000 4000 Hopper-v3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 Time Steps (1e6) 0 2000 4000 Average Return Humanoid-v3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 Time Steps (1e6) 2000 0 2000 4000 6000 Ant-v3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 Time Steps (1e6) 80 60 40 20 Pusher-v2 Ours OAC PPO DDPG TD3 SAC Figure 5: Learning curves for 6 MuJoCo continuous control tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='For better visualization,the curves are smoothed uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The bolded line represents the average evaluation over 10 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The shaded region represents a standard deviation of the average evaluation over 10 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' (1) Learning Steps = 100000 (2) Learning Steps = 200000 (3) Learning Steps = 300000 POLICY No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1 POLICY No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2 POLICY No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='3 POLICY No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='4 Figure 6: The states visited by each style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' For better visualization, the states get dimension reduction by t-SNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The points with different color represents the states visited by the policy with the style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The distance between points represents the difference between states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='4 Ablation Study We perform an ablation study to understand the contribution of the cooperation between policies for message delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The results are shown in Table 2 where we compare the performance of training policies by removing policy cooperation and training them sepa- rately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We perform interactions for 1 million time steps for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The results show that without cooperation, the policy net- work not only trains 4 times more network parameters but also fails to reduce performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' And this performance degradation is even more pronounced on Walker2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Additional learning curves can be found in Supplementary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 6 RELATED WORK This section discusses several methods proposed recently for im- proving the exploration of deep reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' A range of works take an effort in encouraging explorations with (1) Ours (2) TD3 (3) SAC Figure 7: Measuring the exploration region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Comparison of exploration capabilities of TD3 (green), SAC (blue) and Ours (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The points represent region explored by each method in 10 episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' All the states get dimension reduction by the same t-SNE transformation for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Table 2: Max Average Return over 5 trials of 1 million time steps, comparing ablation over cooperation for message de- livery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The maximum value for each task is bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Method HCheetah Hopper Walker2d Ant CCEP 11969 3672 4789 5488 CCEP-Cooperation 11384 3583 4087 4907 TD3 9792 3531 4190 4810 the use of randomness over model parameters [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Another preva- lent series of works propose to enhance exploration by simulta- neously maximizing the expected return and entropy of the pol- icy [15, 21, 22, 39, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Whereas, these methods do not provide heuristic knowledge to guide the exploration, which can be consid- ered to be insufficient and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' To achieve effective exploration, the curiosity mechanism [19, 38] has been proposed in recent works, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=', the counted-based ap- proaches [33] which quantify the “novelty” of a state by the times visited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' However, these methods maintain the state-action visitation counts which make it challenging in solving high-dimensional or continuous tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Other works rely on errors in predicting dynam- ics, which have been used to address the difficulties in complex environments [5, 37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Though the Intrinsic Curiosity Module (ICM) [37] maintains a predictor on state transitions and considers the prediction error as an intrinsic reward, Random Network Distil- lation (RND) [5] utilizes the prediction errors of networks trained on historical trajectories to quantify the novelty of states, which is effective and easy to implement in real applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Another direction in previous work is to study exploration in hierarchical reinforcement learning (HRL) [3, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' These methods are insight from the fact that developers prefer to divide the com- prehensive and knotty problems into several solvable sub-problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' There are some further studies on hierarchy in terms of tasks, rep- resentative of which are goal-based reinforcement learning and skill discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The similarity of these approaches is that they both identify different policies by utilizing latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In goal-based RL, the latent variables are defined by the policy’s goal, which aims to complete several sub-goals and accomplish the whole task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' These methods introduce prior human knowledge, causing them to work brilliantly on some tasks but fail when unaware of human knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Despite our method also introducing latent variables to represent different styles of policies, all the policies share the same objective, nevertheless differing in the road to reach the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Skill discovery methods, which adopt the latent variable to repre- sent the skill learned from the policy, introduce mutual information to organize relationships between the latent variable 𝑧 and some aspects of the trajectories to acquire diverse skills (also known as style) [1, 11, 13, 20, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Nevertheless, these methods train the policy in an unsupervised way [11, 13, 43], suggesting that the skills trained are unaware of task-driven, and they cannot rep- resent the optimal policies when adapted to downstream tasks illustrated in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Our method avoids this issue because we train the policy task-oriented and demonstrate the benefit brought by the attention of these policies to the state value making them differ considerably in exploration style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' For task relevance, some related works that learn skills by jointly learning a set of skills and a meta- controller [3, 10, 13, 14, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The options of the meta-controller control different attentions of each policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' However, these meth- ods usually choose the best option to explore and rarely execute sub-optimal options, leading to the drawback – the algorithm tends to ignore sub-optimal actions that maybe fail in most states but are effective in a few critical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Our proposed approach randomly selects different styles of policies for directed coopera- tive exploration, which are improved accordingly with the value function and produce different styles due to differences in attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 7 CONCLUSION In the value-based method, value estimation bias has been a com- mon problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' While different estimation bias in double value func- tions lead to value function controversy, the controversy can be utilized to encourage policies to yield multiple styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In this paper, we aim at encouraging explorations by multi-styled policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We start by analysis on estimation bias during the value function train- ing process and its effect on the exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We then encourage this controversy between the value functions and generate four critics for producing multi-styled policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Finally, we apply these policies with diverse styles for centralized cooperative exploration which perform superior sample efficiency in the test environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Though there are a lot of works focusing on reducing the estimation bias for an accurate value estimation, few works try to utilize these inevitable errors to make improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Our results show that it is also an option to use the errors to encourage explorations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' For future work, it is an exciting avenue for focusing on more expres- sive policy styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' A style that can be represented as a continuous distribution may be more efficient and more expressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' REFERENCES [1] Joshua Achiam, Harrison Edwards, Dario Amodei, and Pieter Abbeel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Variational option discovery algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' CoRR abs/1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='10299 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' http: //arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org/abs/1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='10299 [2] Marcin Andrychowicz, Filip Wolski, Alex Ray, Jonas Schneider, Rachel Fong, Peter Welinder, Bob McGrew, Josh Tobin, OpenAI Pieter Abbeel, and Wojciech Zaremba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Hindsight experience replay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 30 (2017), 5048–5058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [3] Pierre-Luc Bacon, Jean Harb, and Doina Precup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The option-critic archi- tecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 1726–1734.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [4] Yunpeng Bai, Chen Gong, Bin Zhang, Guoliang Fan, Xinwen Hou, and Yu Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Cooperative Multi-Agent Reinforcement Learning with Hypergraph Convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In 2022 International Joint Conference on Neural Networks (IJCNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1109/IJCNN55064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='9891942 [5] Yuri Burda, Harrison Edwards, Amos Storkey, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Exploration by random network distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [6] Olivier Chapelle and Lihong Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' An Empirical Evaluation of Thompson Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Shawe-Taylor, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Zemel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Bartlett, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Pereira, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Weinberger (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' ), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=', 2249–2257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='neurips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='cc/paper/2011/file/ e53a0a2978c28872a4505bdb51db06dc-Paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='pdf [7] Jongwook Choi, Archit Sharma, Honglak Lee, Sergey Levine, and Shixiang Shane Gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Variational empowerment as representation learning for goal- conditioned reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' PMLR, 1953–1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [8] Kamil Ciosek, Quan Vuong, Robert Loftin, and Katja Hofmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Better Exploration with Optimistic Actor Critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In Advances in Neural Information Pro- cessing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, Hanna M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Wal- lach, Hugo Larochelle, Alina Beygelzimer, Florence d’Alché-Buc, Emily B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Fox, and Roman Garnett (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 1785–1796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='neurips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='cc/paper/ 2019/hash/a34bacf839b923770b2c360eefa26748-Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='html [9] Erwin Coumans and Yunfei Bai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Pybullet, a python module for physics simulation for games, robotics and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [10] Peter Dayan and Geoffrey E Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Feudal reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Ad- vances in neural information processing systems 5 (1992), 271–278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [11] Benjamin Eysenbach, Abhishek Gupta, Julian Ibarz, and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Diversity is all you need: Learning skills without a reward function, In 7th International Conference on Learning Representations, ICLR 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' arXiv preprint arXiv:1802.' metadata={'source': 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+page_content='net, Toulon, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [14] Kevin Frans, Jonathan Ho, Xi Chen, Pieter Abbeel, and John Schulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Meta learning shared hierarchies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In 6th International Conference on Learning Representations, ICLR 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' OpenReview.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Addressing function ap- proximation error in actor-critic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In Proceedings of the 35th International Conference on Machine Learning, ICML 2018 (Proceedings of Machine Learning Re- search, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' PMLR, PMLR, Stockholmsmässan, Stockholm, Sweden, 1582–1591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' http://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='mlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='press/v80/fujimoto18a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='html [17] Chen Gong, Qiang He, Yunpeng Bai, Xinwen Hou, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Wide-Sense Stationary Policy Optimization with Bellman Residual on Video Games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In 2021 IEEE International Conference on Multimedia and Expo, ICME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' IEEE, 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [18] Chen Gong, Zhou Yang, Yunpeng Bai, Junda He, Jieke Shi, Arunesh Sinha, Bowen Xu, Xinwen Hou, Guoliang Fan, and David Lo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Mind Your Data!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Hiding Backdoors in Offline Reinforcement Learning Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' CoRR abs/2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='04688 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='04688 arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='04688 [19] Chen Gong, Zhou Yang, Yunpeng Bai, Jieke Shi, Arunesh Sinha, Bowen Xu, David Lo, Xinwen Hou, and Guoliang Fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Curiosity-Driven and Victim- Aware Adversarial Policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In Proceedings of the 38th Annual Computer Security Applications Conference (Austin, TX, USA) (ACSAC ’22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 186–200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1145/3564625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='3564636 [20] Karol Gregor, Danilo Jimenez Rezende, and Daan Wierstra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Variational intrinsic control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In 5th International Conference on Learning Representations, ICLR 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https://openreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='net/forum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='id=Skc-Fo4Yg [21] Tuomas Haarnoja, Haoran Tang, Pieter Abbeel, and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Rein- forcement Learning with Deep Energy-Based Policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In Proceedings of the 34th International Conference on Machine Learning (Sydney, NSW, Australia) (ICML’17, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' JMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org, 1352–1361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [22] Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In Proceedings of the 35th International Conference on Machine Learning, ICML 2018 (Proceedings of Machine Learning Research, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' PMLR, PMLR, Stockholmsmässan, Stockholm, Sweden, 1856–1865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [23] Hado Hasselt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Double Q-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In Advances in Neural Information Pro- cessing Systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Lafferty, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Williams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Shawe-Taylor, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Zemel, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Culotta (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' ), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='neurips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='cc/paper/ 2010/file/091d584fced301b442654dd8c23b3fc9-Paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='pdf [24] Hado van Hasselt, Arthur Guez, and David Silver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Deep Reinforcement Learning with Double Q-Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence (Phoenix, Arizona) (AAAI’16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' AAAI Press, 2094–2100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [25] Qiang He and Xinwen Hou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' WD3: Taming the Estimation Bias in Deep Reinforcement Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In 32nd IEEE International Conference on Tools with Artificial Intelligence, ICTAI 2020, Baltimore, MD, USA, November 9-11, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' IEEE, 391–398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1109/ICTAI50040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00068 [26] Qiang He, Xinwen Hou, and Yu Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' POPO: Pessimistic Offline Policy Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2022, Virtual and Singapore, 23-27 May 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' IEEE, 4008–4012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1109/ICASSP43922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='9747886 [27] Qiang He, Huangyuan Su, Jieyu Zhang, and Xinwen Hou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Representation Gap in Deep Reinforcement Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' CoRR abs/2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='14557 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='14557 arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='14557 [28] Nicolas Heess, Greg Wayne, Yuval Tassa, Timothy Lillicrap, Martin Riedmiller, and David Silver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Learning and transfer of modulated locomotor controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' CoRR abs/1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='05182 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org/abs/1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='05182 [29] Sanjay Krishnan, Roy Fox, Ion Stoica, and Ken Goldberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Ddco: Discovery of deep continuous options for robot learning from demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In Conference on robot learning, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' PMLR, PMLR, 418–437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' http://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='mlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='press/ v78/krishnan17a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='html [30] Qingfeng Lan, Yangchen Pan, Alona Fyshe, and Martha White.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Maxmin Q-learning: Controlling the Estimation Bias of Q-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https://openreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='net/forum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='id=Bkg0u3Etwr [31] Timothy P Lillicrap, Jonathan J Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Continuous control with deep reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [32] Long-Ji Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Self-improving reactive agents based on reinforcement learning, planning and teaching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Machine Learning 8, 3 (1992), 293–321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1007/BF00992699 [33] Marlos C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Machado, Marc G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Bellemare, and Michael Bowling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Count- Based Exploration with the Successor Representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Proceedings of the AAAI Conference on Artificial Intelligence 34, 04, 5125–5133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [34] Volodymyr Mnih, Adria Puigdomenech Badia, Mehdi Mirza, Alex Graves, Tim- othy Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Asyn- chronous methods for deep reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In International conference on machine learning (JMLR Workshop and Conference Proceedings, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 48), Maria- Florina Balcan and Kilian Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Weinberger (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' PMLR, 1928–1937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [35] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Rusu, Joel Veness, Marc G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Bellemare, Alex Graves, Martin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Riedmiller, Andreas Fidjeland, Georg Ostrovski, Stig Petersen, Charles Beattie, Amir Sadik, Ioannis Antonoglou, Helen King, Dharshan Kumaran, Daan Wierstra, Shane Legg, and Demis Hassabis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Human-level control through deep reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' nature 518, 7540 (2015), 529–533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [36] Shakir Mohamed and Danilo Jimenez Rezende.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Variational information maximisation for intrinsically motivated reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Advances in neural information processing systems 28 (2015), 2125–2133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [37] Deepak Pathak, Pulkit Agrawal, Alexei A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Efros, and Trevor Darrell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Curiosity-driven Exploration by Self-supervised Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In Proceedings of the 34th International Conference on Machine Learning (Proceedings of Machine Learning Research, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' PMLR, 2778–2787.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [38] Roberta Raileanu and Tim Rocktäschel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' RIDE: Rewarding Impact-Driven Exploration for Procedurally-Generated Environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In 8th International Con- ference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [39] Konrad Rawlik, Marc Toussaint, and Sethu Vijayakumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' On Stochastic Optimal Control and Reinforcement Learning by Approximate Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In Proceedings of Robotics: Science and Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Sydney, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='15607/RSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='045 [40] Martin Riedmiller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Neural Fitted Q Iteration – First Experiences with a Data Efficient Neural Reinforcement Learning Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In Machine Learning: ECML 2005, João Gama, Rui Camacho, Pavel B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Brazdil, Alípio Mário Jorge, and Luís Torgo (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Springer Berlin Heidelberg, Berlin, Heidelberg, 317–328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [41] Herbert Robbins and Sutton Monro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 1951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' A stochastic approximation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The annals of mathematical statistics (1951), 400–407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [42] John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Proximal Policy Optimization Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' CoRR abs/1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='06347 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='06347 http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org/abs/1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='06347 [43] Archit Sharma, Shixiang Gu, Sergey Levine, Vikash Kumar, and Karol Hausman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Dynamics-aware unsupervised discovery of skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In 8th International Conference on Learning Representations, ICLR 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='net, Addis Ababa, Ethiopia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https://openreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='net/forum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='id=HJgLZR4KvH [44] David Silver, Guy Lever, Nicolas Heess, Thomas Degris, Daan Wierstra, and Mar- tin Riedmiller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Deterministic Policy Gradient Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In Proceedings of the 31st International Conference on International Conference on Machine Learning Volume 32 (Beijing, China) (ICML’14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' JMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org, I–387–I–395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [45] Richard S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Sutton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Learning to predict by the methods of temporal differ- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Machine Learning 3, 1 (1988), 9–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1007/BF00115009 [46] Richard S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Sutton and Andrew G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Barto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Reinforcement learning: An intro- duction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' MIT press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [47] Richard S Sutton, Doina Precup, and Satinder Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Between MDPs and semi-MDPs: A framework for temporal abstraction in reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Artificial intelligence 112, 1-2 (1999), 181–211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [48] Emanuel Todorov, Tom Erez, and Yuval Tassa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' MuJoCo: A physics engine for model-based control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 5026–5033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1109/IROS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='6386109 [49] Laurens van der Maaten and Geoffrey Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Visualizing Data using t-SNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Journal of Machine Learning Research 9, 86 (2008), 2579–2605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' http: //jmlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org/papers/v9/vandermaaten08a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='html [50] Wei Wei, Yujia Zhang, Jiye Liang, Lin Li, and Yyuze Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Controlling Under- estimation Bias in Reinforcement Learning via Quasi-median Operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Proceed- ings of the AAAI Conference on Artificial Intelligence 36, 8 (Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2022), 8621–8628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1609/aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='v36i8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='20840 [51] Dongming Wu, Xingping Dong, Jianbing Shen, and Steven C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Hoi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Reducing Estimation Bias via Triplet-Average Deep Deterministic Policy Gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' IEEE Transactions on Neural Networks and Learning Systems 31, 11 (2020), 4933– 4945.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1109/TNNLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2959129 [52] Zhiwei Xu, Yunpeng Bai, Dapeng Li, Bin Zhang, and Guoliang Fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' SIDE: State Inference for Partially Observable Cooperative Multi-Agent Reinforcement Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' In Proceedings of the 21st International Conference on Autonomous Agents and Multiagent Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' International Foundation for Autonomous Agents and Multiagent Systems (IFAAMAS), 1400–1408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [53] Yingjun Ye, Xiaohui Zhang, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Automated vehicle’s behavior decision making using deep reinforcement learning and high-fidelity simulation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Transportation Research Part C: Emerging Technologies 107 (2019), 155–170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' [54] Xianjie Zhang, Yu Liu, Hangyu Mao, and Chao Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Common belief multi- agent reinforcement learning based on variational recurrent models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Neurocom- puting 513 (2022), 341–350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='neucom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='144 [55] Brian D Ziebart, Andrew L Maas, J Andrew Bagnell, Anind K Dey, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Maximum entropy inverse reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='. In Aaai, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Chicago, IL, USA, 1433–1438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Supplementary Materials A PROOF OF LEMMA 1 (Lemma 1)(Performance Gap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Let 𝑉 ∗ be the ground truth state value in Bellman value iterations, 𝑄∗ be the ground truth state action value, 𝑉 𝜋𝑓 be the state value when applying learned policy 𝜋𝑓 , 𝑓 be the value function approximator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The performance gap of the policy between the optimal policy 𝜋∗ and the learned policy 𝜋𝑓 is defined by an infinity norm ∥𝑉 ∗ − 𝑉 𝜋𝑓 ∥∞ and we have ∥𝑉 ∗ − 𝑉 𝜋𝑓 ∥∞ ≤ 2∥𝑓 −𝑄∗ ∥∞ 1−𝛾 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' For any 𝑠 ∈ S 𝑉 ∗(𝑠) − 𝑉 𝜋𝑓 (𝑠) =𝑄∗(𝑠, 𝜋∗(𝑠)) − 𝑄∗(𝑠, 𝜋𝑓 (𝑠)) + 𝑄∗(𝑠, 𝜋𝑓 (𝑠)) − 𝑄∗(𝑠, 𝜋𝑓 (𝑠)) ≤𝑄∗(𝑠, 𝜋∗(𝑠)) − 𝑓 (𝑠, 𝜋∗(𝑠)) + 𝑓 (𝑠, 𝜋𝑓 (𝑠)) − 𝑄∗(𝑠, 𝜋𝑓 (𝑠)) + 𝛾E𝑠′∼𝑃 (𝑠,𝜋𝑓 (𝑠)) [𝑉 ∗(𝑠′) − 𝑉 𝜋𝑓 (𝑠′)] ≤2∥𝑓 − 𝑄∗∥∞ + 𝛾∥𝑉 ∗ − 𝑉 𝜋𝑓 ∥∞ B EXPERIMENTAL DETAILS B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1 Environments We evaluate the performance of CCEP on environments from Mu- joCo Control Suite [48]which can be listed as HalfCheetah-v3, Ant- v3, Walker2d-v3, Humanoid-v3, Hopper-v3, and Pusher-v2, and the specific parameters of these environments are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We use the publicly available environments without any modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2 Implementation and Hyper-parameters Here, we describe certain implementation details of CCEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' For our implementation of CCEP, we follows a standard actor-critic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='3 Soft Actor-Critic Implementation Details For implementation of SAC, we use the code the author provided and use the parameters the author recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We use a single Gaussian distribution and use the environment-dependent reward scaling as described by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' For a fair comparison, we apply the version of soft target update and train one iteration per time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We use the reward scales as the author recommended (except for Pusher-v2 which is not mentioned by the author in the article).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Considering that there are similar action dimensions between Pusher-v2 and HalfCheetah-v3, we set the same reward scale for Pusher-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The specific reward scales for each environment is shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Table 3: Environment Specific Parameters Environment State Dimensions Action Dimensions Ant-v3 111 8 HalfCheetah-v3 17 6 Hopper-v3 11 3 Humanoid-v3 376 17 Pusher-v2 23 7 Walker2d-v3 17 6 Table 4: SAC Environment Specific Parameters Environment Reward Scale Ant-v3 5 HalfCheetah-v3 5 Hopper-v3 5 Humanoid-v3 20 Pusher-v2 5 Walker2d 5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='4 Optimistic Actor-Critic Implementation Details The implementation of OAC is mainly based on the open source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We set the hyper-parameters the same as OAC used in MuJoCo which is listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' And for fair comparison, we train with 1 training gradient per environment step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We use the same reward scales as SAC, listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Table 5: SAC Environment Specific Parameters Parameter Value shift multiplier √ 2𝛿 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='86 𝛽𝑈 𝐵 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='66 𝛽𝐿𝐵 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='65 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='5 Reproducing Other Baselines For reproduction of TD3, we use the official implementation ( https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='com/sfujim/TD3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' For reproduction of DDPG and PPO we use OpenAI’s baselines repository and apply default hyper- parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Table 6: CCEP Parameters settings Parameter Value Exploration policy N (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1),𝑧 ∼ 𝑝(𝑧) Number of policy 4 Variance of exploration noise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2 Random starting exploration time steps 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='5 × 104 Optimizer Adam[30] Learning rate for actor 3 × 10−4 Learning rate for critic 3 × 10−4 Replay buffer size 1 × 106 Batch size 256 Discount (𝛾) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='99 Number of hidden layers 2 Number of hidden units per layer 256 Activation function ReLU Iterations per time step 1 Target smoothing coefficient (𝜂) 5 × 10−3 Variance of target policy smoothing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2 Noise clip range [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='5] Target critic update interval 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 Time Steps (1e6) 0 2000 4000 6000 8000 10000 12000 Average Return (a) HalfCheetah-v3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 Time Steps (1e6) 0 1000 2000 3000 4000 5000 (b) Walker2d-v3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 Time Steps (1e6) 0 1000 2000 3000 (c) Hopper-v3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 Time Steps (1e6) 0 1000 2000 3000 4000 5000 6000 Average Return (d) Ant-v3 CCEP CCEP-Cooperation TD3 Figure 8: Ablation over the use of cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Comparison of CCEP, TD3 and the subtraction of cooperation (CCEP- cooperation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 Time Steps (1e6) 0 500 1000 1500 2000 Average Return (a) Walker2D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 Time Steps (1e6) 0 1000 2000 3000 (b) Ant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 Time Steps (1e6) 0 500 1000 1500 2000 2500 (c) Hopper 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='00 Time Steps (1e6) 1000 0 1000 2000 3000 Average Return (d) HalfCheetah TD3 Ours SAC DDPG PPO Figure 9: Learning curves for 4 PyBullet continuous control tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' For better visualization, the curves are smoothed uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The bolded line represents the average evaluation over 10 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The shaded region represents the standard deviation of the average evaluation over 10 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Table 7: Evaluation in PyBullet control suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The highest average return over 10 trials of 1 million time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The maximum value for each task is bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Pybullet Environment Ours SAC TD3 DDPG PPO HalfCheetah 2670 ± 275 2494 ± 266 2415 ± 236 1120 ± 373 465 ± 30 Hopper 2254 ± 186 2167 ± 323 1860 ± 288 1762 ± 368 623 ± 131 Walker2d 1829 ± 418 1369 ± 408 1676 ± 342 929 ± 345 509 ± 106 Ant 3175 ± 184 2423 ± 680 2711 ± 253 483 ± 70 578 ± 19 C ADDITIONAL EXPERIMENTS C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1 Additional Evaluation For an additional Evaluation, We conduct experiments on the state- based PyBullet [9] suite which is based on the well-known open- source physics engine bullet and is packaged as a Python module for robot simulation and learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The suite of Pybullet is considered to be a harder environment than MuJoCo [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We choose TD3 [16], SAC [22], PPO [42], DDPG [31] as our baselines due to their superior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We perform interactions for 1 million steps in 10 different seeds and evaluate the algorithm over 10 episodes every 5k steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We evaluate our algorithm in HalfCheetah, Hopper, Walker2d and ant in the suite of pybullet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Our results report the mean scores and standard deviations in the 10 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We show the learning curves in Figure 9 and the max average return over 10 trials in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2 Additional Ablation Results We compare the learning curves of CCEP, TD3 and the subtraction of cooperation (CCEP-cooperation) for better understanding the contribution of policy cooperation (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We perform inter- actions for 1 million steps in 10 different seeds and evaluate over 10 episodes every 5k steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Our results report the mean scores and standard deviations in the 10 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' We show the learning curves in Figure 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='3 Supplementary Results We provide supplementary results for Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' Figure 10 shows the states visited by each style over 1M time steps with intervals of 100k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The results show that different styles get consistent but new styles emerges as well, which brings enduring exploration capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' (1) Learning Steps = 100000 (2) Learning Steps = 200000 (3) Learning Steps = 300000 (4) Learning Steps = 400000 (5) Learning Steps = 500000 (6) Learning Steps = 600000 (7) Learning Steps = 700000 (8) Learning Steps = 800000 (9) Learning Steps = 900000 POLICY No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='1 POLICY No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='2 POLICY No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='3 POLICY No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content='4 Figure 10: The states visited by each style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' For better visualization, the states get dimension reduction by t-SNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The points with different color represents the states visited by the policy with the style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} +page_content=' The distance between points represents the difference between states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf'} diff --git a/0NFQT4oBgHgl3EQfDTUM/content/tmp_files/2301.13233v1.pdf.txt b/0NFQT4oBgHgl3EQfDTUM/content/tmp_files/2301.13233v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7673dd79bdf565f75e976d2e6f50d0f471589ae0 --- /dev/null +++ b/0NFQT4oBgHgl3EQfDTUM/content/tmp_files/2301.13233v1.pdf.txt @@ -0,0 +1,1466 @@ +MNRAS 000, 1–11 (2022) +Preprint 1 February 2023 +Compiled using MNRAS LATEX style file v3.0 +Observations of the Planetary Nebula SMP LMC 058 with the +JWST MIRI Medium Resolution Spectrometer +O. C. Jones1★ , J. Álvarez-Márquez2 , G. C. Sloan3,4 , P. J. Kavanagh5 , I. Argyriou6 , +A. Labiano7 , D. R. Law3 , P. Patapis8 , Michael Mueller9 , Kirsten L. Larson3 , +Stacey N. Bright3 , P. D. Klaassen1 , O. D. Fox3 +3, Danny Gasman6 +V. C. Geers1 , +Adrian M. Glauser7 , Pierre Guillard10,11 , Omnarayani Nayak3 , A. Noriega-Crespo3 , +Michael E. Ressler12 , B. Sargent3,13 , T. Temim14 , B. Vandenbussche6 , +Macarena García Marín3 +1 UK Astronomy Technology Centre, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK +2 Centro de Astrobiología (CSIC-INTA), Carretera de Ajalvir, 28850 Torrejón de Ardoz, Madrid, Spain +3Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA +4Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599-3255, USA +5Dublin Institute for Advanced Studies, School of Cosmic Physics, Astronomy & Astrophysics Section, 31 Fitzwilliam Place, Dublin 2, Ireland +6Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium +7Telespazio UK for the European Space Agency (ESA), ESAC, Spain +8ETH Zurich, Institute for Particle Physics and Astrophysics, Wolfgang-Paulistr. 27, CH-8093 Zurich, Switzerland +9Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands +10Sorbonne Université, CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98bis bd Arago, +75014 Paris, France +11Institut Universitaire de France, Ministére de l’Enseignement Supérieur et de la Recherche, 1 rue Descartes, 75231 Paris Cedex 05, France +12Jet Propulsion Laboratory, California Institute of Technology,4800 Oak Grove Drive, Pasadena, CA 91109 +13Center for Astrophysical Sciences, The William H. Miller III Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA +14Princeton University, 4 Ivy Ln, Princeton, NJ 08544, USA +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +During the commissioning of JWST, the Medium-Resolution Spectrometer (MRS) on the +Mid-Infrared Instrument (MIRI) observed the planetary nebula SMP LMC 058 in the Large +Magellanic Cloud. The MRS was designed to provide medium resolution (R = 𝜆/Δ𝜆) 3D +spectroscopy in the whole MIRI range. SMP LMC 058 is the only source observed in JWST +commissioning that is both spatially and spectrally unresolved by the MRS and is a good test +of JWST’s capabilities. The new MRS spectra reveal a wealth of emission lines not previously +detected in this metal-poor planetary nebula. From these lines, the spectral resolving power +(𝜆/Δ𝜆) of the MRS is confirmed to be in the range R = 4000 to 1500, depending on the MRS +spectral sub-band. In addition, the spectra confirm that the carbon-rich dust emission is from +SiC grains and that there is little to no time evolution of the SiC dust and emission line strengths +over a 16-year epoch. These commissioning data reveal the great potential of the MIRI MRS. +Key words: instrumentation: spectrographs; infrared: general; Astrophysics - Instrumentation +and Methods for Astrophysics +1 +INTRODUCTION +The succession of increasingly powerful mid-infrared spectrographs +(e.g., the Short Wavelength Spectrometer (SWS) and the Infrared +Spectrograph (IRS) on board the Infrared Space Observatory and +★ E-mail: olivia.jones@stfc.ac.uk +the Spitzer Space Telescope) launched into space has revolutionised +our knowledge of the cool universe (e.g., Waters et al. 1996). +The Mid-Infrared Instrument (MIRI; Wright et al. (submitted)) +on the James Webb Space Telescope (JWST) includes, in addition +to the imager and coronographs, both a low-resolution spectrometer +(LRS) covering wavelengths from 5 to 14 𝜇m (Kendrew et al. 2015) +and a medium-resolution spectrometer (MRS; Wells et al. 2015, +Argyriou et al. (in prep.)), which is an Integral Field Unit (IFU), +© 2022 The Authors +arXiv:2301.13233v1 [astro-ph.IM] 30 Jan 2023 + +2 +O. C. Jones et al. +that has a field of view ranging from 3.2′′ × 3.7′′ to 6.6′′ × 7.7′′ +(Law et al. (in prep.)) and can spatially resolve spectroscopic data +between 4.9 and 27.9 𝜇m. This is the first time a mid-IR IFU has +been deployed outside our atmosphere and will enable resolved +spectroscopic studies of individual stars at the beginning and end +of their evolution, diffuse structure in galaxies and planets. +The Large Magellanic Cloud (LMC) is a gas-rich, metal-poor, +star-forming, irregular galaxy, which is a satellite of the Milky Way, +hosts ∼103 planetary nebulae (PNe) (Reid & Parker 2010; Reid +2014), and is at a uniform distance of ∼50 kpc (Pietrzyński et al. +2013). In lower metallicity environments like the LMC, which has +about half the metallicity of the Milky Way (Westerlund 1997; +Choudhury et al. 2016), significant dust production is expected to +occur in the outflows of asymptotic giant branch (AGB) stars with +further processing as these objects become planetary nebula (PNe). +The gas and dust ejected into the interstellar medium (ISM) by a +strong stellar wind from this phase of evolution contains elements +synthesised in the stellar interior and dredged up to the surface by +convection (e.g., Karakas & Lattanzio 2007, 2014). Their chemical +composition is expected to primarily depend upon the initial stel- +lar mass and the interstellar elemental abundance at the time the +progenitor stars were formed (Kwok 2000; Gonçalves et al. 2014; +Kwitter & Henry 2022). +As such, the infrared spectra of PNe host a rich variety of +features; forbidden emission lines arising from ionisation from the +hot central star (e.g., Stanghellini et al. 2007), complex organic +molecules (e.g., Ziurys 2006), polycyclic aromatic hydrocarbons +(PAHs), and inorganic and organic solids (e.g., Stanghellini et al. +2007; Bernard-Salas et al. 2009; Guzman-Ramirez et al. 2011; +García-Hernández & Górny 2014) with the frequency of carbona- +ceous features higher in the LMC than in Galactic PNe. This is likely +due to the increased efficiency of third dredge-up (TDU) and the +increased C/O ratio at low metallicities (Karakas et al. 2002). Pro- +cessing by an external ambient UV radiation field which is stronger +in the LMC (Gordon et al. 2008) may also affect the circumstellar +chemistry. Detailed examination of PNe at low-metallicity, there- +fore, provides a unique insight into chemical abundances and their +effect on late-stage stellar evolution, dust production, and the for- +mation of PNe in conditions comparable to those during the epoch +of peak star formation in the Universe (Madau et al. 1996). Fur- +thermore, due to their compact nature and brightness over a broad +wavelength range, PNe are also useful calibration sources (e.g., +Swinyard et al. 1996; Feuchtgruber et al. 1997; Perley & Butler +2013; Brown et al. 2014). +SMP LMC 058 was observed by JWST as part of commission- +ing and calibration activities for MIRI. First identified by Sanduleak +et al. (1978), SMP LMC 058 is a carbon-rich planetary nebula (PN) +in the LMC, with a heliocentric radial velocity of 278±7 km s−1 +(Margon et al. 2020). The central star of SMP 058 is a likely C ii +emitter (Margon et al. 2020), consistent with a very early Wolf- +Rayet type star on the carbon sequence (WC). Several dozen very +strong, common emission lines of PNe were also detected in its +optical spectra (Margon et al. 2020). SMP LMC 058 has also been +observed with the Spitzer Infrared Spectrograph (IRS) at both low- +resolution (R∼60–127) and high-resolution (R∼600). The Spitzer +spectra show SMP LMC 058 has unusual dust chemistry with a +strong SiC feature at ∼11.3 𝜇m (Bernard-Salas et al. 2009) and +other associated features, including emission from PAHs at 6–9 +𝜇m, and a shoulder at 18 𝜇m from an unidentified carrier. However, +the Spitzer-IRS data show no clear evidence of fullerenes (Sloan +et al. 2014). SiC is rarely seen in Galactic PNe, in spite of the +higher Si abundance in the Milky Way compared to the Magellanic +Clouds (Jones et al. 2017). Its strength may be due to photoexci- +tation, or because at a high C/O ratio SiC forms on the surface of +carbon grains (Sloan et al. 2014). +In this paper, we describe the observations and calibration of +JWST MIRI MRS commissioning data of SMP LMC 058 (Sec- +tion 2). We then present its MRS spectra in Section 3 and determine +the resolving power of the MRS in Section 4. In Section 5 we +identify and analyse the new emission lines and solid-state features +detected in this carbon-rich planetary nebula and compare this with +Spitzer IRS data. The potential of the MRS and our conclusions are +discussed in Section 6. +2 +OBSERVATIONS AND CALIBRATIONS +The observations were taken as part of the MIRI MRS commis- +sioning program, program ID 1049 (the commissioning purpose of +these observations was PSF characterization). They use the standard +MRS observing template, with 4-point dither patterns optimized for +channels 2, 3, and 4 respectively. Each dither pattern was used twice, +in the ‘positive’ and ‘negative’ direction. Target acquisition was ac- +tivated, with the science target itself serving as an acquisition target. +All three bands (SHORT, MEDIUM, LONG) in all channels were +observed in all dithers. Simultaneous MIRI imaging in filter F770W +was taken in the dither optimized for channel 2. +A dedicated background observation was taken, employing +a 2-point dither optimised for all channels, on a field roughly 3 +arcmin away. The background field was chosen to be relatively +clear of astronomical sources based on archival WISE imaging data +(Wright et al. 2010). +A total of 45 FASTR1 frames were taken per integration. In +target observations, a single integration was taken per dither point. +The background observation had two integrations (to match the total +integration time on source, accounting for the use of only a two- +point dither on the background). The integration time per MRS sub- +band and complete dither were therefore 499.5s or roughly 1,500s +to cover the entire wavelength range (bands SHORT, MEDIUM, +and LONG). Between the six dithers on-target and the single back- +ground, the total integration time was approximately 2.9 hours (6.9 +hr including all overheads). +The MRS observations were processed with version 1.7.3 of +the JWST calibration pipeline and context 0995 of the Calibra- +tion Reference Data System (CRDS). In general, we follow the +standard MRS pipeline procedure (Labiano et al. 2016; Bushouse +et al. 2022; and see Álvarez-Márquez et al. 2022 for an in-flight +example of MRS data calibration). The background subtraction +has been performed using the dedicated background observation. +We have generated twelve 3D spectral cubes, one for each of +the MRS channels and bands, with a spatial and spectral sam- +pling of 0.13" × 0.13" × 0.001 𝜇m, 0.17" × 0.17" × 0.002 𝜇m, +0.20" × 0.20" × 0.003 𝜇m, and 0.35" × 0.35" × 0.006 𝜇m for chan- +nels 1, 2, 3, and 4, respectively. We have performed 1D spectral ex- +tractions individually in each of the MRS cubes using a circular aper- +ture of radius equal to 1.5 × 𝐹𝑊𝐻𝑀(𝜆), where 𝐹𝑊𝐻𝑀(𝜆) = 0.3 +arcsec for 𝜆 < 8𝜇m and 𝐹𝑊𝐻𝑀(𝜆) = 0.31 × 𝜆[𝜇𝑚]/8 arcsec for +𝜆 > 8𝜇m. The selected FWHM (𝜆) values follow the MRS PSF +Full Width at Half Maximum (FWHM). NIRCam observation (see +Figure 1), and MRS observations suggest that SMP LMC 058 is an +unresolved source. We use the MRS PSF models (Patapis et al. in +prep.) to correct the aperture losses in the 1D spectra. The percent- +age of flux that is lost out of the selected aperture is 17% for channel +1 and increases to 30% in channel 4. +MNRAS 000, 1–11 (2022) + +JWST MRS observations of SMP LMC 058 +3 +5h24m21.5s +21.0s +20.5s +20.0s +70 04′57′′ +05′00′′ +03′′ +06′′ +RA (ICRS) +Dec (ICRS) +F356W +0.5 pc +Figure 1. NIRCam F356W image of SMP LMC 058 shown in an Asinh +stretch. At this spatial resolution (0.063′′) SMP LMC 058 is an unresolved +point source. +The 12 spectral segments extracted from these cubes were +corrected for residual fringing using a post-pipeline spectral-level +correction which is a modified version of the detector-level correc- +tion available in the JWST calibration pipeline. The residual fringe +contrasts are reduced by employing an empirical multi-component +sine fitting method (e.g. Kester et al. 2003), under the assumption +that the pipeline fringe flat correction has reduced fringe contrasts +to the point where this multi-component sine approximation is valid +(Kavanagh et al., in prep.). +Finally, each of the 12 individual spectral segments was +stitched together to remove minor flux discontinuities. This was +done by determining a scaling factor between the median flux (ex- +cluding spectral lines) in the overlapping MRS segments; then ap- +plying this multiplicative factor to the longer wavelength segments, +in turn, to effectively shift the spectrum to match the flux of its +neighbouring shorter wavelength segment. This factor was typi- +cally on the order of <5 per cent. The flux data in the overlapping +spectral regions were then averaged. The final stitched spectrum was +inspected to ensure there were no remaining discontinuities which +may affect the continuum and model fitting. +3 +SMP LMC 058 SPECTRUM +Figure 2 shows the extracted spectrum of SMP LMC 058 which +exhibits a rich variety of atomic, molecular and solid state features, +including PAHs and silicon carbide, characteristic of carbon-rich +material, and a strong continuum which rises towards the longest +wavelengths. Due to the superior sensitivity and spectral resolution +(see Section 4) of the MRS, the MIRI spectrum of SMP LMC +058 shows features that are not seen in the Spitzer IRS data (see +Section 5), notably in the number of emission lines detected. +In the spectrum presented here, there is a large amount of fine- +structure line emission present, from the strong nebular forbidden +lines of [Ar ii], [Ar iii], [S iv], [Ne ii], [Ne iii], [S iii] to weak +H recombination lines (Hi) from the Pfund and Humphreys series, +and beyond. To ensure we measure and identify all the emission +lines in the spectra we fit a pseudo-continuum to the broadband +spectral features using a piece-wise spline model. Obvious narrow +band features were identified and masked in the fitting based on +their amplitudes exceeding a threshold value. We used an outlier +rejection fitter to flag and ignore any weaker narrow-band features +that may compromise the continuum fit. After visual inspection of +the fit, it was subtracted to isolate any narrow-band features present. +Figure 3 shows the spectrum of SMP LMC 058 after subtraction +of the pseudocontinuum from the total spectrum. The spectrum is +extremely rich in emission lines. In total 51 lines were detected. +Using the 12 original MRS segments, we identified and ana- +lyzed all detected emission lines with a signal-to-noise ratio (SNR) +greater than 3 in the SMP LMC 058 spectra. Depending on the +line profiles (see Figure 4), we performed one-component and two- +component Gaussian fits, plus a second-order polynomial to simul- +taneously fit the continuum and emission line.1 The uncertainties +on the derived emission line parameters, like the line FWHM, flux, +central wavelength, etc, were estimated using a Monte Carlo simu- +lation (following the same methodology as Álvarez-Márquez et al. +2021, 2022). Systemic velocity shifts were removed using a he- +liocentric radial velocity of 278 ± 7 km/s (Reid & Parker 2006; +Margon et al. 2020). Table 1 presents the measured wavelengths +and fluxes together with the identification of the mid-IR emission +lines in SMP LMC 058 spectra. Small wavelength offsets are con- +sistent with known errors in the MRS FLT-4 wavelength solution +(see discussion by Argyriou et al. (in prep.)) and should be reduced +further by ongoing calibration efforts later in Cycle 1. Weak lines are +more prevalent at shorter wavelengths in channels 1 and 2 where the +MRS sensitivity is higher and the uncertainties in the flux are better +constrained. Furthermore, we find that the current MRS wavelength +calibration is <40 km/s for all spectral sub-bands, better than the +FWHM of the MRS line spread function (75-200 km/s, Labiano +et al. 2021, Argyriou et al. (in prep.)). +4 +MRS RESOLVING POWER +The resolving power (R) is defined as 𝜆/Δ𝜆, where Δ𝜆 is the mini- +mum distance to distinguish two features in a spectrum. We define +the Δ𝜆 as the FWHM of an unresolved emission line. SMP LMC +058 is the only source observed in the JWST commissioning datasets +that is considered both spatially and spectrally unresolved with the +MRS, this makes it an excellent target for determining the inflight +MRS resolving power. Here, we assume the intrinsic width of the +emission lines in SMP LMC 058 to be negligible, as we do not have +high-resolution spectroscopy to characterize its intrinsic velocity +dispersion. Nearby planetary nebula eject gas with typical velocity +dispersions of about 10–51 kms−1 (Reid & Parker 2006). If this is +the case for SMP LMC 058, then assuming a velocity of 25 kms−1 +we might underestimate the MRS resolving power by up to 5% for +channel 1, and up to 1% for channel 4 (see e.g., Law et al. 2021). +The pre-launch MRS resolving power has been established +from MIRI ground-based test and calibration campaigns, using a +set of etalons which provided lines in all MRS bands. It was de- +termined to be in the range of about 4000 to 1500 (Labiano et al. +2021). Figure 5 shows the comparison between the ground-based +MRS resolving power estimates and the inflight estimates derived +from the SMP LMC 058 spectra. The inflight MRS resolving power +has been determined using only emission lines with SNR higher +than 6, and following the FWHM results obtained in the one- and +1 We used the mpfit (Markwardt 2009) Python routine to perform the fits, +the code is publicly available here. +MNRAS 000, 1–11 (2022) + +4 +O. C. Jones et al. +5 +10 +15 +20 +25 +Wavelength (microns) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Flux (Jy) +MIRI MRS +Figure 2. The MIRI MRS spectrum of SMP LMC 058. Numerous emission lines, PAH features and dust features are clearly seen on a rising continuum. These +features are much better resolved in the MRS spectra due to the higher spectral resolution. +two-component Gaussian fits (see Section 3). In the case of Hi +emission lines, we used the FWHM of the narrow gaussian compo- +nent. The errors in the resolving power are, on average, larger for +the Hi emission lines due to the uncertainty in the two-component +Gaussian fit. Given the uncertainties, the inflight MRS resolving +power agrees with the ground-based estimates, and it presents a +trend followed by the equation 4401 − 112 × 𝜆[𝜇𝑚] + 10−9×𝜆[𝜇𝑚]. +The ground-based estimates consider the width of the etalon +emission lines to be negligible, which could imply an underesti- +mation of the MRS resolving power by a factor of 10%. A similar +situation is potentially happening with the inflight estimations due +to the lack of the intrinsic velocity dispersion of SMP LMC 058. We +conclude that the estimations of the ground-based MRS resolving +power (Labiano et al. 2021) are valid, within a 10% of uncertainty, +for the MRS inflight performance. As JWST observes more sources +with spatially and spectrally unresolved spectral lines, their char- +acterisation will provide a more comprehensive understanding of +the inflight variations of the resolving power within each of the 12 +spectral bands. As of now, the continuous "trend" curve in Figure 5 +presents the state of knowledge of the MRS resolving power. +5 +DISCUSSION +5.1 +Emission Lines +Short-High and Long-High Spitzer spectroscopic data of SMP LMC +058 were published in Bernard-Salas et al. (2008). A comparison of +the MRS line fluxes with those found by Bernard-Salas et al. (2008) +is given in Table 2. In general, there is good agreement between our +measurements of the forbidden emission line strengths of [S iv], +[Ne ii], [Ne iii], [S iii]. Furthermore, the high-excitation lines of +[Ar v] at 13.10 𝜇m and [O iv] with ionisation potentials of 60 and +55 eV, respectively, are not detected in either spectrum. These lines +are excited by high-temperature stars with Teff between 140,000 +– 180,000 K. The highest ionisation species in the MRS spectra +are [K iv] (46 eV) and [Na iii] (47 eV), these lines have not been +previously detected by Spitzer. Thus, we consider SMP LMC 058 +to be a low-excitation source. +Given the superior sensitivity of JWST (Rigby et al. 2022) and +the MRS, we detect a line at 14.38 𝜇m an order of magnitude below +the upper-limit of [Ne v] reported by Bernard-Salas et al. (2008). +The ionisation potential of [Ne v] is 97 eV, thus it is unlikely given +the absence of other high-excitation lines in the MRS spectra of SMP +LMC 058, that this emission is from [Ne v], instead, we attribute +this line to [Cl ii] which has an ionisation potential of 13 eV. +As seen in Table 1 higher ionisation potential species expand +at a lower velocity than the lower ionisation potential species (e.g., +Reid & Parker 2006). This is due to ionisation occurring at a greater +distance from the centre of the PN where velocities are greater, and +can cause lower excitation species to expand to larger radii in the +PN. +Hydrogen recombination lines are abundant in the spectrum +of SMP LMC 058, all are new detections. Hi emission lines more +closely trace the ionized regions, compared to molecular hydrogen. +As shown in Figure 4, the line profiles of the bright Hi emission lines +are asymmetric, exhibiting a blue tail, whereas the forbidden emis- +sion lines present symmetric unresolved profiles. Hi emission lines +are composed of a spectrally unresolved main component contain- +MNRAS 000, 1–11 (2022) + +JWST MRS observations of SMP LMC 058 +5 +5 +6 +7 +8 +9 +Wavelength (microns) +0.002 +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +Flux (Jy) +H Hu +H Hu +[ArII] +H Pf +H Hu +H10-7 +[ArIII] +[NaIII] +H16-7 +H15-7 +H13-7 +H12-7 +H15-8 +H13-8 +10 +12 +14 +16 +18 +20 +22 +Wavelength (microns) +0.002 +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +Flux (Jy) +[SIV] +H9-7 +H Hu +[NeII] +[NeV] +[NeIII] +H10-8 +[SIII] +H8-7 +Figure 3. Continuum-subtracted MRS spectrum of SMP LMC 058 (where the continuum includes dust and PAH features), highlighting the atomic emission +lines. The identification of key species are marked on the spectrum. The top panel shows lines in channels 1 and 2 of the MRS, the lower panels show channels +3 and 4. The flux axis is truncated to highlight lower contrast lines. +MNRAS 000, 1–11 (2022) + +6 +O. C. Jones et al. +Table 1. Measured central wavelengths, line flux, line widths, and line identification for SMP LMC 058. The systemic velocity was removed prior to calculating +the velocity shift of a line. If a line is present in multiple MRS bands, measurements are provided for each individual MRS segment. Uncertain line identifications +are denoted by a ‘?’. +Band +Line +𝜆lab +𝜆observed +𝜎𝜆observed +𝜆offset +FWHM +Flux (×10−15) +𝜎 (×10−15) +Identification +𝜇m +𝜇m +𝜇m +km s−1 +nm +erg s−1 cm−2 +erg s−1 cm−2 +1S +Hi 23−7 +4.924 +4.92903 +0.00004 +-46.025 +50.005 +0.11 +0.02 +1S +Hi 22−7 +4.971 +4.97588 +0.00004 +-23.351 +49.997 +0.09 +0.01 +1S +Hi 21−7 +5.026 +5.03086 +0.00007 +-6.428 +63.362 +0.07 +0.01 +1S +Hi 20−7 +5.091 +5.09600 +0.00003 +2.418 +50.000 +0.13 +0.01 +1S +Hi 10−6 +5.129 +5.13342 +0.00001 +-0.165 +94.556 +1.67 +0.02 +1S +Hi 19−7 +5.169 +5.17401 +0.00002 +4.175 +51.268 +0.16 +0.02 +1S +Hi 18−7 +5.264 +5.26856 +0.00006 +0.558 +77.922 +0.16 +0.01 +1S +[Fe ii] +5.340 +5.34504 +0.00011 +5.043 +92.348 +0.06 +0.01 +1S +Hi 17−7 +5.380 +5.38499 +0.00002 +-12.237 +77.919 +0.18 +0.01 +1S +Hi 16−7 +5.525 +5.53072 +0.00006 +-21.881 +92.047 +0.27 +0.02 +1S +Hi 15−7 +5.711 +5.71692 +0.00002 +-8.044 +50.000 +0.28 +0.02 +1M +Hi 15−7 +5.711 +5.71726 +0.00005 +-26.023 +89.264 +0.28 +0.02 +1M +Hi 9−6 +5.908 +5.91340 +0.00001 +15.046 +99.683 +2.31 +0.03 +1M +Hi 14−7 +5.957 +5.96199 +0.00002 +19.244 +73.871 +0.33 +0.02 +1M +[K iv] +5.982 +5.98750 +0.0001 +2.606 +110.497 +0.12 +0.01 +1M +Hi 13−7 +6.292 +6.29807 +0.00005 +-14.725 +74.449 +0.43 +0.04 +1L +Hi 12−7 +6.772 +6.77852 +0.00002 +-10.975 +85.586 +0.58 +0.02 +1L +Hi 21−8 +6.826 +6.83241 +0.00013 +-8.356 +77.331 +0.06 +0.02 +1L +H2(0,0) S(5) +6.910 +6.91588 +0.00013 +2.424 +95.459 +0.13 +0.02 +1L +Hi 20−8 +6.947 +6.95306 +0.00009 +6.515 +75.574 +0.08 +0.02 +1L +[Ar ii] +6.985 +6.99181 +0.00001 +-2.238 +89.682 +1.17 +0.01 +1L +Hi 19−8 +7.093 +7.09935 +0.00017 +-2.424 +76.633 +0.11 +0.01 +1L +Hi 18−8 +7.272 +7.27877 +0.00011 +-14.968 +69.767 +0.10 +0.02 +1L +[Na iii] +7.318 +7.32485 +0.00006 +-14.817 +136.625 +0.46 +0.02 +1L +Hi 6−5 +7.460 +7.46690 +0 +-4.607 +79.653 +11.51 +0.04 +1L +Hi 8−6 +7.502 +7.50949 +0.00001 +-1.396 +79.577 +12.33 +0.07 +1L +Hi 11−7 +7.508 +7.51515 +0.00003 +-2.922 +74.553 +0.69 +0.02 +2S +Hi 15−8 +8.155 +8.16375 +0.00053 +-47.407 +49.995 +0.15 +0.05 +2S +Hi 14−8 +8.665 +8.67220 +0.00019 +11.971 +53.004 +0.28 +0.03 +2M +Hi 10−7 +8.760 +8.76815 +0.00006 +1.510 +110.130 +1.04 +0.05 +2M +[Ar iii] +8.991 +8.99859 +0 +37.731 +101.834 +20.61 +0.07 +2M +Hi 13−8 +9.392 +9.40091 +0.00012 +-5.539 +84.050 +0.25 +0.04 +2M +H2(0,0) S(3) +9.665 +9.67502 +0.00023 +-35.450 +151.677 +0.23 +0.04 +2M +Hi 18−9 +9.847 +9.85887 +0.00037 +-82.099 +75.445 +0.06 +0.01 +2L +[S iv] +10.511 +10.52014 +0.00001 +3.428 +96.022 +30.07 +0.12 +2L +Hi 16−9 +10.804 +10.81253 +0.00068 +30.378 +140.073 +0.13 +0.03 +2L +Hi 9−7 +11.309 +11.31929 +0.00025 +-2.583 +82.684 +1.41 +0.17 +3S +Hi 7−6 +12.372 +12.38265 +0.00003 +17.586 +94.410 +2.90 +0.03 +3S +Hi 11−8 +12.387 +12.39758 +0.00005 +26.303 +107.676 +0.68 +0.02 +3S +Hi 14−9 +12.587 +12.59863 +0.00023 +3.125 +93.158 +0.15 +0.03 +3S +[Ne ii] +12.814 +12.82631 +0 +-20.232 +98.379 +17.63 +0.02 +3M +Hi 13−9 +14.183 +14.19616 +0.00035 +1.955 +69.993 +0.13 +0.05 +3M +[Cl ii]? +14.368 +14.38034 +0.00019 +16.544 +82.249 +0.34 +0.02 +3M +Hi 16−10 +14.962 +14.97556 +0.00064 +11.773 +50.002 +0.04 +0.02 +3L +[Ne iii] +15.555 +15.56957 +0.00001 +-0.617 +130.542 +179.26 +0.55 +3L +Hi 10−8 +16.209 +16.22334 +0.00014 +14.869 +97.309 +0.60 +0.08 +3L +Hi 12−9 +16.881 +16.89664 +0.0002 +-6.106 +91.592 +0.29 +0.04 +4S +[S iii] +18.713 +18.72914 +0.00002 +19.709 +136.931 +11.34 +0.07 +4S +Hi 8−7 +19.062 +19.07898 +0.00005 +9.654 +148.238 +2.40 +0.05 +4M +[Ar iii] +21.830 +21.85033 +0.00032 +1.852 +155.363 +0.82 +0.06 +4M +Hi 13−10+Hi 11−9 +22.340 +22.35612 +0.00055 +68.040 +194.363 +0.64 +0.05 +ing the majority of the line flux (>95%)), and a spectrally resolved +blue-shifted component possibly due to thermal broadening (Chu +et al. 1984) or from condensation outside the main core which may +be evident as a marginally resolved envelope like structure in the +MRS cube at 7.466𝜇m. +Two molecular hydrogen lines (H2) have been detected in the +MRS data, the ortho-H2 𝑣 = 0–0 S(3) and S(5) lines. The S(1) line at +17.055 𝜇m may also be present, although this is not easily discerned +above the continuum and we do not measure its flux. The S(3) and +S(5) rotational line emission probably originate from irradiated, +and perhaps also shocked, dense molecular clumps, torus structures +(e.g., Kastner et al. 1996; Hora et al. 1999; Akras et al. 2017; +MNRAS 000, 1–11 (2022) + +JWST MRS observations of SMP LMC 058 +7 +7.455 +7.460 +7.465 +7.470 +7.475 +Wavelength (microns) +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +Flux (Jy) +Pf +15.54 +15.56 +15.58 +15.60 +Wavelength (microns) +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Flux (Jy) +[Ne III] +Figure 4. Top: The Pf 𝛼 H i emission line profile shows a spatially unre- +solved main component and a weaker spectrally resolved blue-shifted wing. +Bottom: The [Ne iii] line profile is spectrally unresolved and symmetric. +This shape is typical of all the forbidden emission lines in SMP LMC 058. +The dashed line marks the lines observed central wavelength. +Table 2. Comparison of SMP LMC 058 MRS line fluxes with those of +Bernard-Salas et al. (2008) taken with the high-resolution modules on the +Spitzer IRS. All line strengths reported by Bernard-Salas et al. (2008) have +a 10% error except for [S iv] which has a 10–20% error. Errors in the MRS +flux are <1% and are provided for each line in Table 1. +Line +Wavelength +MRS Flux +Spitzer Flux +Ionisation +(Rest) +×10−15 +×10−15 +potential +𝜇m +erg s−1 cm−2 +erg s−1 cm−2 +(eV) +[S iv] +10.511 +30.07 +29.2 +35 +[Ne ii] +12.814 +17.63 +20.6 +22 +[Ar v] +13.099 +<0.029 +<2.4 +60 +[Ne v] +14.323 +<0.005 +<3.8 +97 +[Ne iii] +15.555 +179.26 +200.6 +41 +[S iii] +18.713 +11.34 +11.0 +23 +[O iv] +25.883 +<0.23 +<21.6 +55 +Fang et al. 2018), or from the outer regions of the PNe where the +temperature is about 1000K (Aleman & Gruenwald 2004; Matsuura +et al. 2007b). +5.2 +Dust and PAH Features +The dust in SMP LMC 058 is carbon-rich. Amongst the most promi- +nent features is the strong silicon carbide (SiC) emission at 11 𝜇m +and the rising continuum due to the thermal emission of warm dust. +Strong emission features from PAHs also appear in the spectrum at +5.2, 5.7, 6.2, 7.7, 8.6, 11.2 and 12.7 𝜇m. +At sub-solar metallicities (∼ 0.2 − 0.5 Z⊙), SiC is commonly +observed in PNe, yet it is rarely seen in Galactic PNe or indeed +during the earlier AGB evolutionary phase of metal-poor carbon +stars (Casassus et al. 2001; Zijlstra et al. 2006; Matsuura et al. +2007a; Stanghellini et al. 2007; Bernard-Salas et al. 2008; Woods +et al. 2011, 2012; Sloan et al. 2014; Ruffle et al. 2015; Jones et al. +2017). The strength of the SiC flux in metal-poor PNe is highly +sensitive to the radiation field (Bernard-Salas et al. 2009). This is +likely due to a lower abundance of Si affecting the carbonaceous dust +condensation sequence on the AGB. In this case, rather than SiC +forming first, it instead forms in a mantle surrounding an amorphous +carbon core (Lagadec et al. 2007; Leisenring et al. 2008). Then as +the PNe dust becomes heated and photo-processed, the amorphous +carbon evaporates increasing the SiC surface area and consequently +its feature strength, until a critical ionisation potential of >55 eV +occurs at which point the SiC features disappear (Bernard-Salas +et al. 2009; Sloan et al. 2014). +Following Bernard-Salas et al. (2009), we measure the strength +of the SiC feature by integrating the flux above a continuum- +subtracted spectrum from 9 to 13.2 𝜇m and then subtracting the flux +contributions from the 11.2 𝜇m PAH feature and the [Ne ii] line. +Due to the resolution of the MRS compared to the Spitzer spectra of +SMP LMC 058, we detect several additional lines which contribute +to the integrated flux in the SiC region; these lines include [S iv] +and H i. Thus to obtain a reliable measurement of the SiC feature +strength we also subtract the flux contribution from all emission +lines in the 9 – 13.2 𝜇m region listed in Table 1. A PAH feature at +∼12.6 𝜇m likely contributes a small amount of flux to the measured +SiC feature, however isolating and subtracting this emission contri- +bution from the wing of the SiC feature is challenging even with +the MRS spectral resolution. Additionally, an artefact at ∼12.2 𝜇m +due to a spectral leak (e.g., Gasman et al. 2022) may also affect the +integrated flux. Table 3 gives the measured SiC centroid and cor- +rected feature strength. The latter agrees exceptionally well with the +value of 29.72 ± 0.31 ×10−16 W m−2 measured by Bernard-Salas +et al. (2009) in the Spitzer data of SMP LMC 058. This suggests +there is little to no evolution in the SiC dust on the 16-year time +scales between the observations. Furthermore, the agreement be- +tween the measurements verifies the overall flux calibration of the +MRS instrument (Gasman et al. 2022). +In astronomical sources, the structure, wavelengths and relative +strength of the PAHs can differ strongly between objects, with PNe +showing the most pronounced variations in PAH profiles due to +photoprocessing altering the ratio of aliphatics to aromatics (Peeters +et al. 2002; Pino et al. 2008; Matsuura et al. 2014; Sloan et al. 2014; +Jensen et al. 2022). Figure 6 shows the PAHs in SMP LMC 058. The +PAHs in SMP LMC 058 are considered to have a class B profile by +Bernard-Salas et al. (2009) and Sloan et al. (2014). In this schema +devised by Peeters et al. (2002) and van Diedenhoven et al. (2004) +the 6.2 PAH feature for class B objects has a peak between 6.24 +and 6.28 𝜇m; the dominant 7.7 PAH feature peaks between 7.8 to +MNRAS 000, 1–11 (2022) + +8 +O. C. Jones et al. +4.5 +5 +6 +7 +8 +9 +10 +12 +15 +20 +25 +30 +Wavelength [ m] +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +5000 +MRS Resolving Power +Ground (Labiano+21) +Fit to inflight lines +inflight forbidden lines +inflight HI lines +Figure 5. Comparison between the ground-based and inflight MRS resolving power. Gray filled area and black line: ground-based MRS resolving power +estimates (Labiano et al. 2021). Filled red circles: inflight MRS resolving power calculation using forbidden emission lines identified in this paper. Open red +circles: inflight MRS resolving power calculation using Hi emission lines. +6 +8 +10 +12 +14 +Wavelength (microns) +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +Flux (Jy) +PAH 5.2 m +PAH 5.7 m +PAH 6.2 m +PAH 7.7 m +PAH 8.6 m +PAH 11.2 m +SiC +Local continuum +Figure 6. The SiC and PAH features are highlighted in the spectra of SMP LMC 058. A local continuum fit to the 11.3 𝜇m feature which is superimposed +on the broad SiC emission feature is also shown. The colours highlight the spectral region for each feature, to which a local continuum was fit and the flux +measured over. +MNRAS 000, 1–11 (2022) + +JWST MRS observations of SMP LMC 058 +9 +Table 3. PAH and SiC Fluxes and Centroids. +Centroid +Integrated Flux +Integrated Flux Error +𝜇m +W m−2 +W m−2 +PAH +5.262 +1.64×10−18 +7.0×10−20 +PAH +5.698 +5.62×10−18 +1.1×10−19 +PAH +6.274 +9.259×10−17 +2.9×10−19 +PAH +7.834 +3.427×10−16 +1.2×10−18 +PAH +8.665 +6.231×10−17 +8.5×10−19 +PAH +11.298 +1.047×10−16 +2.0×10−18 +SiC +11.097 +3.067×10−15 +4.5×10−18 +8.0 𝜇m; and the 8.6 PAH band is red-shifted. These values agree +well with our measured centroids listed in Table 3. Furthermore, the +PAHs observed in SMP LMC 058 closely resemble those observed +in the ISO SWS spectrum of the Galactic post-AGB star, HD 44179 +(the Red Rectangle) which also shows strong aromatic features on +top of a continuum (Waters et al. 1998). +The relative strength of the PAH features depends on a number +of factors including the degree of ionisation of the radiation field +(e.g., Allamandola et al. 1999). The strength of the PAH features +in SMP LMC 058 was determined by integrating the flux of the +feature above an adopted local continuum, fit to each side of the +feature and measured using specutils line_flux. Particular care +was taken in fitting a continuum, too, and then measuring the 11.25 +𝜇m band (produced by the out-of-plane solo C–H bending mode) +as this is superimposed on top of the broad SiC feature. Table 3 +presents the central wavelength of the features and the integrated +flux. The ratio of the PAH strengths correlates with the source type +and hence its physical conditions (Hony et al. 2001); ionized PAHs +have strong features at 6.2, 7.7 and 8.6 𝜇m whilst the 11.2 𝜇m PAH +feature is stronger for neutral PAHs. From the PAH line strengths +given in Table 3 it is evident that the 7.7𝜇m feature dominates the +total PAH emission, and thus the dust around SMP LMC 058 is +likely experiencing a high degree of ionisation. +Carbon-rich PNe can show a rich variety of solid-state material +in their spectra in addition to PAHs. The C60 fullerene molecule +typically exhibits features at ∼7.0, 8.5, 17.4 and 18.9 𝜇m, and all +four were first identified in the spectrum of the Galactic PN TC-1 +(Cami et al. 2010). Fullerenes have since been detected in several +other PNe (e.g., García-Hernández et al. 2010, 2011; Sloan et al. +2014). The still-unidentified 21 𝜇m emission feature, first detected +by Kwok et al. 1989, can also appear in carbon-rich PNe, often +associated with unusual PAH emission and aliphatic hydrocarbons +(Cerrigone et al. 2011; Matsuura et al. 2014; Sloan et al. 2014; Volk +et al. 2020). +The spectra of SMP LMC 058 from the IRS on Spitzer did +not show any of these unusual hydrocarbon-related features, but the +improved spectral resolution of the MRS allows for a much more +careful examination. Nonetheless, these additional features remain +too weak to be detected. SMP LMC 058 presents a classic Class +B PAH spectrum, as expected for objects which have evolved to +the young PN stage (Sloan et al. 2014). Younger objects which +could still be described as post-AGB objects would show the 21 𝜇m +feature and/or aliphatics. Sloan et al. (2014) identified SMP LMC +058 as a member of the Big-11 group because of the combination +of a strong SiC emission feature and the 11.2 𝜇m PAH feature and +the absence of fullerenes. This group is actually related to the PNe +that show fullerenes, and the presence or absence of fullerenes may +be due to something as simple as which have a clear line of sight to +the interior of the dust shells where the fullerenes are expected to +be present. +6 +SUMMARY AND CONCLUSIONS +We have presented MIRI/MRS spectra of the carbon-rich planetary +nebula SMP LMC 058 located in the metal-poor Large Magellanic +Cloud. SMP LMC 058 is a point source in the MRS data and +its spectrum contains the only spatially and spectrally unresolved +emission lines observed during the commissioning of the JWST +Medium-Resolution Spectrometer. In the MRS spectrum, we de- +tected 51 emission lines, of which 47 were previously undetected +in this source. The strongest emission lines were used to determine +the spectral resolutions of the MIRI MRS instrument. The resolving +power is R > 3960 in channel 1, R > 3530 in channel 2, R > 3200 +in channel 3, and R > 1920 in channel 4. This on-sky performance +is comparable to the resolution determined from the ground cali- +bration of the MRS which provides resolving powers from 4000 at +channel 1 to 1500 at channel 4. Furthermore, a comparison of the +line strengths and spectral continuum to previous observations of +SMP LMC 058 with the IRS on the Spitzer was used to verify the +absolute flux calibration of the MRS instrument. The MRS spectra +confirm that the carbon-rich dust emission is from grains and not +isolated molecules and that there is little to no time evolution of the +SiC dust and emission line strengths in the 16 years between the +observations. The PAH emission is dominated by the 7.7𝜇m feature. +The strong PAHs and SiC in the spectra are consistent with the lack +of high-excitation lines detected in the spectra, which if present, +would indicate a hard radiation field that would likely destroy these +grains. These commissioning data reveal the great potential and +resolving power of the MIRI MRS to study line, molecular and +solid-state features in individual sources in nearby galaxies. +ACKNOWLEDGEMENTS +We thank Kay Justtanont for her insights, comments and dis- +cussions. This work is based on observations made with the +NASA/ESA/CSA James Webb Space Telescope. The data were ob- +tained from the Mikulski Archive for Space Telescopes at the Space +Telescope Science Institute, which is operated by the Association of +Universities for Research in Astronomy, Inc., under NASA contract +NAS 5-03127 for JWST. These observations are associated with +program #1049. This work is based in part on observations made +with the Spitzer Space Telescope, which was operated by the Jet +Propulsion Laboratory, California Institute of Technology under a +contract with NASA +O.C.J acknowledge support from an STFC Webb fellowship. +J.A.M. and A.L acknowledge support by grant PIB2021-127718NB- +100 by the Spanish Ministry of Science and Innovation/State Agency +of Research (MCIN/AEI). P.J.K acknowledges financial support +from the Science Foundation Ireland/Irish Research Council Path- +way programme under Grant Number 21/PATH-S/9360. I.A., D.G., +and B.V. thank the European Space Agency (ESA) and the Belgian +Federal Science Policy Office (BELSPO) for their support in the +framework of the PRODEX Programme. PG would like to thank +the University Pierre and Marie Curie, the Institut Universitaire de +France, the Centre National d’Etudes Spatiales (CNES), the "Pro- +gramme National de Cosmologie and Galaxies" (PNCG) and the +"Physique Chimie du Milieu Interstellaire" (PCMI) programs of +MNRAS 000, 1–11 (2022) + +10 +O. C. Jones et al. +CNRS/INSU, with INC/INP co-funded by CEA and CNES, for +there financial supports. +MIRI draws on the scientific and technical expertise of the +following organisations: Ames Research Center, USA; Airbus De- +fence and Space, UK; CEA-Irfu, Saclay, France; Centre Spatial +de Liége, Belgium; Consejo Superior de Investigaciones Científi- +cas, Spain; Carl Zeiss Optronics, Germany; Chalmers University of +Technology, Sweden; Danish Space Research Institute, Denmark; +Dublin Institute for Advanced Studies, Ireland; European Space +Agency, Netherlands; ETCA, Belgium; ETH Zurich, Switzerland; +Goddard Space Flight Center, USA; Institute d’Astrophysique Spa- +tiale, France; Instituto Nacional de Técnica Aeroespacial, Spain; In- +stitute for Astronomy, Edinburgh, UK; Jet Propulsion Laboratory, +USA; Laboratoire d’Astrophysique de Marseille (LAM), France; +Leiden University, Netherlands; Lockheed Advanced Technology +Center (USA); NOVA Opt-IR group at Dwingeloo, Netherlands; +Northrop Grumman, USA; Max-Planck Institut für Astronomie +(MPIA), Heidelberg, Germany; Laboratoire d’Etudes Spatiales et +d’Instrumentation en Astrophysique (LESIA), France; Paul Scher- +rer Institut, Switzerland; Raytheon Vision Systems, USA; RUAG +Aerospace, Switzerland; Rutherford Appleton Laboratory (RAL +Space), UK; Space Telescope Science Institute, USA; Toegepast- +Natuurwetenschappelijk Onderzoek (TNO-TPD), Netherlands; UK +Astronomy Technology Centre, UK; University College London, +UK; University of Amsterdam, Netherlands; University of Arizona, +USA; University of Bern, Switzerland; University of Cardiff, UK; +University of Cologne, Germany; University of Ghent; University +of Groningen, Netherlands; University of Leicester, UK; University +of Leuven, Belgium; University of Stockholm, Sweden; Utah State +University, USA. A portion of this work was carried out at the Jet +Propulsion Laboratory, California Institute of Technology, under a +contract with the National Aeronautics and Space Administration. +The following National and International Funding Agencies +funded and supported the MIRI development: NASA; ESA; Bel- +gian Science Policy Office (BELSPO); Centre Nationale d’Etudes +Spatiales (CNES); Danish National Space Centre; Deutsches Zen- +trum fur Luftund Raumfahrt (DLR); Enterprise Ireland; Ministerio +De Economia y Competividad; Netherlands Research School for As- +tronomy (NOVA); Netherlands Organisation for Scientific Research +(NWO); Science and Technology Facilities Council; Swiss Space +Office; Swedish National Space Agency; and UK Space Agency. +Facilities: JWST (MIRI/MRS) - James Webb Space Telescope. +DATA AVAILABILITY +JWST data were obtained from the Mikulski Archive for +Space Telescopes at the Space Telescope Science Institute +(https://archive.stsci.edu/). +REFERENCES +Akras S., Gonçalves D. R., Ramos-Larios G., 2017, MNRAS, 465, 1289 +Aleman I., Gruenwald R., 2004, ApJ, 607, 865 +Allamandola L. J., Hudgins D. M., Sandford S. A., 1999, ApJ, 511, L115 +Álvarez-Márquez J., Marques-Chaves R., Colina L., Pérez-Fournon I., 2021, +A&A, 647, A133 +Álvarez-Márquez J., et al., 2022, arXiv e-prints, p. arXiv:2209.01695 +Bernard-Salas J., Pottasch S. R., Gutenkunst S., Morris P. W., Houck J. R., +2008, ApJ, 672, 274 +Bernard-Salas J., Peeters E., Sloan G. C., Gutenkunst S., Matsuura M., +Tielens A. G. G. M., Zijlstra A. A., Houck J. R., 2009, ApJ, 699, 1541 +Brown M. J. I., Jarrett T. H., Cluver M. E., 2014, Publ. Astron. Soc. Australia, +31, e049 +Bushouse +H., +et +al., +2022, +spacetelescope/jwst: +JWST +1.6.2, +doi:10.5281/zenodo.6984366, +https://doi.org/10.5281/ +zenodo.6984366 +Cami J., Bernard-Salas J., Peeters E., Malek S. E., 2010, Science, 329, 1180 +Casassus S., Roche P. F., Aitken D. K., Smith C. H., 2001, MNRAS, 327, +744 +Cerrigone L., Hora J. L., Umana G., Trigilio C., Hart A., Fazio G., 2011, +ApJ, 738, 121 +Choudhury S., Subramaniam A., Cole A. A., 2016, MNRAS, 455, 1855 +Chu Y. H., Kwitter K. B., Kaler J. B., Jacoby G. H., 1984, PASP, 96, 598 +Fang X., Zhang Y., Kwok S., Hsia C.-H., Chau W., Ramos-Larios G., Guer- +rero M. A., 2018, ApJ, 859, 92 +Feuchtgruber H., et al., 1997, ApJ, 487, 962 +García-Hernández D. A., Górny S. K., 2014, A&A, 567, A12 +García-Hernández D. A., Manchado A., García-Lario P., Stanghellini L., +Villaver E., Shaw R. A., Szczerba R., Perea-Calderón J. V., 2010, ApJ, +724, L39 +García-Hernández D. A., et al., 2011, ApJ, 737, L30 +Gasman D., et al., 2022, arXiv e-prints, p. arXiv:2212.03596 +Gonçalves D. R., Magrini L., Teodorescu A. M., Carneiro C. M., 2014, +MNRAS, 444, 1705 +Gordon K. D., Engelbracht C. W., Rieke G. H., Misselt K. A., Smith J. D. T., +Kennicutt Robert C. J., 2008, ApJ, 682, 336 +Guzman-Ramirez L., Zijlstra A. A., Níchuimín R., Gesicki K., Lagadec E., +Millar T. J., Woods P. M., 2011, MNRAS, 414, 1667 +Hony S., Van Kerckhoven C., Peeters E., Tielens A. G. G. M., Hudgins +D. M., Allamandola L. J., 2001, A&A, 370, 1030 +Hora J. L., Latter W. B., Deutsch L. K., 1999, ApJS, 124, 195 +Jensen P. A., Shannon M. J., Peeters E., Sloan G. C., Stock D. J., 2022, +A&A, 665, A153 +Jones O. C., et al., 2017, MNRAS, 470, 3250 +Karakas A., Lattanzio J. C., 2007, Publ. Astron. Soc. Australia, 24, 103 +Karakas A. I., Lattanzio J. C., 2014, Publ. Astron. Soc. Australia, 31, e030 +Karakas A. I., Lattanzio J. C., Pols O. R., 2002, Publ. Astron. Soc. Australia, +19, 515 +Kastner J. H., Weintraub D. A., Gatley I., Merrill K. M., Probst R. G., 1996, +ApJ, 462, 777 +Kendrew S., et al., 2015, Publications of the Astronomical Society of the +Pacific, 127, 623 +Kester D. J. M., Beintema D. A., Lutz D., 2003, in Metcalfe L., Salama A., +Peschke S. B., Kessler M. F., eds, ESA Special Publication Vol. 481, +The Calibration Legacy of the ISO Mission. p. 375 +Kwitter K. B., Henry R. B. C., 2022, PASP, 134, 022001 +Kwok S., 2000, The Origin and Evolution of Planetary Nebulae +Kwok S., Volk K. M., Hrivnak B. J., 1989, ApJ, 345, L51 +Labiano +A., +et +al., +2016, +in +Observatory +Operations: +Strategies, +Processes, and Systems VI. p. 99102W (arXiv:1608.05312), +doi:10.1117/12.2232554 +Labiano A., et al., 2021, A&A, 656, A57 +Lagadec E., et al., 2007, MNRAS, 376, 1270 +Law D. R., et al., 2021, AJ, 161, 52 +Leisenring J. M., Kemper F., Sloan G. C., 2008, ApJ, 681, 1557 +Madau P., Ferguson H. C., Dickinson M. E., Giavalisco M., Steidel C. C., +Fruchter A., 1996, MNRAS, 283, 1388 +Margon B., Massey P., Neugent K. F., Morrell N., 2020, ApJ, 898, 85 +Markwardt C. B., 2009, in Bohlender D. A., Durand D., Dowler P., +eds, Astronomical Society of the Pacific Conference Series Vol. 411, +Astronomical Data Analysis Software and Systems XVIII. p. 251 +(arXiv:0902.2850) +Matsuura M., et al., 2007a, MNRAS, 382, 1889 +Matsuura M., et al., 2007b, MNRAS, 382, 1447 +Matsuura M., et al., 2014, MNRAS, 439, 1472 +Peeters E., Hony S., Van Kerckhoven C., Tielens A. G. G. M., Allamandola +L. J., Hudgins D. M., Bauschlicher C. W., 2002, A&A, 390, 1089 +MNRAS 000, 1–11 (2022) + +JWST MRS observations of SMP LMC 058 +11 +Perley R. A., Butler B. J., 2013, ApJS, 204, 19 +Pietrzyński G., et al., 2013, Nature, 495, 76 +Pino T., et al., 2008, A&A, 490, 665 +Reid W. A., 2014, MNRAS, 438, 2642 +Reid W. A., Parker Q. A., 2006, MNRAS, 373, 521 +Reid W. A., Parker Q. A., 2010, MNRAS, 405, 1349 +Rigby J., et al., 2022, arXiv e-prints, p. arXiv:2207.05632 +Ruffle P. M. E., et al., 2015, MNRAS, 451, 3504 +Sanduleak N., MacConnell D. J., Philip A. G. D., 1978, PASP, 90, 621 +Sloan G. C., et al., 2014, ApJ, 791, 28 +Stanghellini L., García-Lario P., García-Hernández D. A., Perea-Calderón +J. V., Davies J. E., Manchado A., Villaver E., Shaw R. A., 2007, ApJ, +671, 1669 +Swinyard B. M., et al., 1996, A&A, 315, L43 +Volk K., Sloan G. C., Kraemer K. E., 2020, Ap&SS, 365, 88 +Waters L. B. F. M., et al., 1996, A&A, 315, L361 +Waters L. B. F. M., et al., 1998, Nature, 391, 868 +Wells M., et al., 2015, Publications of the Astronomical Society of the +Pacific, 127, 646–664 +Westerlund B. E., 1997, Cambridge Astrophysics Series, 29 +Woods P. M., et al., 2011, MNRAS, 411, 1597 +Woods P. M., Walsh C., Cordiner M. A., Kemper F., 2012, MNRAS, 426, +2689 +Wright E. L., et al., 2010, AJ, 140, 1868 +Zijlstra A. A., et al., 2006, MNRAS, 370, 1961 +Ziurys L. M., 2006, Proceedings of the National Academy of Science, 103, +12274 +van Diedenhoven B., Peeters E., Van Kerckhoven C., Hony S., Hudgins +D. M., Allamandola L. J., Tielens A. G. G. M., 2004, ApJ, 611, 928 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–11 (2022) + diff --git a/0NFQT4oBgHgl3EQfDTUM/content/tmp_files/load_file.txt b/0NFQT4oBgHgl3EQfDTUM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e151107df9237fb179eac456e104d39e428aa4db --- /dev/null +++ b/0NFQT4oBgHgl3EQfDTUM/content/tmp_files/load_file.txt @@ -0,0 +1,1486 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf,len=1485 +page_content='MNRAS 000, 1–11 (2022) Preprint 1 February 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='0 Observations of the Planetary Nebula SMP LMC 058 with the JWST MIRI Medium Resolution Spectrometer O.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Celestijnenlaan 200D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 3001 Leuven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Belgium 7Telespazio UK for the European Space Agency (ESA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' ESAC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Spain 8ETH Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Institute for Particle Physics and Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Wolfgang-Paulistr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 27, CH-8093 Zurich, Switzerland 9Kapteyn Astronomical Institute, University of Groningen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Box 800, 9700 AV Groningen, The Netherlands 10Sorbonne Université, CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98bis bd Arago, 75014 Paris, France 11Institut Universitaire de France, Ministére de l’Enseignement Supérieur et de la Recherche, 1 rue Descartes, 75231 Paris Cedex 05, France 12Jet Propulsion Laboratory, California Institute of Technology,4800 Oak Grove Drive, Pasadena, CA 91109 13Center for Astrophysical Sciences, The William H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Miller III Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA 14Princeton University, 4 Ivy Ln, Princeton, NJ 08544, USA Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' in original form ZZZ ABSTRACT During the commissioning of JWST, the Medium-Resolution Spectrometer (MRS) on the Mid-Infrared Instrument (MIRI) observed the planetary nebula SMP LMC 058 in the Large Magellanic Cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The MRS was designed to provide medium resolution (R = 𝜆/Δ𝜆) 3D spectroscopy in the whole MIRI range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' SMP LMC 058 is the only source observed in JWST commissioning that is both spatially and spectrally unresolved by the MRS and is a good test of JWST’s capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The new MRS spectra reveal a wealth of emission lines not previously detected in this metal-poor planetary nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' From these lines, the spectral resolving power (𝜆/Δ𝜆) of the MRS is confirmed to be in the range R = 4000 to 1500, depending on the MRS spectral sub-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' In addition, the spectra confirm that the carbon-rich dust emission is from SiC grains and that there is little to no time evolution of the SiC dust and emission line strengths over a 16-year epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' These commissioning data reveal the great potential of the MIRI MRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Key words: instrumentation: spectrographs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' infrared: general;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Astrophysics - Instrumentation and Methods for Astrophysics 1 INTRODUCTION The succession of increasingly powerful mid-infrared spectrographs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', the Short Wavelength Spectrometer (SWS) and the Infrared Spectrograph (IRS) on board the Infrared Space Observatory and ★ E-mail: olivia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='jones@stfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='uk the Spitzer Space Telescope) launched into space has revolutionised our knowledge of the cool universe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Waters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The Mid-Infrared Instrument (MIRI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (submitted)) on the James Webb Space Telescope (JWST) includes, in addition to the imager and coronographs, both a low-resolution spectrometer (LRS) covering wavelengths from 5 to 14 𝜇m (Kendrew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2015) and a medium-resolution spectrometer (MRS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Wells et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2015, Argyriou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' )), which is an Integral Field Unit (IFU), © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='13233v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='IM] 30 Jan 2023 2 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' that has a field of view ranging from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2′′ × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='7′′ to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='6′′ × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='7′′ (Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=')) and can spatially resolve spectroscopic data between 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='9 and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='9 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' This is the first time a mid-IR IFU has been deployed outside our atmosphere and will enable resolved spectroscopic studies of individual stars at the beginning and end of their evolution, diffuse structure in galaxies and planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The Large Magellanic Cloud (LMC) is a gas-rich, metal-poor, star-forming, irregular galaxy, which is a satellite of the Milky Way, hosts ∼103 planetary nebulae (PNe) (Reid & Parker 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Reid 2014), and is at a uniform distance of ∼50 kpc (Pietrzyński et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' In lower metallicity environments like the LMC, which has about half the metallicity of the Milky Way (Westerlund 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Choudhury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2016), significant dust production is expected to occur in the outflows of asymptotic giant branch (AGB) stars with further processing as these objects become planetary nebula (PNe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The gas and dust ejected into the interstellar medium (ISM) by a strong stellar wind from this phase of evolution contains elements synthesised in the stellar interior and dredged up to the surface by convection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Karakas & Lattanzio 2007, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Their chemical composition is expected to primarily depend upon the initial stel- lar mass and the interstellar elemental abundance at the time the progenitor stars were formed (Kwok 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Gonçalves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Kwitter & Henry 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' As such, the infrared spectra of PNe host a rich variety of features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' forbidden emission lines arising from ionisation from the hot central star (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Stanghellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2007), complex organic molecules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Ziurys 2006), polycyclic aromatic hydrocarbons (PAHs), and inorganic and organic solids (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Stanghellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Bernard-Salas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Guzman-Ramirez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' García-Hernández & Górny 2014) with the frequency of carbona- ceous features higher in the LMC than in Galactic PNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' This is likely due to the increased efficiency of third dredge-up (TDU) and the increased C/O ratio at low metallicities (Karakas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Pro- cessing by an external ambient UV radiation field which is stronger in the LMC (Gordon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2008) may also affect the circumstellar chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Detailed examination of PNe at low-metallicity, there- fore, provides a unique insight into chemical abundances and their effect on late-stage stellar evolution, dust production, and the for- mation of PNe in conditions comparable to those during the epoch of peak star formation in the Universe (Madau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Fur- thermore, due to their compact nature and brightness over a broad wavelength range, PNe are also useful calibration sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Swinyard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Feuchtgruber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Perley & Butler 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' SMP LMC 058 was observed by JWST as part of commission- ing and calibration activities for MIRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' First identified by Sanduleak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (1978), SMP LMC 058 is a carbon-rich planetary nebula (PN) in the LMC, with a heliocentric radial velocity of 278±7 km s−1 (Margon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The central star of SMP 058 is a likely C ii emitter (Margon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2020), consistent with a very early Wolf- Rayet type star on the carbon sequence (WC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Several dozen very strong, common emission lines of PNe were also detected in its optical spectra (Margon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' SMP LMC 058 has also been observed with the Spitzer Infrared Spectrograph (IRS) at both low- resolution (R∼60–127) and high-resolution (R∼600).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The Spitzer spectra show SMP LMC 058 has unusual dust chemistry with a strong SiC feature at ∼11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='3 𝜇m (Bernard-Salas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2009) and other associated features, including emission from PAHs at 6–9 𝜇m, and a shoulder at 18 𝜇m from an unidentified carrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' However, the Spitzer-IRS data show no clear evidence of fullerenes (Sloan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' SiC is rarely seen in Galactic PNe, in spite of the higher Si abundance in the Milky Way compared to the Magellanic Clouds (Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Its strength may be due to photoexci- tation, or because at a high C/O ratio SiC forms on the surface of carbon grains (Sloan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' In this paper, we describe the observations and calibration of JWST MIRI MRS commissioning data of SMP LMC 058 (Sec- tion 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' We then present its MRS spectra in Section 3 and determine the resolving power of the MRS in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' In Section 5 we identify and analyse the new emission lines and solid-state features detected in this carbon-rich planetary nebula and compare this with Spitzer IRS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The potential of the MRS and our conclusions are discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2 OBSERVATIONS AND CALIBRATIONS The observations were taken as part of the MIRI MRS commis- sioning program, program ID 1049 (the commissioning purpose of these observations was PSF characterization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' They use the standard MRS observing template, with 4-point dither patterns optimized for channels 2, 3, and 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Each dither pattern was used twice, in the ‘positive’ and ‘negative’ direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Target acquisition was ac- tivated, with the science target itself serving as an acquisition target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' All three bands (SHORT, MEDIUM, LONG) in all channels were observed in all dithers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Simultaneous MIRI imaging in filter F770W was taken in the dither optimized for channel 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A dedicated background observation was taken, employing a 2-point dither optimised for all channels, on a field roughly 3 arcmin away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The background field was chosen to be relatively clear of astronomical sources based on archival WISE imaging data (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A total of 45 FASTR1 frames were taken per integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' In target observations, a single integration was taken per dither point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The background observation had two integrations (to match the total integration time on source, accounting for the use of only a two- point dither on the background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The integration time per MRS sub- band and complete dither were therefore 499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='5s or roughly 1,500s to cover the entire wavelength range (bands SHORT, MEDIUM, and LONG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Between the six dithers on-target and the single back- ground, the total integration time was approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='9 hours (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='9 hr including all overheads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The MRS observations were processed with version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='3 of the JWST calibration pipeline and context 0995 of the Calibra- tion Reference Data System (CRDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' In general, we follow the standard MRS pipeline procedure (Labiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Bushouse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' and see Álvarez-Márquez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2022 for an in-flight example of MRS data calibration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The background subtraction has been performed using the dedicated background observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' We have generated twelve 3D spectral cubes, one for each of the MRS channels and bands, with a spatial and spectral sam- pling of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='13" × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='13" × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='001 𝜇m, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='17" × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='17" × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='002 𝜇m, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='20" × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='20" × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='003 𝜇m, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='35" × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='35" × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='006 𝜇m for chan- nels 1, 2, 3, and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' We have performed 1D spectral ex- tractions individually in each of the MRS cubes using a circular aper- ture of radius equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='5 × 𝐹𝑊𝐻𝑀(𝜆), where 𝐹𝑊𝐻𝑀(𝜆) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='3 arcsec for 𝜆 < 8𝜇m and 𝐹𝑊𝐻𝑀(𝜆) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='31 × 𝜆[𝜇𝑚]/8 arcsec for 𝜆 > 8𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The selected FWHM (𝜆) values follow the MRS PSF Full Width at Half Maximum (FWHM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' NIRCam observation (see Figure 1), and MRS observations suggest that SMP LMC 058 is an unresolved source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' We use the MRS PSF models (Patapis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=') to correct the aperture losses in the 1D spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The percent- age of flux that is lost out of the selected aperture is 17% for channel 1 and increases to 30% in channel 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' MNRAS 000, 1–11 (2022) JWST MRS observations of SMP LMC 058 3 5h24m21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='5s 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='0s 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='5s 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='0s 70 04′57′′ 05′00′′ 03′′ 06′′ RA (ICRS) Dec (ICRS) F356W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='5 pc Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' NIRCam F356W image of SMP LMC 058 shown in an Asinh stretch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' At this spatial resolution (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='063′′) SMP LMC 058 is an unresolved point source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The 12 spectral segments extracted from these cubes were corrected for residual fringing using a post-pipeline spectral-level correction which is a modified version of the detector-level correc- tion available in the JWST calibration pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The residual fringe contrasts are reduced by employing an empirical multi-component sine fitting method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Kester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2003), under the assumption that the pipeline fringe flat correction has reduced fringe contrasts to the point where this multi-component sine approximation is valid (Kavanagh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Finally, each of the 12 individual spectral segments was stitched together to remove minor flux discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' This was done by determining a scaling factor between the median flux (ex- cluding spectral lines) in the overlapping MRS segments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' then ap- plying this multiplicative factor to the longer wavelength segments, in turn, to effectively shift the spectrum to match the flux of its neighbouring shorter wavelength segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' This factor was typi- cally on the order of <5 per cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The flux data in the overlapping spectral regions were then averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The final stitched spectrum was inspected to ensure there were no remaining discontinuities which may affect the continuum and model fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 3 SMP LMC 058 SPECTRUM Figure 2 shows the extracted spectrum of SMP LMC 058 which exhibits a rich variety of atomic, molecular and solid state features, including PAHs and silicon carbide, characteristic of carbon-rich material, and a strong continuum which rises towards the longest wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Due to the superior sensitivity and spectral resolution (see Section 4) of the MRS, the MIRI spectrum of SMP LMC 058 shows features that are not seen in the Spitzer IRS data (see Section 5), notably in the number of emission lines detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' In the spectrum presented here, there is a large amount of fine- structure line emission present, from the strong nebular forbidden lines of [Ar ii], [Ar iii], [S iv], [Ne ii], [Ne iii], [S iii] to weak H recombination lines (Hi) from the Pfund and Humphreys series, and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' To ensure we measure and identify all the emission lines in the spectra we fit a pseudo-continuum to the broadband spectral features using a piece-wise spline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Obvious narrow band features were identified and masked in the fitting based on their amplitudes exceeding a threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' We used an outlier rejection fitter to flag and ignore any weaker narrow-band features that may compromise the continuum fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' After visual inspection of the fit, it was subtracted to isolate any narrow-band features present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Figure 3 shows the spectrum of SMP LMC 058 after subtraction of the pseudocontinuum from the total spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The spectrum is extremely rich in emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' In total 51 lines were detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Using the 12 original MRS segments, we identified and ana- lyzed all detected emission lines with a signal-to-noise ratio (SNR) greater than 3 in the SMP LMC 058 spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Depending on the line profiles (see Figure 4), we performed one-component and two- component Gaussian fits, plus a second-order polynomial to simul- taneously fit the continuum and emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='1 The uncertainties on the derived emission line parameters, like the line FWHM, flux, central wavelength, etc, were estimated using a Monte Carlo simu- lation (following the same methodology as Álvarez-Márquez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Systemic velocity shifts were removed using a he- liocentric radial velocity of 278 ± 7 km/s (Reid & Parker 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Margon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Table 1 presents the measured wavelengths and fluxes together with the identification of the mid-IR emission lines in SMP LMC 058 spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Small wavelength offsets are con- sistent with known errors in the MRS FLT-4 wavelength solution (see discussion by Argyriou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=')) and should be reduced further by ongoing calibration efforts later in Cycle 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Weak lines are more prevalent at shorter wavelengths in channels 1 and 2 where the MRS sensitivity is higher and the uncertainties in the flux are better constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Furthermore, we find that the current MRS wavelength calibration is <40 km/s for all spectral sub-bands, better than the FWHM of the MRS line spread function (75-200 km/s, Labiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2021, Argyriou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 4 MRS RESOLVING POWER The resolving power (R) is defined as 𝜆/Δ𝜆, where Δ𝜆 is the mini- mum distance to distinguish two features in a spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' We define the Δ𝜆 as the FWHM of an unresolved emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' SMP LMC 058 is the only source observed in the JWST commissioning datasets that is considered both spatially and spectrally unresolved with the MRS, this makes it an excellent target for determining the inflight MRS resolving power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Here, we assume the intrinsic width of the emission lines in SMP LMC 058 to be negligible, as we do not have high-resolution spectroscopy to characterize its intrinsic velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Nearby planetary nebula eject gas with typical velocity dispersions of about 10–51 kms−1 (Reid & Parker 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' If this is the case for SMP LMC 058, then assuming a velocity of 25 kms−1 we might underestimate the MRS resolving power by up to 5% for channel 1, and up to 1% for channel 4 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The pre-launch MRS resolving power has been established from MIRI ground-based test and calibration campaigns, using a set of etalons which provided lines in all MRS bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' It was de- termined to be in the range of about 4000 to 1500 (Labiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Figure 5 shows the comparison between the ground-based MRS resolving power estimates and the inflight estimates derived from the SMP LMC 058 spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The inflight MRS resolving power has been determined using only emission lines with SNR higher than 6, and following the FWHM results obtained in the one- and 1 We used the mpfit (Markwardt 2009) Python routine to perform the fits, the code is publicly available here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' MNRAS 000, 1–11 (2022) 4 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 5 10 15 20 25 Wavelength (microns) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='5 Flux (Jy) MIRI MRS Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The MIRI MRS spectrum of SMP LMC 058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Numerous emission lines, PAH features and dust features are clearly seen on a rising continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' These features are much better resolved in the MRS spectra due to the higher spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' two-component Gaussian fits (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' In the case of Hi emission lines, we used the FWHM of the narrow gaussian compo- nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The errors in the resolving power are, on average, larger for the Hi emission lines due to the uncertainty in the two-component Gaussian fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Given the uncertainties, the inflight MRS resolving power agrees with the ground-based estimates, and it presents a trend followed by the equation 4401 − 112 × 𝜆[𝜇𝑚] + 10−9×𝜆[𝜇𝑚].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The ground-based estimates consider the width of the etalon emission lines to be negligible, which could imply an underesti- mation of the MRS resolving power by a factor of 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A similar situation is potentially happening with the inflight estimations due to the lack of the intrinsic velocity dispersion of SMP LMC 058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' We conclude that the estimations of the ground-based MRS resolving power (Labiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2021) are valid, within a 10% of uncertainty, for the MRS inflight performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' As JWST observes more sources with spatially and spectrally unresolved spectral lines, their char- acterisation will provide a more comprehensive understanding of the inflight variations of the resolving power within each of the 12 spectral bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' As of now, the continuous "trend" curve in Figure 5 presents the state of knowledge of the MRS resolving power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 5 DISCUSSION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='1 Emission Lines Short-High and Long-High Spitzer spectroscopic data of SMP LMC 058 were published in Bernard-Salas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A comparison of the MRS line fluxes with those found by Bernard-Salas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (2008) is given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' In general, there is good agreement between our measurements of the forbidden emission line strengths of [S iv], [Ne ii], [Ne iii], [S iii].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Furthermore, the high-excitation lines of [Ar v] at 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='10 𝜇m and [O iv] with ionisation potentials of 60 and 55 eV, respectively, are not detected in either spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' These lines are excited by high-temperature stars with Teff between 140,000 – 180,000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The highest ionisation species in the MRS spectra are [K iv] (46 eV) and [Na iii] (47 eV), these lines have not been previously detected by Spitzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Thus, we consider SMP LMC 058 to be a low-excitation source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Given the superior sensitivity of JWST (Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2022) and the MRS, we detect a line at 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='38 𝜇m an order of magnitude below the upper-limit of [Ne v] reported by Bernard-Salas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The ionisation potential of [Ne v] is 97 eV, thus it is unlikely given the absence of other high-excitation lines in the MRS spectra of SMP LMC 058, that this emission is from [Ne v], instead, we attribute this line to [Cl ii] which has an ionisation potential of 13 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' As seen in Table 1 higher ionisation potential species expand at a lower velocity than the lower ionisation potential species (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Reid & Parker 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' This is due to ionisation occurring at a greater distance from the centre of the PN where velocities are greater, and can cause lower excitation species to expand to larger radii in the PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Hydrogen recombination lines are abundant in the spectrum of SMP LMC 058, all are new detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Hi emission lines more closely trace the ionized regions, compared to molecular hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' As shown in Figure 4, the line profiles of the bright Hi emission lines are asymmetric, exhibiting a blue tail, whereas the forbidden emis- sion lines present symmetric unresolved profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Hi emission lines are composed of a spectrally unresolved main component contain- MNRAS 000, 1–11 (2022) JWST MRS observations of SMP LMC 058 5 5 6 7 8 9 Wavelength (microns) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='010 Flux (Jy) H Hu H Hu [ArII] H Pf H Hu H10-7 [ArIII] [NaIII] H16-7 H15-7 H13-7 H12-7 H15-8 H13-8 10 12 14 16 18 20 22 Wavelength (microns) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='010 Flux (Jy) [SIV] H9-7 H Hu [NeII] [NeV] [NeIII] H10-8 [SIII] H8-7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Continuum-subtracted MRS spectrum of SMP LMC 058 (where the continuum includes dust and PAH features), highlighting the atomic emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The identification of key species are marked on the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The top panel shows lines in channels 1 and 2 of the MRS, the lower panels show channels 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The flux axis is truncated to highlight lower contrast lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' MNRAS 000, 1–11 (2022) 6 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Measured central wavelengths, line flux, line widths, and line identification for SMP LMC 058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The systemic velocity was removed prior to calculating the velocity shift of a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' If a line is present in multiple MRS bands, measurements are provided for each individual MRS segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Uncertain line identifications are denoted by a ‘?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Band Line 𝜆lab 𝜆observed 𝜎𝜆observed 𝜆offset FWHM Flux (×10−15) 𝜎 (×10−15) Identification 𝜇m 𝜇m 𝜇m km s−1 nm erg s−1 cm−2 erg s−1 cm−2 1S Hi 23−7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='924 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='92903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='00004 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='025 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='02 1S Hi 22−7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='971 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='97588 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='00004 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='351 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='01 1S Hi 21−7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='026 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='03086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='00007 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='428 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='362 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='01 1S Hi 20−7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='091 5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='13342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='00001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='165 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='556 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='02 1S Hi 19−7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='169 5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='654 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='238 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='05 4M [Ar iii] 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='830 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='85033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='00032 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='852 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='06 4M Hi 13−10+Hi 11−9 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='340 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='35612 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='00055 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='040 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='05 ing the majority of the line flux (>95%)), and a spectrally resolved blue-shifted component possibly due to thermal broadening (Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 1984) or from condensation outside the main core which may be evident as a marginally resolved envelope like structure in the MRS cube at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='466𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Two molecular hydrogen lines (H2) have been detected in the MRS data, the ortho-H2 𝑣 = 0–0 S(3) and S(5) lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The S(1) line at 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='055 𝜇m may also be present, although this is not easily discerned above the continuum and we do not measure its flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The S(3) and S(5) rotational line emission probably originate from irradiated, and perhaps also shocked, dense molecular clumps, torus structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Kastner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Hora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Akras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' MNRAS 000, 1–11 (2022) JWST MRS observations of SMP LMC 058 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='455 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='460 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='465 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='470 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='475 Wavelength (microns) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='10 Flux (Jy) Pf 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='54 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='56 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='58 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='60 Wavelength (microns) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='00 Flux (Jy) [Ne III] Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Top: The Pf 𝛼 H i emission line profile shows a spatially unre- solved main component and a weaker spectrally resolved blue-shifted wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Bottom: The [Ne iii] line profile is spectrally unresolved and symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' This shape is typical of all the forbidden emission lines in SMP LMC 058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The dashed line marks the lines observed central wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Comparison of SMP LMC 058 MRS line fluxes with those of Bernard-Salas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (2008) taken with the high-resolution modules on the Spitzer IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' All line strengths reported by Bernard-Salas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (2008) have a 10% error except for [S iv] which has a 10–20% error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Errors in the MRS flux are <1% and are provided for each line in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Line Wavelength MRS Flux Spitzer Flux Ionisation (Rest) ×10−15 ×10−15 potential 𝜇m erg s−1 cm−2 erg s−1 cm−2 (eV) [S iv] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='511 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='07 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2 35 [Ne ii] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='814 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='63 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='6 22 [Ar v] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='099 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='029 <2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='4 60 [Ne v] 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='323 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='005 <3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='8 97 [Ne iii] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='555 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='26 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='6 41 [S iii] 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='713 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='34 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='0 23 [O iv] 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='883 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='23 <21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='6 55 Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2018), or from the outer regions of the PNe where the temperature is about 1000K (Aleman & Gruenwald 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Matsuura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2007b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2 Dust and PAH Features The dust in SMP LMC 058 is carbon-rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Amongst the most promi- nent features is the strong silicon carbide (SiC) emission at 11 𝜇m and the rising continuum due to the thermal emission of warm dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Strong emission features from PAHs also appear in the spectrum at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='7, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='7, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='6, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='7 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' At sub-solar metallicities (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='5 Z⊙), SiC is commonly observed in PNe, yet it is rarely seen in Galactic PNe or indeed during the earlier AGB evolutionary phase of metal-poor carbon stars (Casassus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Zijlstra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Matsuura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2007a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Stanghellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Bernard-Salas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Woods et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2011, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Sloan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Ruffle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The strength of the SiC flux in metal-poor PNe is highly sensitive to the radiation field (Bernard-Salas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' This is likely due to a lower abundance of Si affecting the carbonaceous dust condensation sequence on the AGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' In this case, rather than SiC forming first, it instead forms in a mantle surrounding an amorphous carbon core (Lagadec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Leisenring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Then as the PNe dust becomes heated and photo-processed, the amorphous carbon evaporates increasing the SiC surface area and consequently its feature strength, until a critical ionisation potential of >55 eV occurs at which point the SiC features disappear (Bernard-Salas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Sloan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Following Bernard-Salas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (2009), we measure the strength of the SiC feature by integrating the flux above a continuum- subtracted spectrum from 9 to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2 𝜇m and then subtracting the flux contributions from the 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2 𝜇m PAH feature and the [Ne ii] line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Due to the resolution of the MRS compared to the Spitzer spectra of SMP LMC 058, we detect several additional lines which contribute to the integrated flux in the SiC region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' these lines include [S iv] and H i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Thus to obtain a reliable measurement of the SiC feature strength we also subtract the flux contribution from all emission lines in the 9 – 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2 𝜇m region listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A PAH feature at ∼12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='6 𝜇m likely contributes a small amount of flux to the measured SiC feature, however isolating and subtracting this emission contri- bution from the wing of the SiC feature is challenging even with the MRS spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Additionally, an artefact at ∼12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2 𝜇m due to a spectral leak (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Gasman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2022) may also affect the integrated flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Table 3 gives the measured SiC centroid and cor- rected feature strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The latter agrees exceptionally well with the value of 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='31 ×10−16 W m−2 measured by Bernard-Salas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (2009) in the Spitzer data of SMP LMC 058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' This suggests there is little to no evolution in the SiC dust on the 16-year time scales between the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Furthermore, the agreement be- tween the measurements verifies the overall flux calibration of the MRS instrument (Gasman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' In astronomical sources, the structure, wavelengths and relative strength of the PAHs can differ strongly between objects, with PNe showing the most pronounced variations in PAH profiles due to photoprocessing altering the ratio of aliphatics to aromatics (Peeters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Pino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Matsuura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Sloan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Jensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Figure 6 shows the PAHs in SMP LMC 058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The PAHs in SMP LMC 058 are considered to have a class B profile by Bernard-Salas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (2009) and Sloan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' In this schema devised by Peeters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (2002) and van Diedenhoven et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (2004) the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2 PAH feature for class B objects has a peak between 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='24 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='28 𝜇m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' the dominant 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='7 PAH feature peaks between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='8 to MNRAS 000, 1–11 (2022) 8 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='5 5 6 7 8 9 10 12 15 20 25 30 Wavelength [ m] 1000 1500 2000 2500 3000 3500 4000 4500 5000 MRS Resolving Power Ground (Labiano+21) Fit to inflight lines inflight forbidden lines inflight HI lines Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Comparison between the ground-based and inflight MRS resolving power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Gray filled area and black line: ground-based MRS resolving power estimates (Labiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Filled red circles: inflight MRS resolving power calculation using forbidden emission lines identified in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Open red circles: inflight MRS resolving power calculation using Hi emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 6 8 10 12 14 Wavelength (microns) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='200 Flux (Jy) PAH 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2 m PAH 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='7 m PAH 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2 m PAH 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='7 m PAH 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='6 m PAH 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2 m SiC Local continuum Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The SiC and PAH features are highlighted in the spectra of SMP LMC 058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A local continuum fit to the 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='3 𝜇m feature which is superimposed on the broad SiC emission feature is also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The colours highlight the spectral region for each feature, to which a local continuum was fit and the flux measured over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' MNRAS 000, 1–11 (2022) JWST MRS observations of SMP LMC 058 9 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' PAH and SiC Fluxes and Centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Centroid Integrated Flux Integrated Flux Error 𝜇m W m−2 W m−2 PAH 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='262 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='64×10−18 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='0×10−20 PAH 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='698 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='62×10−18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='1×10−19 PAH 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='274 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='259×10−17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='9×10−19 PAH 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='834 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='427×10−16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2×10−18 PAH 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='665 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='231×10−17 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='5×10−19 PAH 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='298 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='047×10−16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='0×10−18 SiC 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='097 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='067×10−15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='5×10−18 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='0 𝜇m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' and the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='6 PAH band is red-shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' These values agree well with our measured centroids listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Furthermore, the PAHs observed in SMP LMC 058 closely resemble those observed in the ISO SWS spectrum of the Galactic post-AGB star, HD 44179 (the Red Rectangle) which also shows strong aromatic features on top of a continuum (Waters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The relative strength of the PAH features depends on a number of factors including the degree of ionisation of the radiation field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Allamandola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The strength of the PAH features in SMP LMC 058 was determined by integrating the flux of the feature above an adopted local continuum, fit to each side of the feature and measured using specutils line_flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Particular care was taken in fitting a continuum, too, and then measuring the 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='25 𝜇m band (produced by the out-of-plane solo C–H bending mode) as this is superimposed on top of the broad SiC feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Table 3 presents the central wavelength of the features and the integrated flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The ratio of the PAH strengths correlates with the source type and hence its physical conditions (Hony et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' ionized PAHs have strong features at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='6 𝜇m whilst the 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2 𝜇m PAH feature is stronger for neutral PAHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' From the PAH line strengths given in Table 3 it is evident that the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='7𝜇m feature dominates the total PAH emission, and thus the dust around SMP LMC 058 is likely experiencing a high degree of ionisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Carbon-rich PNe can show a rich variety of solid-state material in their spectra in addition to PAHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The C60 fullerene molecule typically exhibits features at ∼7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='0, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='5, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='4 and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='9 𝜇m, and all four were first identified in the spectrum of the Galactic PN TC-1 (Cami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Fullerenes have since been detected in several other PNe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', García-Hernández et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2010, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Sloan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The still-unidentified 21 𝜇m emission feature, first detected by Kwok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 1989, can also appear in carbon-rich PNe, often associated with unusual PAH emission and aliphatic hydrocarbons (Cerrigone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Matsuura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Sloan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Volk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The spectra of SMP LMC 058 from the IRS on Spitzer did not show any of these unusual hydrocarbon-related features, but the improved spectral resolution of the MRS allows for a much more careful examination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Nonetheless, these additional features remain too weak to be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' SMP LMC 058 presents a classic Class B PAH spectrum, as expected for objects which have evolved to the young PN stage (Sloan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Younger objects which could still be described as post-AGB objects would show the 21 𝜇m feature and/or aliphatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Sloan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' (2014) identified SMP LMC 058 as a member of the Big-11 group because of the combination of a strong SiC emission feature and the 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2 𝜇m PAH feature and the absence of fullerenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' This group is actually related to the PNe that show fullerenes, and the presence or absence of fullerenes may be due to something as simple as which have a clear line of sight to the interior of the dust shells where the fullerenes are expected to be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 6 SUMMARY AND CONCLUSIONS We have presented MIRI/MRS spectra of the carbon-rich planetary nebula SMP LMC 058 located in the metal-poor Large Magellanic Cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' SMP LMC 058 is a point source in the MRS data and its spectrum contains the only spatially and spectrally unresolved emission lines observed during the commissioning of the JWST Medium-Resolution Spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' In the MRS spectrum, we de- tected 51 emission lines, of which 47 were previously undetected in this source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The strongest emission lines were used to determine the spectral resolutions of the MIRI MRS instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The resolving power is R > 3960 in channel 1, R > 3530 in channel 2, R > 3200 in channel 3, and R > 1920 in channel 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' This on-sky performance is comparable to the resolution determined from the ground cali- bration of the MRS which provides resolving powers from 4000 at channel 1 to 1500 at channel 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Furthermore, a comparison of the line strengths and spectral continuum to previous observations of SMP LMC 058 with the IRS on the Spitzer was used to verify the absolute flux calibration of the MRS instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The MRS spectra confirm that the carbon-rich dust emission is from grains and not isolated molecules and that there is little to no time evolution of the SiC dust and emission line strengths in the 16 years between the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The PAH emission is dominated by the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='7𝜇m feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The strong PAHs and SiC in the spectra are consistent with the lack of high-excitation lines detected in the spectra, which if present, would indicate a hard radiation field that would likely destroy these grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' These commissioning data reveal the great potential and resolving power of the MIRI MRS to study line, molecular and solid-state features in individual sources in nearby galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank Kay Justtanont for her insights, comments and dis- cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The data were ob- tained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', under NASA contract NAS 5-03127 for JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' These observations are associated with program #1049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' This work is based in part on observations made with the Spitzer Space Telescope, which was operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='J acknowledge support from an STFC Webb fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='L acknowledge support by grant PIB2021-127718NB- 100 by the Spanish Ministry of Science and Innovation/State Agency of Research (MCIN/AEI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='K acknowledges financial support from the Science Foundation Ireland/Irish Research Council Path- way programme under Grant Number 21/PATH-S/9360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' thank the European Space Agency (ESA) and the Belgian Federal Science Policy Office (BELSPO) for their support in the framework of the PRODEX Programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' PG would like to thank the University Pierre and Marie Curie, the Institut Universitaire de France, the Centre National d’Etudes Spatiales (CNES), the "Pro- gramme National de Cosmologie and Galaxies" (PNCG) and the "Physique Chimie du Milieu Interstellaire" (PCMI) programs of MNRAS 000, 1–11 (2022) 10 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' CNRS/INSU, with INC/INP co-funded by CEA and CNES, for there financial supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' MIRI draws on the scientific and technical expertise of the following organisations: Ames Research Center, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Airbus De- fence and Space, UK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' CEA-Irfu, Saclay, France;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Centre Spatial de Liége, Belgium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Consejo Superior de Investigaciones Científi- cas, Spain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Carl Zeiss Optronics, Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Chalmers University of Technology, Sweden;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Danish Space Research Institute, Denmark;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Dublin Institute for Advanced Studies, Ireland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' European Space Agency, Netherlands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' ETCA, Belgium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' ETH Zurich, Switzerland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Goddard Space Flight Center, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Institute d’Astrophysique Spa- tiale, France;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Instituto Nacional de Técnica Aeroespacial, Spain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' In- stitute for Astronomy, Edinburgh, UK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Jet Propulsion Laboratory, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Laboratoire d’Astrophysique de Marseille (LAM), France;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Leiden University, Netherlands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Lockheed Advanced Technology Center (USA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' NOVA Opt-IR group at Dwingeloo, Netherlands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Northrop Grumman, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Max-Planck Institut für Astronomie (MPIA), Heidelberg, Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Laboratoire d’Etudes Spatiales et d’Instrumentation en Astrophysique (LESIA), France;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Paul Scher- rer Institut, Switzerland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Raytheon Vision Systems, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' RUAG Aerospace, Switzerland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Rutherford Appleton Laboratory (RAL Space), UK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Space Telescope Science Institute, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Toegepast- Natuurwetenschappelijk Onderzoek (TNO-TPD), Netherlands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' UK Astronomy Technology Centre, UK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' University College London, UK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' University of Amsterdam, Netherlands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' University of Arizona, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' University of Bern, Switzerland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' University of Cardiff, UK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' University of Cologne, Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' University of Ghent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' University of Groningen, Netherlands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' University of Leicester, UK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' University of Leuven, Belgium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' University of Stockholm, Sweden;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Utah State University, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A portion of this work was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' The following National and International Funding Agencies funded and supported the MIRI development: NASA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' ESA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Bel- gian Science Policy Office (BELSPO);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Centre Nationale d’Etudes Spatiales (CNES);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Danish National Space Centre;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Deutsches Zen- trum fur Luftund Raumfahrt (DLR);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Enterprise Ireland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Ministerio De Economia y Competividad;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Netherlands Research School for As- tronomy (NOVA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Netherlands Organisation for Scientific Research (NWO);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Science and Technology Facilities Council;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Swiss Space Office;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Swedish National Space Agency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' and UK Space Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Facilities: JWST (MIRI/MRS) - James Webb Space Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' DATA AVAILABILITY JWST data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute (https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='edu/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' REFERENCES Akras S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Gonçalves D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Ramos-Larios G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2017, MNRAS, 465, 1289 Aleman I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Gruenwald R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2004, ApJ, 607, 865 Allamandola L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Hudgins D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Sandford S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 1999, ApJ, 511, L115 Álvarez-Márquez J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Marques-Chaves R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Colina L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Pérez-Fournon I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2021, A&A, 647, A133 Álvarez-Márquez J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='01695 Bernard-Salas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Pottasch S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Gutenkunst S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Morris P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Houck J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2008, ApJ, 672, 274 Bernard-Salas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Peeters E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Sloan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Gutenkunst S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Matsuura M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Tielens A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Zijlstra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Houck J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2009, ApJ, 699, 1541 Brown M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Jarrett T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Cluver M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2014, Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Australia, 31, e049 Bushouse H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2022, spacetelescope/jwst: JWST 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='6984366, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='5281/ zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='6984366 Cami J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Bernard-Salas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Peeters E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Malek S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2010, Science, 329, 1180 Casassus S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Roche P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Aitken D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Smith C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2001, MNRAS, 327, 744 Cerrigone L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Hora J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Umana G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Trigilio C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Hart A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Fazio G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2011, ApJ, 738, 121 Choudhury S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Subramaniam A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Cole A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2016, MNRAS, 455, 1855 Chu Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Kwitter K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Kaler J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Jacoby G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 1984, PASP, 96, 598 Fang X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Zhang Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Kwok S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Hsia C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Chau W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Ramos-Larios G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Guer- rero M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2018, ApJ, 859, 92 Feuchtgruber H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 1997, ApJ, 487, 962 García-Hernández D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Górny S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2014, A&A, 567, A12 García-Hernández D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Manchado A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', García-Lario P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Stanghellini L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Villaver E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Shaw R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Szczerba R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Perea-Calderón J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2010, ApJ, 724, L39 García-Hernández D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2011, ApJ, 737, L30 Gasman D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='03596 Gonçalves D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Magrini L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Teodorescu A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Carneiro C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2014, MNRAS, 444, 1705 Gordon K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Engelbracht C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Rieke G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Misselt K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Smith J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Kennicutt Robert C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2008, ApJ, 682, 336 Guzman-Ramirez L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Zijlstra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Níchuimín R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Gesicki K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Lagadec E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Millar T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Woods P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2011, MNRAS, 414, 1667 Hony S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Van Kerckhoven C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Peeters E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Tielens A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Hudgins D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Allamandola L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2001, A&A, 370, 1030 Hora J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Latter W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Deutsch L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 1999, ApJS, 124, 195 Jensen P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Shannon M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Peeters E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Sloan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Stock D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2022, A&A, 665, A153 Jones O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2017, MNRAS, 470, 3250 Karakas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Lattanzio J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2007, Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Australia, 24, 103 Karakas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Lattanzio J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2014, Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Australia, 31, e030 Karakas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Lattanzio J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Pols O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2002, Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' Australia, 19, 515 Kastner J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Weintraub D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Gatley I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Merrill K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Probst R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 1996, ApJ, 462, 777 Kendrew S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2015, Publications of the Astronomical Society of the Pacific, 127, 623 Kester D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Beintema D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Lutz D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2003, in Metcalfe L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Salama A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Peschke S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Kessler M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', eds, ESA Special Publication Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 481, The Calibration Legacy of the ISO Mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 375 Kwitter K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Henry R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2022, PASP, 134, 022001 Kwok S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2000, The Origin and Evolution of Planetary Nebulae Kwok S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Volk K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Hrivnak B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 1989, ApJ, 345, L51 Labiano A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2016, in Observatory Operations: Strategies, Processes, and Systems VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 99102W (arXiv:1608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='05312), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2232554 Labiano A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2021, A&A, 656, A57 Lagadec E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2007, MNRAS, 376, 1270 Law D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2021, AJ, 161, 52 Leisenring J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Kemper F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Sloan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2008, ApJ, 681, 1557 Madau P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Ferguson H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Dickinson M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Giavalisco M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Steidel C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Fruchter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 1996, MNRAS, 283, 1388 Margon B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Massey P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Neugent K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Morrell N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2020, ApJ, 898, 85 Markwardt C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2009, in Bohlender D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Durand D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Dowler P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', eds, Astronomical Society of the Pacific Conference Series Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 411, Astronomical Data Analysis Software and Systems XVIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' 251 (arXiv:0902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='2850) Matsuura M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2007a, MNRAS, 382, 1889 Matsuura M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2007b, MNRAS, 382, 1447 Matsuura M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2014, MNRAS, 439, 1472 Peeters E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Hony S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Van Kerckhoven C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Tielens A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Allamandola L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Hudgins D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Bauschlicher C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2002, A&A, 390, 1089 MNRAS 000, 1–11 (2022) JWST MRS observations of SMP LMC 058 11 Perley R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Butler B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2013, ApJS, 204, 19 Pietrzyński G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2013, Nature, 495, 76 Pino T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2008, A&A, 490, 665 Reid W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2014, MNRAS, 438, 2642 Reid W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Parker Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2006, MNRAS, 373, 521 Reid W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Parker Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2010, MNRAS, 405, 1349 Rigby J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content='05632 Ruffle P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2015, MNRAS, 451, 3504 Sanduleak N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', MacConnell D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Philip A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 1978, PASP, 90, 621 Sloan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2014, ApJ, 791, 28 Stanghellini L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', García-Lario P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', García-Hernández D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Perea-Calderón J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Davies J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Manchado A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Villaver E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Shaw R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2007, ApJ, 671, 1669 Swinyard B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 1996, A&A, 315, L43 Volk K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Sloan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Kraemer K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2020, Ap&SS, 365, 88 Waters L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 1996, A&A, 315, L361 Waters L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 1998, Nature, 391, 868 Wells M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2015, Publications of the Astronomical Society of the Pacific, 127, 646–664 Westerlund B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 1997, Cambridge Astrophysics Series, 29 Woods P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2011, MNRAS, 411, 1597 Woods P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Walsh C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Cordiner M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Kemper F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2012, MNRAS, 426, 2689 Wright E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2010, AJ, 140, 1868 Zijlstra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2006, MNRAS, 370, 1961 Ziurys L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2006, Proceedings of the National Academy of Science, 103, 12274 van Diedenhoven B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Peeters E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Van Kerckhoven C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Hony S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Hudgins D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Allamandola L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', Tielens A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=', 2004, ApJ, 611, 928 This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} +page_content=' MNRAS 000, 1–11 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfDTUM/content/2301.13233v1.pdf'} diff --git a/1NAzT4oBgHgl3EQfDfoM/vector_store/index.faiss b/1NAzT4oBgHgl3EQfDfoM/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..6731f9d3d7de8b781e45052c8fe067dcbca92f41 --- /dev/null +++ b/1NAzT4oBgHgl3EQfDfoM/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:69d3ff663888866b5b518e6f3246b0725f8e040048fe33bf2cf607e9e5b618c5 +size 5832749 diff --git a/1dE0T4oBgHgl3EQfdgBe/content/tmp_files/2301.02377v1.pdf.txt b/1dE0T4oBgHgl3EQfdgBe/content/tmp_files/2301.02377v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e684818d96ced031397a7962711afe9537bf0082 --- /dev/null +++ b/1dE0T4oBgHgl3EQfdgBe/content/tmp_files/2301.02377v1.pdf.txt @@ -0,0 +1,1126 @@ +Communications in Mathematics n (2023), no. m, 00–12 +DOI: https://doi.org/10.46298/cm.ABCD +©2023 G´abor Rom´an +This is an open access article licensed under the CC BY-SA 4.0 +1 +On square-free numbers generated from given sets of primes +II +G´abor Rom´an +Abstract. We progress with the investigation started in article [7], namely the anal- +ysis of the asymptotic behaviour of QP(x) for different sets P, where QP(x) is the +element count of the set containing those positive square-free integers, which are +smaller than-, or equal to x, and which are only divisible by the elements of P. We +study how QP(x) behaves when we require that χ(p) = 1 must hold for every p ∈ P, +where χ is a Dirichlet character. +1 +Introduction +Let’s take a set of prime numbers P, and denote with QP(x) the element count of the +set of all those positive square-free integers, which are smaller than-, or equal to x; and +which are only divisible by the elements of P. +In article [7] we examined how QP(x) behaves asymptotically based on the structure +of P in two scenarios. During the first scenario, P contained those primes which are not +greater than an x dependent bound λ(x), see [7, Prop. 1]. In the second scenario, P +contained those primes which are not greater than an x dependent bound λ(x), and which +fall into certain congruence classes modulo q, see [7, Prop. 2]. +In this article, we go further, and render the structure of P more complex. We would +want to restrict ourselves to primes p, for which a certain integer a is quadratic residue +modulo p, but to generalise, we are going to require that χ(p) = 1 holds, where χ is a +Dirichlet character. This covers our goal, as the only real valued primitive non-principal +χ(n) are given for positive n by the Kronecker symbol (D|n), where D is a fundamental +discriminant, see [9]. +MSC 2020: 11M06, 11M20, 11N36, 11N37, 11N69 +Keywords: Square-free numbers, Combinatorial sieve, Dirichlet character, Square-free numbers in +arithmetic progressions, L-functions, Euler product +Affiliation: +G´abor Rom´an – E¨otv¨os Lor´and University, Budapest, Hungary +E-mail: rogpaai@inf.elte.hu +arXiv:2301.02377v1 [math.NT] 6 Jan 2023 + +=P sciences2 +G´abor Rom´an +The results will be of course similar to the results in article [7], heavily depending on +the conductor q(χ) of the Dirichlet character χ in context. +Proposition 1.1. Let χ be a real valued non-principal Dirichlet character. Furthermore, let +λ : R → [1, +∞) be a monotone increasing function which is in o(x1/2), and let P contain +all the primes p which are not greater than λ(x), and for which χ(p) = 1. +Then for every ε > 0, there exit real constants a1 and a2 such that +ea1 +ln q(χ) +ln λ(x) x +ln x +� +ln λ(x) +� +q(χ)ε ≪ QP(x) ≪ ea2 +ln q(χ) +ln λ(x) x +ln x +� +ln q(χ) +� +ln λ(x) +(1) +as x → +∞. In addition, if χ is primitive, q(χ) is big enough, and L(s, χ) has no real +zero in the interval (1 − cχ, 1), then there exists a real constant a3 such that +ea3 +ln q(χ) +ln λ(x) x +ln x +� +ln λ(x) +� +ln q(χ) +≪ QP(x) +(2) +where cχ ≤ 1 is a χ dependant positive constant. +We can see that for primitive χ, when q(χ) is big enough, and L(s, χ) doesn’t have a +real zero close to 1, then we can cancel the ln q(χ) terms in the lower bounds by choosing +a λ(x) which contains q(χ)ε, for ε > 0. +Assuming the Riemann hypothesis for L(s, χ) we can drop the primitiveness, and the +bounds are perturbed only by expressions containing ln ln q(χ) instead of ln q(χ). +Proposition 1.2. Let χ be a real valued non-principal Dirichlet character. Then choose a +function λ, and with it define the set P as in the case of proposition 1.1. +If the Riemann hypothesis holds for L(s, χ), then there exist real constants a4 and a5 +such that +ea4 +ln ln q(χ) +ln λ(x) +x +ln x +� +ln λ(x) +� +ln ln q(χ) +≪ QP(x) ≪ ea5 +ln ln q(χ) +ln λ(x) +x +ln x +� +ln ln q(χ) +� +ln λ(x) +(3) +as x → +∞ and q(χ) → +∞. +A natural extension of these results would be to only allow P to contain primes which +are congruent to some mi modulo q, where q > 0 is an integer, and the m1, . . . , mk naturals +are pairwise distinct relative primes to q. We are going to restrict ourselves to the case +when q = 4; and either m = 1, or m = 3. Concerning the technicalities of this restriction +see section 3. +Proposition 1.3. Let χ be a real valued primitive non-principal Dirichlet character; and +either let m = 1, or m = 3. Furthermore, let λ : R → [1, +∞) be a monotone increasing +function which is in o(x1/2), and let P contain all those primes p which are not greater +than λ(x), for which χ(p) = 1, and for which p ≡ m (mod 4) holds. + +On square-free numbers generated from given sets of primes II +3 +When m = 1, then for every ε > 0, there exit real constants b1 and b2 such that +eb1 +ln q(χ) +ln λ(x) x +ln x +4� +ln λ(x) +� +q(χ)ε ≪ QP(x) ≪ eb2 +ln q(χ) +ln λ(x) x +ln x +� +ln q(χ) +4� +ln λ(x) +(4) +as x → +∞. In addition, if q(χ) is big enough, and L(s, χ) has no real zero in the interval +(1 − cχ, 1), then there exists a real constant b3 such that +eb3 +ln q(χ) +ln λ(x) x +ln x +4� +ln λ(x) +� +ln q(χ) +≪ QP(x) +(5) +where cχ ≤ 1 is a χ dependant positive constant. +When m = 3, then for every ε > 0, there exit real constants b4 and b5 such that +eb4 +ln q(χ) +ln λ(x) x +ln x +4� +ln λ(x) +4� +q(χ)ε ln q(χ) +≪ QP(x) ≪ eb5 +ln q(χ) +ln λ(x) x +ln x +4� +q(χ)ε ln q(χ) +4� +ln λ(x) +(6) +as x → +∞. In addition, if q(χ) is big enough, L(s, χ) has no real zero in the interval +(1 − cχ, 1), then there exists a real constant b6 such that +QP(x) ≍ eb6 +ln q(χ) +ln λ(x) x +ln x +4� +ln λ(x) +(7) +where cχ ≤ 1 is a χ dependant positive constant. +As in the previous case, we can get much better results assuming that the Riemann +hypothesis holds for L(s, χ). +Proposition 1.4. Let χ be a real valued primitive non-principal Dirichlet character; and +either let m = 1, or m = 3. Then choose a function λ, and with it define the set P as in +the case of proposition 1.3. +If the Riemann hypothesis holds for L(s, χ), and when m = 1, then there exist real +constants b7, and b8 such that +eb7 +ln ln q(χ) +ln λ(x) +x +ln x +4� +ln λ(x) +� +ln ln q(χ) +≪ QP(x) ≪ eb8 +ln ln q(χ) +ln λ(x) +x +ln x +� +ln ln q(χ) +4� +ln λ(x) +(8) +as x → +∞ and q(χ) → +∞. +In the same setting, but with m = 3, there exists a real constant b9 such that +QP(x) ≍ eb9 +ln ln q(χ) +ln λ(x) +x +ln x +4� +ln λ(x) +(9) +as x → +∞ and q(χ) → +∞. + +4 +G´abor Rom´an +2 +Proofs +Throughout the proofs, when the index of a summation is p, or the index of a product +is p, then p takes its values from the set of primes. +Lemma 2.1. Let χ be a real valued non-principal Dirichlet character. +Then we have +� +p≤y +χ(p) ln +� +1 − 1 +p +� += − ln L(1, χ) + O +� 1 +ln y +L′ +L (1, χ) +� ++ O(1) +as y → +∞. +Proof. Fix a real valued non-principal Dirichlet character χ. We begin the proof by showing +that +� +p≤y +χ(p) +p += ln L(1, χ) + O +� 1 +ln y +L′ +L (1, χ) +� ++ O(1). +(10) +holds as y → +∞. +• First we show that the equality +� +p +χ(p) +p += ln L(1, χ) + O(1) +(11) +holds. Based on [5, Sec. 5.9], when χ is a non-principal (or in their case non-trivial) +character, then L(s, χ) is entire. By this, there is no pole at s = 1, so we have that +the Euler product form +L(1, χ) = +� +p +� +1 − χ(p) +p +�−1 +(12) +see [5, Sec. 5.1, Sec. 5.9], or [2, Sec. 11.5], converges. For every non-principal +Dirichlet character, L(1, χ) ̸= 0, see [2, Thm. 6.20, Lem. 7.7], and as χ is real +valued, L(1, χ) is a positive real number, so we can take the logarithm of both sides +of (12) to get +ln L(1, χ) = − +� +p +ln +� +1 − χ(p) +p +� += +� +p +χ(p) +p ++ +� +p +∞ +� +k=2 +χ(p)k +kpk +where we could use the Mercator series, see [1, 4.1.24], as |χ(p)/p| < 1. Based on +the sum of the geometric series, see [1, 3.6.10], we have +� +p +∞ +� +k=2 +���� +χ(p)k +kpk +���� ≤ +� +p +∞ +� +k=2 +1 +pk = +� +p +1 +p(p − 1) < +� +p +1 +p2 +(13) +which is finite. + +On square-free numbers generated from given sets of primes II +5 +• Next we show that +� +y

1. +Now we prove proposition 1.1. +Proof. Fix a real valued non-principal Dirichlet character χ, and select a function λ sat- +isfying the requirements of proposition 1.1. According article [7], we have to bound the +product +� +p≤x1/2 +� +1 − +1 +p + 1 +� +α(λ(x))−1 ≍ +1 +ln x +� +L(1, χ) +� +ln λ(x)eO +� +1 +ln λ(x) +L′ +L (1,χ) +� +(21) +as x → +∞, where we have used [7, Lem. 2], and lemma 2.2. When χ is a non-principal +character, then based on article [3] we have +cεq(χ)−ε < L(1, χ) < ln q(χ) +(22) +where ε is any positive number and cε is a positive number depending on ε. Also, based +on [5, Prop. 5.7] we have +L′ +L (1, χ) ≪ ln q(χ) +(23) +where the implied constant being absolute. Using these bounds in expression (21) and the +method from article [7] we can get the bounds in expression (1). +Assuming that χ is primitive, q(χ) is big enough, and that there exists a positive +constant cχ ≤ 1 such that L(s, χ) has no real zero in the interval (1 − cχ, 1), we have +1 +ln q(χ) ≪ L(1, χ) +(24) +where the implied constant being positive and absolute, see article [4]. Using this bound +in expression (21) and the method from article [7] we get the bound in expression (2). + +8 +G´abor Rom´an +The proof of proposition 1.2 is the following. +Proof. Fix a real valued non-principal Dirichlet character χ, and select a function λ sat- +isfying the requirements of proposition 1.2. We are going to bound expression (21), but +with bounds based on the Riemann hypothesis. If χ is a real valued non-principal Dirichlet +character, and we assume that the Riemann hypothesis holds for L(s, χ), then based on +[6, Thm. 1] we have +1 + o(1) +ε1 ln ln q(χ) < L(1, χ) < (1 + o(1))ε1 ln ln q(χ) +(25) +as q(χ) → ∞, where ε1 and ε2 are real constants. +(As a side note, infinitely many +real primitive characters χ satisfy these inequalities without assuming that the Riemann +hypothesis holds for L(s, χ), see article [3].) In the same setting, based on [5, Thm. 5.17] +we have +L′ +L (1, χ) ≪ ln ln q(χ) +(26) +where the implied constant being absolute. Using these bounds in expression (21) and the +method from article [7] we get the bounds in expression (3). +To prove proposition 1.3 and proposition 1.4 we introduce the following function. +βm(y) := +� +p≤y +p≡m(4) +χ(p)=1 +� +1 − +1 +p + 1 +� +Lemma 2.3. Let χ be a real valued non-principal Dirichlet character. +Then we have +β1(y) ≍ +1 +4� +L(1, χ)L(1, χχ4,3) +1 +4√ln yeO +� +1 +ln y +L′ +L (1,χ) +� ++O +� +1 +ln y +L′ +L (1,χχ4,3) +� +(27) +and +β3(y) ≍ +4 +� +L(1, χχ4,3) +L(1, χ) +1 +4√ln yeO +� +1 +ln y +L′ +L (1,χ) +� ++O +� +1 +ln y +L′ +L (1,χχ4,3) +� +(28) +as y → +∞, where χ4,3 is the non-principal Dirichlet character modulo 4. +Proof. Fix a real valued non-principal Dirichlet character χ, and let either m = 1, or +m = 3. We can rewrite βm(y) as +� +p≤y +p≡m(4) +χ(p)=1 +� +1 − +1 +p + 1 +� += +� +p≤y +p≡m(4) +χ(p)=1 +� +1 − 1 +p2 +�−1 � +p≤y +p≡m(4) +χ(p)=1 +� +1 − 1 +p +� + +On square-free numbers generated from given sets of primes II +9 +where the first product on the right hand side can be bounded by a small positive constant. +Taking the logarithm of the second product on the right hand side we get +� +p≤y +p≡m(4) +χ(p)=1 +ln +� +1 − 1 +p +� += 1 +2 +� +p≤y +p≡m(4) +(1 + χ(p)) ln +� +1 − 1 +p +� +where we can split the finite sum on the right hand side as +1 +2 +� +p≤y +p≡m(4) +ln +� +1 − 1 +p +� ++ 1 +2 +� +p≤y +p≡m(4) +χ(p) ln +� +1 − 1 +p +� +. +(29) +Based on [2, Thm. 6.16] we can write the second sum in expression (29) as +1 +2 +� +p≤y +χ(p) ln +� +1 − 1 +p +� +1 +ϕ(4) +� +χ4 +χ4(p)χ4(m) +(30) +where the internal sum iterates through the ϕ(4) Dirichlet characters modulo 4. There +are two Dirichlet characters modulo 4; we are going to denote them as χ4,1 (the principal +character), and as χ4,3 (the non-principal character). Splitting the internal sum, we get +χ4,1(m) +4 +� +p≤y +χ(p)χ4,1(p) ln +� +1 − 1 +p +� ++ χ4,3(m) +4 +� +p≤y +χ(p)χ4,3(p) ln +� +1 − 1 +p +� +. +(31) +Concerning the sum on the left hand side of expression (31), as χ4,1(m) = 1; and as +χ4,1(p) = 1 when (p, 4) = 1, otherwise χ4,1(p) = 0, we have +1 +4 +� +p≤y +(p,4)=1 +χ(p) ln +� +1 − 1 +p +� += 1 +4 +� +p≤y +χ(p) ln +� +1 − 1 +p +� ++ O(1) +where we can use lemma 2.1 to get +−1 +4 ln L(1, χ) + O +� 1 +ln y +L′ +L (1, χ) +� ++ O(1). +Concerning the sum on the right hand side of expression (31), χχ4,3 is a real valued non- +principal character, so we can use lemma 2.1 again to get +−χ4,3(m) +4 +ln L(1, χχ4,3) + O +� 1 +ln y +L′ +L (1, χχ4,3) +� ++ O(1). +Substituting these result in expression (29), and exponentiating, in the case when m = 1, +we get expression (27) as χ4,3(1) = 1, and because +� +p≤y +p≡m(4) +� +1 − 1 +p +� +≍ +1 +√ln y + +10 +G´abor Rom´an +based on the article of Williams [11]. Similarly in the case when m = 3 we get expression +(28) as χ4,3(3) = −1. +Now we proof proposition 1.3. +Proof. Fix a real valued primitive non-principal Dirichlet character χ; and either let m = 1, +or m = 3. Furthermore select a function λ satisfying the requirements of proposition 1.3. +Based on article [7], we have to bound the product +� +p≤x1/2 +� +1 − +1 +p + 1 +� +βm(λ(x))−1 +(32) +when m = 1, and separately when m = 3. +When m = 1, then expression (32) is asymptotic to +1 +ln x +4� +L(1, χ)L(1, χχ4,3) +4� +ln λ(x)eO +� +1 +ln y +L′ +L (1,χ) +� ++O +� +1 +ln y +L′ +L (1,χχ4,3) +� +(33) +as x → +∞, where we have used [7, Lem. 2] and lemma 2.3. Using the bounds from +expression (22) and expression (23) we can bound expression (33) from below as +1 +ln x +4� +ln λ(x) +4� +q(χ)εq(χχ4,3)εeO +� +1 +ln y (ln q(χ)+ln q(χχ4,3)) +� +and as +1 +ln x +4� +ln q(χ) +4� +ln q(χχ4,3) +4� +ln λ(x)eO +� +1 +ln y (ln q(χ)+ln q(χχ4,3)) +� +from above. As χ and χ4,3 are both primitive, their product χχ4,3 is primitive too, see [10, +Ch. 3]. But then q(χχ4,3) ∈ O(q(χ)), see [5, Sec. 3.3]. Based on this and on the method +in article [7] we get the bounds in expression (4). +Assuming that q(χ) is big enough, and that there exists a positive constant cχ ≤ 1 such +that L(s, χ) has no real zero in the interval (1−cχ, 1), we can use bound (24) in expression +(33) to get +1 +ln x +4� +ln λ(x) +4� +ln q(χ) 4� +ln q(χχ4,3) +eO +� +1 +ln y (ln q(χ)+ln q(χχ4,3)) +� +from where we can get bound (5) based on the previous train of thoughts. +When m = 3, then expression (32) is asymptotic to +1 +ln x +4 +� +L(1, χ) +L(1, χχ4,3) +4� +ln λ(x)eO +� +1 +ln y +L′ +L (1,χ) +� ++O +� +1 +ln y +L′ +L (1,χχ4,3) +� +(34) +as x → +∞, where we have used [7, Lem. 2] and lemma 2.3 again. As in the previous +case, we can bound expression (34) from below as +1 +ln x +4 +� +q(χ)−ε +ln q(χχ4,3) +4� +ln λ(x)eO +� +1 +ln y (ln q(χ)+ln q(χχ4,3)) +� + +On square-free numbers generated from given sets of primes II +11 +and from above as +1 +ln x +4 +� +ln q(χ) +q(χχ4,3)−ε +4� +ln λ(x)eO +� +1 +ln y (ln q(χ)+ln q(χχ4,3)) +� +from where we get the bounds in expression (6). +Assuming that q(χ) is big enough, and that there exists a positive constant cχ ≤ 1 such +that L(s, χ) has no real zero in the interval (1−cχ, 1), we can use bound (24) in expression +(34). +The logarithmic contribution of the terms L(1, χ) and L(1, χχ4,3) “cancel” each +other, and we get asymptotic (7). +And finally, the proof of proposition 1.4 is the following. +Proof. Fix a real valued primitive non-principal Dirichlet character χ; and either let m = 1, +or m = 3. We use the same method as in the proof of proposition 1.3, but this time we +assume that the Riemann hypothesis holds for L(s, χ). +When m = 1, then we can use the bounds from expression (25) and expression (26) to +bound expression (33) from below as +1 +ln x +4� +ln λ(x) +4� +ln ln q(χ) 4� +ln ln q(χχ4,3) +eO +� +1 +ln y (ln ln q(χ)+ln ln q(χχ4,3)) +� +and from above as +1 +ln x +4� +ln ln q(χ) +4� +ln ln q(χχ4,3) +4� +ln λ(x)eO +� +1 +ln y (ln ln q(χ)+ln ln q(χχ4,3)) +� +as x → +∞ and as q(χ) → +∞. Using the same train of thought as in the proof of +proposition 1.3, we get the bounds in expression (8). +When m = 3, then using the above applied bounds (25) and (26) we get the asymptotic +(9) from asymptotic (34) via “cancellation” again. +3 +Remarks +As we have already mentioned in section 1, before proposition 1.3, a natural extension +of proposition 1.1 and proposition 1.2 would be to only allow P to contain primes which +are congruent to some mi modulo q, where q > 0 is an integer, and the m1, . . . , mk naturals +are pairwise distinct relative primes to q. In the case of modulo 4, the results were already +different for distinct m, so we can expect a similar outcome for larger moduli. However a +general strategy for the generalisation could go along the following train of thoughts. We +would have to supply an asymptotic for the product +� +p≤y +p≡m(q) +χ(p)=1 +� +1 − +1 +p + 1 +� + +12 +G´abor Rom´an +which could be done by the refinement of lemma 2.3, and its proof. If we follow this path, +then expression (30) will turn into +1 +2 +� +p≤y +χ(p) ln +� +1 − 1 +p +� 1 +ϕ(q) +� +χq +χq(p)χq(m) +where we can separate the principal character from the non-principal ones as +χq,1(m) +ϕ(q) +� +p≤y +χ(p)χq,1(p) ln +� +1 − 1 +p +� +and what remains is +1 +ϕ(q) +� +χq̸=χq,1 +χq(m) +� +p≤y +χ(p)χq(p) ln +� +1 − 1 +p +� +. +Due to the fact that χq,1 is the principal character, the first sum can be handled with our +already presented techniques. The double sum is more problematic. On the one hand, for +the internal sum we would have to refine lemma 2.1 and its proof. We would have to make +sure that when χ is complex valued, then we can take the logarithm of L(1, χ) and its +product form; furthermore that the values of these two logarithms match. On the other +hand, we would have to obtain a good estimation for the external sum. +References +[1] Abramowitz M. and Stegun I. A.: Handbook of Mathematical Functions with Formulas, Graphs, +and Mathematical Tables. Dover publications (1972). +[2] Apostol T. M.: Introduction to Analytic Number Theory. Springer-Verlag (1976). +[3] Bateman P. T. and Chowla S. and Erd˝os P.: Remarks on the size of L(1, χ). Publ. Math. +Debrecen 1 (2-4) (1950) 165–182. +[4] Hoffstein J.: On the Siegel–Tatuzawa theorem. Acta. Arith. 38 (2) (1980) 168–174. +[5] Iwaniec H. and Kowalski E.: Analytic Number Theory. A.M.S. Colloquium Publications (2004). +[6] Littlewood J. E.: On the class-number of the corpus P( +√ +−k). Proc. London Math. Soc. 27 (1) +(1928) 358–372. +[7] Rom´an G.: On square-free numbers generated from given sets of primes. Comm. Math. 30 (1) +(2022) 229–237. +[8] Rosser J. B. and Schoenfeld L.: Approximate formulas for some functions of prime numbers. +Illinois Journal of Mathematics 6 (1) (1962) 64–94. +[9] Walfisz A.: Zur additiven Zahlentheorie II.. Math. Z. 40 (1) (1936) 592–607. +[10] Washington L. C.: Introduction to Cyclotomic Fields. Springer-Verlag (1982). +[11] Williams K.S.: Mertens’ Theorem for Arithmetic Progressions. J. Number Theory 6 (5) (1974) +353–359. +Received: Received date +Accepted for publication: Accepted date +Communicated by: Handling Editor + diff --git a/1dE0T4oBgHgl3EQfdgBe/content/tmp_files/load_file.txt b/1dE0T4oBgHgl3EQfdgBe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..194ef973b1ceb135068ae37b52d8e287c67f3ed6 --- /dev/null +++ b/1dE0T4oBgHgl3EQfdgBe/content/tmp_files/load_file.txt @@ -0,0 +1,304 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf,len=303 +page_content='Communications in Mathematics n (2023), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' m, 00–12 DOI: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='46298/cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='ABCD ©2023 G´abor Rom´an This is an open access article licensed under the CC BY-SA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='0 1 On square-free numbers generated from given sets of primes II G´abor Rom´an Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' We progress with the investigation started in article [7], namely the anal- ysis of the asymptotic behaviour of QP(x) for different sets P, where QP(x) is the element count of the set containing those positive square-free integers, which are smaller than-, or equal to x, and which are only divisible by the elements of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' We study how QP(x) behaves when we require that χ(p) = 1 must hold for every p ∈ P, where χ is a Dirichlet character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' 1 Introduction Let’s take a set of prime numbers P, and denote with QP(x) the element count of the set of all those positive square-free integers, which are smaller than-, or equal to x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' and which are only divisible by the elements of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' In article [7] we examined how QP(x) behaves asymptotically based on the structure of P in two scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' During the first scenario, P contained those primes which are not greater than an x dependent bound λ(x), see [7, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' In the second scenario, P contained those primes which are not greater than an x dependent bound λ(x), and which fall into certain congruence classes modulo q, see [7, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' In this article, we go further, and render the structure of P more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' We would want to restrict ourselves to primes p, for which a certain integer a is quadratic residue modulo p, but to generalise, we are going to require that χ(p) = 1 holds, where χ is a Dirichlet character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' This covers our goal, as the only real valued primitive non-principal χ(n) are given for positive n by the Kronecker symbol (D|n), where D is a fundamental discriminant, see [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' MSC 2020: 11M06, 11M20, 11N36, 11N37, 11N69 Keywords: Square-free numbers, Combinatorial sieve, Dirichlet character, Square-free numbers in arithmetic progressions, L-functions, Euler product Affiliation: G´abor Rom´an – E¨otv¨os Lor´and University, Budapest, Hungary E-mail: rogpaai@inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='elte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='hu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='02377v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='NT] 6 Jan 2023 =P sciences2 G´abor Rom´an The results will be of course similar to the results in article [7], heavily depending on the conductor q(χ) of the Dirichlet character χ in context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Let χ be a real valued non-principal Dirichlet character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Furthermore, let λ : R → [1, +∞) be a monotone increasing function which is in o(x1/2), and let P contain all the primes p which are not greater than λ(x), and for which χ(p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Then for every ε > 0, there exit real constants a1 and a2 such that ea1 ln q(χ) ln λ(x) x ln x � ln λ(x) � q(χ)ε ≪ QP(x) ≪ ea2 ln q(χ) ln λ(x) x ln x � ln q(χ) � ln λ(x) (1) as x → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' In addition, if χ is primitive, q(χ) is big enough, and L(s, χ) has no real zero in the interval (1 − cχ, 1), then there exists a real constant a3 such that ea3 ln q(χ) ln λ(x) x ln x � ln λ(x) � ln q(χ) ≪ QP(x) (2) where cχ ≤ 1 is a χ dependant positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' We can see that for primitive χ, when q(χ) is big enough, and L(s, χ) doesn’t have a real zero close to 1, then we can cancel the ln q(χ) terms in the lower bounds by choosing a λ(x) which contains q(χ)ε, for ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Assuming the Riemann hypothesis for L(s, χ) we can drop the primitiveness, and the bounds are perturbed only by expressions containing ln ln q(χ) instead of ln q(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Let χ be a real valued non-principal Dirichlet character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Then choose a function λ, and with it define the set P as in the case of proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' If the Riemann hypothesis holds for L(s, χ), then there exist real constants a4 and a5 such that ea4 ln ln q(χ) ln λ(x) x ln x � ln λ(x) � ln ln q(χ) ≪ QP(x) ≪ ea5 ln ln q(χ) ln λ(x) x ln x � ln ln q(χ) � ln λ(x) (3) as x → +∞ and q(χ) → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' A natural extension of these results would be to only allow P to contain primes which are congruent to some mi modulo q, where q > 0 is an integer, and the m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' , mk naturals are pairwise distinct relative primes to q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' We are going to restrict ourselves to the case when q = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' and either m = 1, or m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Concerning the technicalities of this restriction see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Let χ be a real valued primitive non-principal Dirichlet character;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' and either let m = 1, or m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Furthermore, let λ : R → [1, +∞) be a monotone increasing function which is in o(x1/2), and let P contain all those primes p which are not greater than λ(x), for which χ(p) = 1, and for which p ≡ m (mod 4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' On square-free numbers generated from given sets of primes II 3 When m = 1, then for every ε > 0, there exit real constants b1 and b2 such that eb1 ln q(χ) ln λ(x) x ln x 4� ln λ(x) � q(χ)ε ≪ QP(x) ≪ eb2 ln q(χ) ln λ(x) x ln x � ln q(χ) 4� ln λ(x) (4) as x → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' In addition, if q(χ) is big enough, and L(s, χ) has no real zero in the interval (1 − cχ, 1), then there exists a real constant b3 such that eb3 ln q(χ) ln λ(x) x ln x 4� ln λ(x) � ln q(χ) ≪ QP(x) (5) where cχ ≤ 1 is a χ dependant positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' When m = 3, then for every ε > 0, there exit real constants b4 and b5 such that eb4 ln q(χ) ln λ(x) x ln x 4� ln λ(x) 4� q(χ)ε ln q(χ) ≪ QP(x) ≪ eb5 ln q(χ) ln λ(x) x ln x 4� q(χ)ε ln q(χ) 4� ln λ(x) (6) as x → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' In addition, if q(χ) is big enough, L(s, χ) has no real zero in the interval (1 − cχ, 1), then there exists a real constant b6 such that QP(x) ≍ eb6 ln q(χ) ln λ(x) x ln x 4� ln λ(x) (7) where cχ ≤ 1 is a χ dependant positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' As in the previous case, we can get much better results assuming that the Riemann hypothesis holds for L(s, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Let χ be a real valued primitive non-principal Dirichlet character;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' and either let m = 1, or m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Then choose a function λ, and with it define the set P as in the case of proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' If the Riemann hypothesis holds for L(s, χ), and when m = 1, then there exist real constants b7, and b8 such that eb7 ln ln q(χ) ln λ(x) x ln x 4� ln λ(x) � ln ln q(χ) ≪ QP(x) ≪ eb8 ln ln q(χ) ln λ(x) x ln x � ln ln q(χ) 4� ln λ(x) (8) as x → +∞ and q(χ) → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' In the same setting, but with m = 3, there exists a real constant b9 such that QP(x) ≍ eb9 ln ln q(χ) ln λ(x) x ln x 4� ln λ(x) (9) as x → +∞ and q(χ) → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' 4 G´abor Rom´an 2 Proofs Throughout the proofs, when the index of a summation is p, or the index of a product is p, then p takes its values from the set of primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Let χ be a real valued non-principal Dirichlet character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Then we have � p≤y χ(p) ln � 1 − 1 p � = − ln L(1, χ) + O � 1 ln y L′ L (1, χ) � + O(1) as y → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Fix a real valued non-principal Dirichlet character χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' We begin the proof by showing that � p≤y χ(p) p = ln L(1, χ) + O � 1 ln y L′ L (1, χ) � + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' (10) holds as y → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' First we show that the equality � p χ(p) p = ln L(1, χ) + O(1) (11) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Based on [5, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='9], when χ is a non-principal (or in their case non-trivial) character, then L(s, χ) is entire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' By this, there is no pole at s = 1, so we have that the Euler product form L(1, χ) = � p � 1 − χ(p) p �−1 (12) see [5, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='1, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='9], or [2, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='5], converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' For every non-principal Dirichlet character, L(1, χ) ̸= 0, see [2, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='20, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='7], and as χ is real valued, L(1, χ) is a positive real number, so we can take the logarithm of both sides of (12) to get ln L(1, χ) = − � p ln � 1 − χ(p) p � = � p χ(p) p + � p ∞ � k=2 χ(p)k kpk where we could use the Mercator series, see [1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='24], as |χ(p)/p| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' Based on the sum of the geometric series, see [1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content='10], we have � p ∞ � k=2 ���� χ(p)k kpk ���� ≤ � p ∞ � k=2 1 pk = � p 1 p(p − 1) < � p 1 p2 (13) which is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'} +page_content=' On square-free numbers generated from given sets of primes II 5 Next we show that � y

1013 Mʘ). +1. Introduction +The basic properties of galaxies, supermassive black holes, and the intra-group/intra-cluster +medium cannot be understood without considering the impact of the return of mass, metals, +energy, and momentum from both populations of massive stars (stellar winds and supernovae) +and supermassive black holes (winds and jets). Examples include the shape of the stellar mass +function, the quenching and subsequent suppression of star-formation in massive galaxies, the +mass-metallicity and mass-radius relations, the Kennicutt-Schmidt law of star-formation, and +the group/cluster X-ray luminosity-temperature relation (see reviews by Somerville & Davé +2015, Naab & Ostriker 2017, Donahue & Voit 2022). + +This input of energy and momentum from massive stars and black holes is generically referred +to as feedback. Even the highest resolution numerical simulations cannot fully include all the +relevant physics ab-initio, and must rely on “sub-grid physics” (essentially, recipes for processes +that cannot be spatially resolved). The same is true of semi-analytic models. This underscores +the importance of using observations to inform the choices that are made in simulations and +models. While there is now a considerable body of data on feedback from both massive stars +and supermassive back holes (e.g. Veilleux et al. 2020; McNamara & Nulsen 2007; Thompson & +Heckman 2023), we still have a very incomplete understanding of the impact of this feedback +on the surrounding gas. +In this paper, we will take a different approach from previous investigations of feedback, and +try to compile a global inventory (that is, integrated over cosmic time) of the amount of kinetic +energy and momentum per co-moving volume element injected by massive stars and +supermassive black holes. We will then compare the respective importance of these feedback +sources as a function of time and of galaxy and black hole mass. +2. Methodology +2.1 Massive Stars +To compute the total amount of kinetic energy injected by massive stars (stellar winds and +supernovae) per unit volume, we start with the present-day amount of stellar mass per unit +volume. We use the compilation in Madau & Dickinson (2016), adjusted to a standard Chabrier +Initial Mass Function (IMF - Chabrier, 2003). This value is 3.4 x 108 Mʘ Mpc-3. To compute the +corresponding amount of kinetic energy we need to first correct this to account for stars that +were formed but are no longer present. This requires multiplication by 1/(1-R), where R is the +so-called Returned Fraction, which is 0.3 for a Chabrier IMF. Thus, the total mass of stars +formed per unit volume is 4.9 x 108 Mʘ Mpc-3. Starburst99 (Leitherer et al. 1999) models for a +Chabrier IMF yield a total kinetic energy in stellar winds and supernova ejecta of 6.9 x 1015 erg +gm-1. This then gives a value for the kinetic energy density due to stars of Ustar = 6.8 x 1057 erg +Mpc-3. +How much of this kinetic energy is available to supply feedback? The stellar ejecta initially +carrying the energy collide and their kinetic energy is converted to thermal energy. This hot gas +can then expand and flow outward with the thermal energy being converted back into kinetic +energy (e.g. Chevalier & Clegg 1985). Some of the initial thermal energy can be lost through +radiative cooling, so that only a fraction εstar remains to provide feedback. Numerical +simulations that represent typical conditions in low-z star-forming galaxies yield εstar ≈ 0.1 (Kim +et al. 2020), with a value that increases with the star-formation rate per unit area (SFR/A). At +the much higher values of SFR/A seen in starbursts (e.g. Kennicutt & Evans 2012), simulations +and models predict far greater efficiency, with εstar ≈ 0.3 to 1.0 (Schneider et al. 2020; Fielding & +Bryan 2022). This is consistent with X-ray observations of the H-like and He-like Fe Kα emission- +lines in starburst galaxies from the very hot (108 K) gas created as the stellar ejecta are +thermalized through shocks (Thompson & Heckman 2023). These results imply that rather little + +of the initial kinetic energy is lost through radiative cooling, and this is substantiated by +estimates of the rate of PΔV work done by the wind on the ambient gas (Thompson & Heckman +2023). While there are no such constraints on galaxies at high (z > 1) redshift, we do know that +these galaxies have values of SFR/A similar to those seen in low-z starbursts (e.g. Forster- +Schreiber & Wuyts 2020), and that galactic winds driven by massive stars at this epoch are both +ubiquitous and very similar to those seen in low-z starburst galaxies (see Thompson & Heckman +2023). Note that roughly 60% of the total present-day stellar mass was formed at z > 1, during +this “windy” epoch (Madau & Dickinson 2016). +The situation for momentum injection is less uncertain because momentum will be conserved +even in the face of significant radiative losses. We can simply use the methodology above but +use Starburst99 to compute the specific injection rate of momentum by massive stars +(supernovae, stellar winds, and radiation pressure). The value is 7.4 x 107 cm s-1, and for a total +stellar mass density of 4.8 x 108 Mʘ Mpc-3, this yields 7.1 x 1049 gm cm s-1 Mpc-3. +2.2 Black-Hole Driven Winds and Radiation Pressure +Winds driven by supermassive black holes are multi-phase and have been measured in a +number of different ways. Molecular outflows have been detected in both emission and +absorption (see the review by Veilleux et al. 2020). Calculating kinetic energy outflow rates is +conceptually straightforward. The luminosity of a CO transition can be converted into a total +molecular gas mass, albeit with uncertainties (Tacconi et al. 2020). The measured outflow +velocity and the radius of the outflow then yields a kinetic energy flux given as ½ Mgas vout3 rout-1. +For absorption, the OH column density and outflow velocity yields an outflow rate (for an +assumed outflow size and OH/H2 conversion factor). The first compilation of molecular outflows +by Fiore et al. (2017) implied typical kinetic energy fluxes of dEwind/dt ≈ 3% LBol, however more +recent compilations of measurements (Lutz et al. 2020, Lamperti et al. 2022 and private +communication) have yielded much smaller values (median of 0.1%). +Similarly, the outflow rates of warm ionized gas can be measured using the Hα or [OIII]5007 +luminosity and measured electron density to derive the total mass of ionized gas and then +measuring the outflow velocity and radius of the outflow to determine dEwind/dt. Different +recent measurements have come to drastically different results, with median values ranging +from as high as 1% of Lbol (Kakkad et al. 2022) to 0.3 % (Fiore et al. 2017), to 0.1% (Revalski et al. +2021), to 0.01% (Dall’Agnol de Oliveira 2021), to 0.0003% (Trindade Falcao et al. 2021). +An independent measurement of the outflow rate in the warm ionized gas comes from +observations of BAL QSOs. Here, the absorption-lines can provide a column density and outflow +velocity. Direct measurements of the electron densities can be made using the ratio of column +densities in lines arising from an excited state vs. the ground state. Photoionization models +using the observed ionizing luminosity (Q) and the inferred value of the ionization parameter +(U) then yield a size for the outflow: rout = (Q/4π ne c U)1/2 (Miller et al. 2020). With a velocity, +radius, and column density, the kinetic energy flux can be estimated. The results span a huge + +range, from 0.001% to 10% Lbol (median value of 0.3%). Highly ionized outflows are also +detected in about 40% of AGN (Tombesi et al. 2011) based on X-ray absorption-lines. However, +because the size scales of these outflows are so uncertain, the kinetic energy outflow rates are +also uncertain (by about two orders-of-magnitude, typically ranging between 0.01 and 1% of +LBol – Tombesi et al. 2012). +It is clear from the above that assigning a value for the ratio of dEwind/dt to Lbol is difficult. If we +take the median values of 0.1%, 0.3%, and 0.1% LBol for the molecular, warm-ionized, and +highly-ionized phases, we get a total value of 0.5% LBol. Multiplying this by the total bolometric +energy density per co-moving volume element volume produced by supermassive black holes +of Urad = 8.6 x 1058 erg Mpc-3 (Hopkins et al. 2007), yields Uwind = 4.3 x 1056 ergs Mpc-3. This is +6% as large as the value derived for massive stars. Using the present-day mass per unit volume +in supermassive black holes of ρBH = 5 x 105 Mʘ Mpc-3 (Hopkins et al. 2007) this wind energy +density can also be expressed as Uwind = 5 x 10-4 ρBH c2. +We can also consider the amount of momentum provided by AGN. An initial estimate is +implied by the momentum carried by radiation (Urad/c) where Urad is the total amount of radiant +energy per unit volume produced over cosmic time by AGN. This yields an amount of +momentum per unit volume of 2.9 x 1048 gm cm s-1 Mpc-3 (about 4% of the value for massive +stars). Since the momentum flux (in the non-relativistic case) is just twice the kinetic energy flux +divided by the outflow velocity, we need only consider the momentum carried by the molecular +and warm ionized flows (since the BAL QSO and X-ray outflows are over an order-of-magnitude +faster, but carry similar kinetic energy fluxes). +For the molecular outflows, the data in Lutz (2020) and Lamperti et al. (2022 and private +communication) yield median values of dpwind/dt = 1.0 and 0.7 LBol/c respectively. The near +equality is consistent with the idea that the molecular outflows are driven by radiation +pressure. If so, then combining radiation pressure and the molecular outflows would be double- +counting in the inventory of momentum. +As noted above, there is a very wide range in the ratio between the kinetic energy flux in the +warm ionized gas and the AGN bolometric luminosity, and this translates directly into +uncertainties in the ratio of momentum flux and radiation pressure for this gas phase. +Estimated median values of this ratio range from ≈10 (Kakkad et al. 2022), to ≈1 (Fiore et al. +2017; Revalski et al. 2021), to ≈0.1 (Dall’Agnol de Oliveira et al. 2021), to ≈0.01 (Trindade Falcao +et al. 2021). It appears that the momentum flux in the warm ionized outflows is not likely to be +significantly larger than those in the molecular gas or to that carried by radiation. +This represents a total injected momentum per unit volume of at most ≈1049 gm cm s-1 Mpc-3, +even if we simply add the three sources (radiation, molecular gas, ionized gas) together. This is +still an order of magnitude below the value for massive stars. + + +2.3 Black Hole-Driven Jets +The earliest evidence for the outflow of kinetic energy driven by supermassive black holes came +from observations of “double lobes” of synchrotron radio emission that straddled massive +elliptical galaxies (Baade & Minkowski 1954). Subsequent radio observations at high angular +resolution showed narrow collimated features (“jets”) linking the two lobes to the galactic +nucleus (see Miley 1980). +It is now possible to quantify the amount of kinetic energy carried by jets as a function of the +luminosity of the radio source that they power. This can be done by joint observations of the +radio and X-ray emission. The expanding radio sources inflate lobes of relativistic plasma, which +in X-rays can be observed as cavities in the surrounding hot gas. Bırzan et al. (2004,2008), Dunn +et al. (2005), Rafferty et al. (2006), and Cavagnolo et al. (2010) derived the pΔV work (energy) +associated with the cavities in a sample of massive galaxies, groups, and clusters, and used the +buoyancy timescale (e.g. Churazov et al. 2001) to estimate their ages. They combined these +cavity powers with the monochromatic 1.4 GHz radio luminosities to show that the two were +well-correlated. The largest uncertainty in this method is the determination of the cavity energy +from the measured pressure and volume: Ecav = fcavpΔV . For the relativistic plasma of the radio +lobes the enthalpy of the cavity is 4pΔV. Taking fcav = 4, Heckman & Best (2014) derived the +following best-fit relation from the cavity data: +1) dEjet/dt = 1.3 × 1038 (L1.4GHz/1026 W Hz−1)0.68 W +This empirical relation is very similar to predictions from theoretical models of radio jets. +Willett et al. (1999) used synchrotron properties to derive the relation: +2) dEjet/dt = 2.8 × 1036 (fW)3/2 (L1.4GHz/1026 W Hz−1)0.84 W +Here fW is a dimensionless factor (in the range 1 to 20) accounting for the uncertainties in the +extrapolation from the population of relativistic electrons that produce the observed radio +synchrotron emission to the total energy. Agreement with the X-ray cavity data implies fW ≈10 +to 20 (see Heckman & Best 2014). +We adopt the theoretical relation (equation 2), but calibrated by the cavity data (i.e. taking fW = +15) and use this to convert the radio luminosity function of AGN between z = 0.1 and 3 (Yuan et +al. 2017) into a measure of the evolution in the rate of kinetic energy injection per unit volume +by radio jets. +The results are shown in Figure 1, and show that the peak rate of kinetic energy injection by jets +occurs at a significantly lower redshift (z ≈ 1) than the peak rate due to massive stars and black- +hole-driven winds (z ≈ 2, as also shown in Figure 1). We then integrate the energy injection rate +by interpolating the values at z = 0.1, 0.5, 1.0, 2.0, and 3.0 and extrapolating from z = 0.1 to 0 +and from z = 3.0 to infinity (this extrapolation does not add significantly to the total - see Figure +1). This then gives a value for the time-integrated total kinetic energy per unit volume due to + +jets of Ujet = 2.6 x 1057 erg Mpc-3. These results are broadly in line with similar estimates derived +from low-frequency radio luminosity functions (Kondapally et al., private communication). +The time-integrated kinetic energy input from jets is ≈6 times larger than the value estimated +about for black-hole-driven winds, and 40% (400%) the total amount of kinetic energy +generated by massive stars for εstar = 1 (0.1). Alternatively, using the present-day mass per unit +volume in supermassive black holes of ρBH = 5 x 105 Mʘ (Hopkins et al. 2007) this jet energy +density can also be expressed as Ujet = 2.9 x 10-3 ρBH c2. +The kinetic energy carried by jets is in the form of relativistic bulk motion. In this case, the +momentum can be taken as p ≈ KE/c. The above value of Ujet then implies a momentum density +of 8.7 x 1046 gm cm s-1 Mpc-3. This is much less than the momentum carried by radiation and +winds from supermassive black holes, and the momentum produced by massive stars. Jets are +therefore far more important feedback sources in terms of kinetic energy than momentum. +2.4 The Bottom Line +For total kinetic energy inventory, the largest single source is either massive stars (for εstar > 0.4) +or jets (for εstar < 0.4). AGN winds are only important at the <10% level. For the total +momentum inventory, massive stars dominate (AGN contribute at the ≈10% level). The peak +rate of kinetic energy injection by jets occurs at a substantially lower redshift than that from +stars or AGN winds (z ≈ 1 and 2, respectively). These results are summarized in Table 1 and +Figure 1. +______________________________________________________________________________ +Table 1 – Summary of Feedback Inventory +1 +2 +3 +4 +5 +6 +Sample +Log ρ +Log sKE +Log ρKE +Log sp +ρp +Massive Stars +8.69 +-5.11 +57.83 +7.87 +49.85 +BH Winds +5.70 +-3.30 +56.63 +10.00 +49.00 +BH Jets +5.70 +-2.54 +57.43 +7.94 +46.94 +______________________________________________________________________________ +Notes: +Column 2 – The log of the present-day mass density of stars (row 3) and supermassive black holes (rows +4 and 5) formed over cosmic time in units of Mʘ Mpc-3. +Column 3 – The log of the specific kinetic energy released: energy per unit mass in stars (row 3 and black +holes (rows 4 and 5). Given in units of c2, and assuming εstar = 1.0. +Column 4 – The log of the amount of kinetic energy created per unit volume (in ergs Mpc-3). +Column 5 – The specific momentum created (momentum per unit mass in stars (row 3) and black holes +(rows 4 and 5). In units of cm s-1. +Column 6 – The log of the amount of momentum created per unit volume (gm cm s-1 Mpc-3). +______________________________________________________________________________ + + +Figure 1 – A plot of the amount of kinetic energy injected per Gyr and co-moving cubic Mpc as a function +of lookback time for massive stars (supernovae and stellar winds; black) and black-hole-driven jets (blue) +and winds (red). For massive stars we show the cases in which 100% (solid line) and 10% (dashed line) of +the kinetic energy created is delivered to the surroundings (i.e. not lost to radiative cooling). Note that +for momentum injection, massive stars dominate at all epochs, with the same time dependence as for +kinetic energy injection (i.e. as given by the solid black line). +3. Implications +3.1 For Galaxies +To assess the implications of these results for galaxy evolution, it is essential to consider the +dependences of feedback on the masses of both galaxies and supermassive black holes. We can +go beyond these simple global values and examine the relative importance of feedback (both +kinetic energy and momentum) as a function of the ratio of supermassive black hole mass to +galaxy stellar mass. In Figure 2 we show a plot of black hole vs. galaxy mass that is similar to +that in Heckman & Best (2014) for the z ≈ 0.1 universe (based on SDSS). In this case, these are +present day stellar masses, and would need to be increased by a factor 1/(1-R) = 1.42 to +represent the total mass of stars ever formed. The masses for the black holes were estimated +from the M-σ relation from McConnell & Ma (2013). In figure 2, we have color-coded the plot +by the fraction of galaxies in which star-formation has been quenched, which we define to be + +Redshift +0 +0.5 +1 +2 +346 +57.5 +57.0 +56.5 +56.0 +55.5 +55.0 +Supernovae (8star = 1.0) + - - Supernovae (star = 0.1) +54.5 +-AGNjets +AGNwinds +54.0 +0 +2 +4 +6 +8 +10 +12 +Lookback time / GyrSFR/Mstar < 10-11 yr-1. It is clear that the quenched fraction depends strongly on both the stellar +and black hole masses. +The mean relation between stellar and black hole mass in Figure 2 can be approximated as log +MBH = 2.0 log Mstar -14.0, implying MBH/Mstar α Mstar1.0 α MBH0.50. Thus, the relative importance of +feedback integrated over cosmic time from massive stars and black holes should be a strong +function of mass. Let us quantify this for kinetic energy and then for momentum. For kinetic +energy, in a given galaxy (and assuming that global averages can be applied to individual +galaxies; see below) the inventories above imply that the ratio KEBH/KEstar = 315 εstar-1 MBH/Mstar +(where Mstar is the present-day stellar mass). For momentum, the corresponding ratio is +pBH/pstar = 100 MBH/Mstar. We can then plot these relations in Figure 2 to see the regimes in +which feedback from supermassive black holes exceeds that from stars. For kinetic energy, we +show this separately for values of εstar = 0.1 and 1.0. + +Figure 2 – A plot of the distribution of SDSS galaxies in the plane of galaxy stellar mass vs. supermassive +black hole mass. The latter were estimated using the MBH vs. σ relation in McConnell & Ma (2013). The +relative numbers of galaxies in each bin are indicated by the green contours (increasing by factors of 2) +and the color-coding represents the fraction of galaxies that are quenched (SFR/Mstar < 10-11 yr-1). The +dark blue dashed line indicates where the momentum injected by black holes equals that from massive +stars. The two light blue dashed lines indicate where the kinetic energy from black holes equals that from +massive stars for values of εstar = 0.1 and 1.0 (see text). The transition from predominantly star forming +to predominantly quenched galaxies occurs near the relationship for εstar = 0.1. + +10 +1.0 +0.8 +Quenched fraction +0.6 +8 +0.4 +0.1 +KEBH +0.2 +6 +0.0 +10.0 +10.5 +11.0 +11.5 +12.0 +log1o(Stellar mass / Msun)In terms of momentum input, stars dominate over black holes in almost all cases. However, +considering kinetic energy, we find that the transition from galaxies that are mostly quenched +to those that are mostly star-forming occurs very near the dividing line between jet-dominated +feedback and stellar dominated feedback for a value of εstar ≈ 0.1. This is suggestive evidence +that quenching is driven by the feedback of kinetic energy from jets driven by supermassive +black holes. However, we caution that the transition from star forming to quiescent galaxies +also occurs at the transition from disk dominated to bulge dominated galaxies, so the causal +connections between galaxy structure, star formation, and black hole feedback are not entirely +clear. +We emphasize that the relations plotted in Figure 2 explicitly assume that the global relations +can be applied to individual galaxies, namely that the amount of feedback from massive stars in +a given galaxy is proportional to stellar mass and that the amount of feedback from jets is +proportional to the black hole mass. The former seems like a safe assumption, but the +dependence of the production of radio jets on black hole mass may be complex. We know that +there are essentially two populations of radio galaxies (e.g. Heckman & Best 2014). In one case +(“radiative mode”) the jets are launched by star-forming galaxies and are accompanied by +strong nuclear radiation (QSO-like). In the other class (“jet-mode”) the jets are launched by +quenched galaxies, with little accompanying nuclear radiation. The radiative mode becomes +more important at higher luminosities and at higher redshifts. For the jet-mode galaxies, +Sabater et al. (2019) find that the probability of producing a jet with a given luminosity depends +on both the stellar and black hole mass (and more strongly on the former). +The situation for radiative-mode radio galaxies is less clear, although the indications are that +any dependence of the ratio of KE/MBH on MBH or Mstar is weaker (e.g. Janssen et al. 2012, +Kondapally et al. 2022). In the context of Figure 2, it may be that the jet-mode is not the +dominant population in terms of actively quenching, since the jet-mode galaxies are already +quenched (instead, these may just ‘maintain’ a quenched state). If quenching is due to jets in +radiative-mode galaxies, the dividing line between quenched and star-forming galaxies in Figure +2 would imply that time-integrated amount of jet energy contributed by a radiative mode +galaxy is proportional to its black hole mass (i.e. the integrated ratio of jet kinetic energy and +energy carried by radiation is independent of black hole mass in these galaxies). +Another way to consider this is to ask how the amount of energy supplied by stars and by black +holes scales with the binding energy of the galaxy. We take Ebind ≈ Mstar vc2 where vc is the galaxy +circular velocity. The Tully-Fischer relation for disk galaxies (McGaugh et al. 2000) and the +Faber-Jackson relation for ellipticals (Bernardi et al. 2003) both imply vc α Mstar1/4. Thus, we +have Ebind α Mstar3/2. Given that KEstar α Mstar and KEBH α MBH α Mstar2, this implies that KEstar/Ebind +α Mstar-1/2, while KEBH/Ebind α Mstar1/2 α MBH1/4. This again underscores the fundamental +difference in the mass-dependence of feedback from massive stars and supermassive black +holes: feedback from stars becomes increasingly impactful on the galaxy as the mass decreases, +while feedback from black holes has greater impact as the mass increases. + +3.2 For the Intra-Group and Intra-Cluster Media +It has long been known that the basic observed properties of the hot gas in groups and clusters +of galaxies (Mhalo > 1013 Mʘ) are not consistent with simple models of purely gravitational +processes operating during the formation of these systems (see Donahue & Voit 2022 and +references therein). A particularly simple example of this is the observed relationship between +the X-ray temperature (a proxy for halo mass) and X-ray luminosity. As the halo masses +decrease, the observed X-ray luminosities fall further below the relationship expected simply +from gravitational infall and heating. These lower luminosities arise because the hot gas in +these less-massive halos is more spatially-extended than the dark matter, with the resulting +drop in gas density leading to lower X-ray luminosities. +This could be due to the feedback of energy injected into the hot gas, which “lifts” the gas +outward. As described above, there is direct observational evidence in the local universe of +radio jets delivering energy to the hot gas in groups and clusters. As discussed in Donahue & +Voit (2022), for this to be responsible for lifting the hot gas, an amount of kinetic energy equal +to ≈0.5% MBH/c2 must be delivered. This is close to the value for jets and AGN winds that we +estimated above of ≈0.34%. Note that this could be supplemented by the kinetic energy from +massive stars (which would be 0.25 to 2.5 the value for jets for εstar = 0.1 and 1.0 respectively). +4. Summary +Based on a global inventory of the amount of kinetic energy and momentum injected by +massive stars (stellar winds and supernovae), and by winds and jets driven by supermassive +black holes, we draw the following conclusions: +i) +The major sources of kinetic energy are massive stars and jets. Winds driven by +supermassive black holes provide <10% of the total. The global ratio of the kinetic +energy injected by massive stars to that injected by jets is 2.5 εstar (where εstar is the +fraction of injected energy from stars that is not lost to radiative cooling). +ii) +Massive stars are the dominant source of momentum injection (90% of the total). +AGN winds provide 10%, and radio jets are negligible. +iii) +The peak in the feedback from jets occurs at z ≈ 1, considerably later than the +contributions of AGN-winds and massive stars (peaking at z ≈ 2). +iv) +Since the ratio of the mass of the supermassive black hole to the galaxy stellar mass +increases steeply with mass, there will be a mass-dependence in the relative +importance of feedback from the two sources. +v) +For the assumptions that the total amount of kinetic energy from massive stars is +proportional to the galaxy’s stellar mass, and that the total amount of kinetic energy +from a supermassive black hole is proportional to its mass, we find that the +populations of quenched and star-forming galaxies occur in the regimes where +supermassive black hole feedback and massive star feedback dominate, respectively +(for a value of εstar ≈ 0.1). + +vi) +By comparing the amount of kinetic energy injected as a function of the binding +energy of a galaxy, we show that feedback becomes more impactful as galaxy mass +decreases for massive stars, but more impactful as galaxy mass increases for black +holes. +vii) +The global amount of kinetic energy injected by radio jets and AGN winds per unit +volume, combined with the supermassive black hole mass function, yields an +efficiency for producing kinetic energy in jets of 0.34% c2. This is very close to the +amount of energy needed to explain X-ray luminosity-temperature relation in groups +and clusters (0.5% c2). + +REFERENCES +Baade, W. & Minkowski, R. 1954, ApJ, 119, 206 +Bernardi, M 2003, AJ, 125, 1849 +Bırzan L., Rafferty D., McNamara B., Wise M., Nulsen P. 2004, ApJ, 607, 800 +Bırzan L., McNamara B., Nulsen P., Carilli C., Wise, M. 2008. ApJ, 686, 859 +Cavagnolo K., McNamara B., Nulsen P. et al. 2010, ApJ, 720, 1066 +Chabrier, G. 2003, PASP, 115, 763 +Chevalier, R., & Clegg, A. 1985, Nature, 317, 44 +Churazov, E., Bruggen, M., Kaiser C., Bohringer H., Forman W. 2001, Ap J, 554,261 +Dall'Agnol de Oliveira, B., Storchi-Bergmann, T., Kraemer, S. et al. 2021, MNRAS, 504, 3890 +Donahue, M., & Voit, G.M, 2022, Physics Reports, 973, 1 +Dunn, R., Fabian, A., Taylor, G. 2005, MNRAS, 364, 1343 +Fielding, D. & Bryan, G. 2022, ApJ, 984, 82 +Fiore, F., Feruglio, C., Shankar, F. et al. 2017, A&A, 601, 143 +Heckman, T. & Best, P.N. 2014, ARA&A, 52, 589 +Hopkins, P., Richards, G., & Hernquist, L. 2007, ApJ, 654, 731 +Janssen, R., Best, P.N., Röttgering, H., Brinchmann, J. 2012, A&A, 541, 62 +Kakkad, D.,Sani, E., Rojas, A. et al. 2022, MNRAS, 511, 2105 +Kennicutt, R., & Evans, N. 2012, ARA&A, 50, 531 +Kim, C.-G., Ostriker, E., Somerville, R. et al. 2020, ApJ, 900, 61 +Kondapally, R., Best, P.N., Cochrane, R.K. et al. 2022, MNRAS, 513, 3742 +Lamperti, I., Pereira-Santaella, M., Perna, M., et al. 2022, A&A, 668, 45 +Leitherer, C., Schaerer, D., Goldader, J. et al. 1999, ApJS, 123, 3 +Lutz, D., Sturm, E., Janssen, A., et al. 2020, A&A, 633, 134 +Madau, P., & Dickinson, M. 2014, ARA&A, 52, 415 +McConnell, M.J., Ma, C.-P., 2013, ApJ, 764, 184 +McGaugh, S. et al. 2000, ApJ, 533, L99 +McNamara, B., & Nulsen, P. 2007, ARA&A, 45, 117 +Miley, G. 1980, ARA&A, 18, 165 +Miller, T, Arav, N., Xu, X. & Kriss, G. 2020, MNRAS, 499, 1522 +Naab, T. & Ostriker, J. 2017, ARA&A, 55, 59 +Rafferty, D., McNamara, B., Nulsen, P., & Wise M. 2006, ApJ, 652, 216 +Revalski, M., Meena, B., Martinez, F. et al. 2021, ApJ, 910, 139 + +Sabater, J., Best, P., Hardcastle, M. et al. 2019, A&A, 622, 17 +Schneider, E., Ostriker, E., Robertson, B., & Thompson, T. 2020, ApJ, 895, 43 +Somerville, R. & Davé, R. 2015, ARA&A, +Tacconi, L., Genzel, R., & Sternberg, A. 2020, ARA&A, 58, 157 +Thompson, T, & Heckman, T. 2023, ARA&A, in press +Tombesi, F., Cappi, M., Reeves, J., et al. 2011, ApJ, 742, 44 +Tombesi, F., Cappi, M., Reeves, J., & Braito, 2012, MNRAS, 422, 1 +Trindade Falcao, A., Kraemer, S., Fischer, T., et al. 2021, MNRAS, 500, 1491 +Veilleux, S., Maiolino, R., Bolatto, A. & Aalto, S. 2020, A&Arv, 28, 2 +Willott, C., Rawlings, S., Blundell, K., & Lacy M. 1999, MNRAS, 309, 1017 +Yuan, Z., Wang, J., Zhou, M. et al. 2017, 846, 78 + diff --git a/6NFKT4oBgHgl3EQf_C4s/content/tmp_files/load_file.txt b/6NFKT4oBgHgl3EQf_C4s/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0afeaab19ef60b8b53f49f6c487e62402c67dd78 --- /dev/null +++ b/6NFKT4oBgHgl3EQf_C4s/content/tmp_files/load_file.txt @@ -0,0 +1,512 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf,len=511 +page_content='A Global Inventory of Feedback Timothy M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Heckman The William H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Miller III Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA Philip N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Best Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK Abstract Feedback from both supermassive black holes and massive stars plays a fundamental role in the evolution of galaxies and the inter-galactic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' In this paper we use available data to estimate the total amount of kinetic energy and momentum created per co-moving volume element over the history of the universe from three sources: massive stars and supernovae, radiation pressure and winds driven by supermassive black holes, and radio jets driven by supermassive black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Kinetic energy and momentum injection from jets peaks at z ≈ 1, while the other two sources peak at z ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Massive stars are the dominant global source of momentum injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' For supermassive black holes, we find that the amount of kinetic energy from jets is about an order-of-magnitude larger than that from winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' We also find that amount of kinetic energy created by massive stars is about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='5 εstar times that carried by jets (where εstar is the fraction of injected energy not lost to radiative cooling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' We discuss the implications of these results for the evolution of galaxies and the IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Because the ratio of black hole mass to galaxy mass is a steeply increasing function of mass, we show that the relative importance of black hole feedback to stellar feedback likewise increases with mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' We show that there is a trend in the present-day universe which, in the simplest picture, is consistent with galaxies that have been dominated by black hole feedback being generally quenched, while galaxies that have been dominated by stellar feedback are star-forming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' We also note that the amount of kinetic energy carried by jets and winds appears sufficient to explain the properties of hot gas in massive halos (> 1013 Mʘ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Introduction The basic properties of galaxies, supermassive black holes, and the intra-group/intra-cluster medium cannot be understood without considering the impact of the return of mass, metals, energy, and momentum from both populations of massive stars (stellar winds and supernovae) and supermassive black holes (winds and jets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Examples include the shape of the stellar mass function, the quenching and subsequent suppression of star-formation in massive galaxies, the mass-metallicity and mass-radius relations, the Kennicutt-Schmidt law of star-formation, and the group/cluster X-ray luminosity-temperature relation (see reviews by Somerville & Davé 2015, Naab & Ostriker 2017, Donahue & Voit 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This input of energy and momentum from massive stars and black holes is generically referred to as feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Even the highest resolution numerical simulations cannot fully include all the relevant physics ab-initio, and must rely on “sub-grid physics” (essentially, recipes for processes that cannot be spatially resolved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The same is true of semi-analytic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This underscores the importance of using observations to inform the choices that are made in simulations and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' While there is now a considerable body of data on feedback from both massive stars and supermassive back holes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Veilleux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' McNamara & Nulsen 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Thompson & Heckman 2023), we still have a very incomplete understanding of the impact of this feedback on the surrounding gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' In this paper, we will take a different approach from previous investigations of feedback, and try to compile a global inventory (that is, integrated over cosmic time) of the amount of kinetic energy and momentum per co-moving volume element injected by massive stars and supermassive black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' We will then compare the respective importance of these feedback sources as a function of time and of galaxy and black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Methodology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1 Massive Stars To compute the total amount of kinetic energy injected by massive stars (stellar winds and supernovae) per unit volume, we start with the present-day amount of stellar mass per unit volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' We use the compilation in Madau & Dickinson (2016), adjusted to a standard Chabrier Initial Mass Function (IMF - Chabrier, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This value is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='4 x 108 Mʘ Mpc-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' To compute the corresponding amount of kinetic energy we need to first correct this to account for stars that were formed but are no longer present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This requires multiplication by 1/(1-R), where R is the so-called Returned Fraction, which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='3 for a Chabrier IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Thus, the total mass of stars formed per unit volume is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='9 x 108 Mʘ Mpc-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Starburst99 (Leitherer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 1999) models for a Chabrier IMF yield a total kinetic energy in stellar winds and supernova ejecta of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='9 x 1015 erg gm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This then gives a value for the kinetic energy density due to stars of Ustar = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='8 x 1057 erg Mpc-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' How much of this kinetic energy is available to supply feedback?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The stellar ejecta initially carrying the energy collide and their kinetic energy is converted to thermal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This hot gas can then expand and flow outward with the thermal energy being converted back into kinetic energy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Chevalier & Clegg 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Some of the initial thermal energy can be lost through radiative cooling, so that only a fraction εstar remains to provide feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Numerical simulations that represent typical conditions in low-z star-forming galaxies yield εstar ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1 (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2020), with a value that increases with the star-formation rate per unit area (SFR/A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' At the much higher values of SFR/A seen in starbursts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Kennicutt & Evans 2012), simulations and models predict far greater efficiency, with εstar ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='3 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0 (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Fielding & Bryan 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This is consistent with X-ray observations of the H-like and He-like Fe Kα emission- lines in starburst galaxies from the very hot (108 K) gas created as the stellar ejecta are thermalized through shocks (Thompson & Heckman 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' These results imply that rather little of the initial kinetic energy is lost through radiative cooling, and this is substantiated by estimates of the rate of PΔV work done by the wind on the ambient gas (Thompson & Heckman 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' While there are no such constraints on galaxies at high (z > 1) redshift, we do know that these galaxies have values of SFR/A similar to those seen in low-z starbursts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Forster- Schreiber & Wuyts 2020), and that galactic winds driven by massive stars at this epoch are both ubiquitous and very similar to those seen in low-z starburst galaxies (see Thompson & Heckman 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Note that roughly 60% of the total present-day stellar mass was formed at z > 1, during this “windy” epoch (Madau & Dickinson 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The situation for momentum injection is less uncertain because momentum will be conserved even in the face of significant radiative losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' We can simply use the methodology above but use Starburst99 to compute the specific injection rate of momentum by massive stars (supernovae, stellar winds, and radiation pressure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The value is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='4 x 107 cm s-1, and for a total stellar mass density of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='8 x 108 Mʘ Mpc-3, this yields 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1 x 1049 gm cm s-1 Mpc-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='2 Black-Hole Driven Winds and Radiation Pressure Winds driven by supermassive black holes are multi-phase and have been measured in a number of different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Molecular outflows have been detected in both emission and absorption (see the review by Veilleux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Calculating kinetic energy outflow rates is conceptually straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The luminosity of a CO transition can be converted into a total molecular gas mass, albeit with uncertainties (Tacconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The measured outflow velocity and the radius of the outflow then yields a kinetic energy flux given as ½ Mgas vout3 rout-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' For absorption, the OH column density and outflow velocity yields an outflow rate (for an assumed outflow size and OH/H2 conversion factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The first compilation of molecular outflows by Fiore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' (2017) implied typical kinetic energy fluxes of dEwind/dt ≈ 3% LBol, however more recent compilations of measurements (Lutz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2020, Lamperti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2022 and private communication) have yielded much smaller values (median of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Similarly, the outflow rates of warm ionized gas can be measured using the Hα or [OIII]5007 luminosity and measured electron density to derive the total mass of ionized gas and then measuring the outflow velocity and radius of the outflow to determine dEwind/dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Different recent measurements have come to drastically different results, with median values ranging from as high as 1% of Lbol (Kakkad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2022) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='3 % (Fiore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2017), to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1% (Revalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2021), to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='01% (Dall’Agnol de Oliveira 2021), to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0003% (Trindade Falcao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' An independent measurement of the outflow rate in the warm ionized gas comes from observations of BAL QSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Here, the absorption-lines can provide a column density and outflow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Direct measurements of the electron densities can be made using the ratio of column densities in lines arising from an excited state vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Photoionization models using the observed ionizing luminosity (Q) and the inferred value of the ionization parameter (U) then yield a size for the outflow: rout = (Q/4π ne c U)1/2 (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' With a velocity, radius, and column density, the kinetic energy flux can be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The results span a huge range, from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='001% to 10% Lbol (median value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='3%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Highly ionized outflows are also detected in about 40% of AGN (Tombesi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2011) based on X-ray absorption-lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' However, because the size scales of these outflows are so uncertain, the kinetic energy outflow rates are also uncertain (by about two orders-of-magnitude, typically ranging between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='01 and 1% of LBol – Tombesi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' It is clear from the above that assigning a value for the ratio of dEwind/dt to Lbol is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' If we take the median values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='3%, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1% LBol for the molecular, warm-ionized, and highly-ionized phases, we get a total value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='5% LBol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Multiplying this by the total bolometric energy density per co-moving volume element volume produced by supermassive black holes of Urad = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='6 x 1058 erg Mpc-3 (Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2007), yields Uwind = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='3 x 1056 ergs Mpc-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This is 6% as large as the value derived for massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Using the present-day mass per unit volume in supermassive black holes of ρBH = 5 x 105 Mʘ Mpc-3 (Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2007) this wind energy density can also be expressed as Uwind = 5 x 10-4 ρBH c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' We can also consider the amount of momentum provided by AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' An initial estimate is implied by the momentum carried by radiation (Urad/c) where Urad is the total amount of radiant energy per unit volume produced over cosmic time by AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This yields an amount of momentum per unit volume of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='9 x 1048 gm cm s-1 Mpc-3 (about 4% of the value for massive stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Since the momentum flux (in the non-relativistic case) is just twice the kinetic energy flux divided by the outflow velocity, we need only consider the momentum carried by the molecular and warm ionized flows (since the BAL QSO and X-ray outflows are over an order-of-magnitude faster, but carry similar kinetic energy fluxes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' For the molecular outflows, the data in Lutz (2020) and Lamperti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' (2022 and private communication) yield median values of dpwind/dt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='7 LBol/c respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The near equality is consistent with the idea that the molecular outflows are driven by radiation pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' If so, then combining radiation pressure and the molecular outflows would be double- counting in the inventory of momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' As noted above, there is a very wide range in the ratio between the kinetic energy flux in the warm ionized gas and the AGN bolometric luminosity, and this translates directly into uncertainties in the ratio of momentum flux and radiation pressure for this gas phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Estimated median values of this ratio range from ≈10 (Kakkad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2022), to ≈1 (Fiore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Revalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2021), to ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1 (Dall’Agnol de Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2021), to ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='01 (Trindade Falcao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' It appears that the momentum flux in the warm ionized outflows is not likely to be significantly larger than those in the molecular gas or to that carried by radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This represents a total injected momentum per unit volume of at most ≈1049 gm cm s-1 Mpc-3, even if we simply add the three sources (radiation, molecular gas, ionized gas) together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This is still an order of magnitude below the value for massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='3 Black Hole-Driven Jets The earliest evidence for the outflow of kinetic energy driven by supermassive black holes came from observations of “double lobes” of synchrotron radio emission that straddled massive elliptical galaxies (Baade & Minkowski 1954).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Subsequent radio observations at high angular resolution showed narrow collimated features (“jets”) linking the two lobes to the galactic nucleus (see Miley 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' It is now possible to quantify the amount of kinetic energy carried by jets as a function of the luminosity of the radio source that they power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This can be done by joint observations of the radio and X-ray emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The expanding radio sources inflate lobes of relativistic plasma, which in X-rays can be observed as cavities in the surrounding hot gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Bırzan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' (2004,2008), Dunn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' (2005), Rafferty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' (2006), and Cavagnolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' (2010) derived the pΔV work (energy) associated with the cavities in a sample of massive galaxies, groups, and clusters, and used the buoyancy timescale (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2001) to estimate their ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' They combined these cavity powers with the monochromatic 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='4 GHz radio luminosities to show that the two were well-correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The largest uncertainty in this method is the determination of the cavity energy from the measured pressure and volume: Ecav = fcavpΔV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' For the relativistic plasma of the radio lobes the enthalpy of the cavity is 4pΔV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Taking fcav = 4, Heckman & Best (2014) derived the following best-fit relation from the cavity data: 1) dEjet/dt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='3 × 1038 (L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='4GHz/1026 W Hz−1)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='68 W This empirical relation is very similar to predictions from theoretical models of radio jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Willett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' (1999) used synchrotron properties to derive the relation: 2) dEjet/dt = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='8 × 1036 (fW)3/2 (L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='4GHz/1026 W Hz−1)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='84 W Here fW is a dimensionless factor (in the range 1 to 20) accounting for the uncertainties in the extrapolation from the population of relativistic electrons that produce the observed radio synchrotron emission to the total energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Agreement with the X-ray cavity data implies fW ≈10 to 20 (see Heckman & Best 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' We adopt the theoretical relation (equation 2), but calibrated by the cavity data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' taking fW = 15) and use this to convert the radio luminosity function of AGN between z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1 and 3 (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2017) into a measure of the evolution in the rate of kinetic energy injection per unit volume by radio jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The results are shown in Figure 1, and show that the peak rate of kinetic energy injection by jets occurs at a significantly lower redshift (z ≈ 1) than the peak rate due to massive stars and black- hole-driven winds (z ≈ 2, as also shown in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' We then integrate the energy injection rate by interpolating the values at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0 and extrapolating from z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1 to 0 and from z = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0 to infinity (this extrapolation does not add significantly to the total - see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This then gives a value for the time-integrated total kinetic energy per unit volume due to jets of Ujet = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='6 x 1057 erg Mpc-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' These results are broadly in line with similar estimates derived from low-frequency radio luminosity functions (Kondapally et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', private communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The time-integrated kinetic energy input from jets is ≈6 times larger than the value estimated about for black-hole-driven winds, and 40% (400%) the total amount of kinetic energy generated by massive stars for εstar = 1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Alternatively, using the present-day mass per unit volume in supermassive black holes of ρBH = 5 x 105 Mʘ (Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2007) this jet energy density can also be expressed as Ujet = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='9 x 10-3 ρBH c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The kinetic energy carried by jets is in the form of relativistic bulk motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' In this case, the momentum can be taken as p ≈ KE/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The above value of Ujet then implies a momentum density of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='7 x 1046 gm cm s-1 Mpc-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This is much less than the momentum carried by radiation and winds from supermassive black holes, and the momentum produced by massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Jets are therefore far more important feedback sources in terms of kinetic energy than momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='4 The Bottom Line For total kinetic energy inventory, the largest single source is either massive stars (for εstar > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='4) or jets (for εstar < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' AGN winds are only important at the <10% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' For the total momentum inventory, massive stars dominate (AGN contribute at the ≈10% level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The peak rate of kinetic energy injection by jets occurs at a substantially lower redshift than that from stars or AGN winds (z ≈ 1 and 2, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' These results are summarized in Table 1 and Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' ______________________________________________________________________________ Table 1 – Summary of Feedback Inventory 1 2 3 4 5 6 Sample Log ρ Log sKE Log ρKE Log sp ρp Massive Stars 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='69 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='11 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='83 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='87 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='85 BH Winds 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='70 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='30 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='63 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='00 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='00 BH Jets 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='54 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='43 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='94 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='94 ______________________________________________________________________________ Notes: Column 2 – The log of the present-day mass density of stars (row 3) and supermassive black holes (rows 4 and 5) formed over cosmic time in units of Mʘ Mpc-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Column 3 – The log of the specific kinetic energy released: energy per unit mass in stars (row 3 and black holes (rows 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Given in units of c2, and assuming εstar = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Column 4 – The log of the amount of kinetic energy created per unit volume (in ergs Mpc-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Column 5 – The specific momentum created (momentum per unit mass in stars (row 3) and black holes (rows 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' In units of cm s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Column 6 – The log of the amount of momentum created per unit volume (gm cm s-1 Mpc-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' ______________________________________________________________________________ Figure 1 – A plot of the amount of kinetic energy injected per Gyr and co-moving cubic Mpc as a function of lookback time for massive stars (supernovae and stellar winds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' black) and black-hole-driven jets (blue) and winds (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' For massive stars we show the cases in which 100% (solid line) and 10% (dashed line) of the kinetic energy created is delivered to the surroundings (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' not lost to radiative cooling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Note that for momentum injection, massive stars dominate at all epochs, with the same time dependence as for kinetic energy injection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' as given by the solid black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Implications 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1 For Galaxies To assess the implications of these results for galaxy evolution, it is essential to consider the dependences of feedback on the masses of both galaxies and supermassive black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' We can go beyond these simple global values and examine the relative importance of feedback (both kinetic energy and momentum) as a function of the ratio of supermassive black hole mass to galaxy stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' In Figure 2 we show a plot of black hole vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' galaxy mass that is similar to that in Heckman & Best (2014) for the z ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1 universe (based on SDSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' In this case, these are present day stellar masses, and would need to be increased by a factor 1/(1-R) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='42 to represent the total mass of stars ever formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The masses for the black holes were estimated from the M-σ relation from McConnell & Ma (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' In figure 2, we have color-coded the plot by the fraction of galaxies in which star-formation has been quenched, which we define to be Redshift 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='5 1 2 346 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='5 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='5 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0 Supernovae (8star = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0) - Supernovae (star = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='5 AGNjets AGNwinds 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0 0 2 4 6 8 10 12 Lookback time / GyrSFR/Mstar < 10-11 yr-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' It is clear that the quenched fraction depends strongly on both the stellar and black hole masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The mean relation between stellar and black hole mass in Figure 2 can be approximated as log MBH = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0 log Mstar -14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0, implying MBH/Mstar α Mstar1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0 α MBH0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Thus, the relative importance of feedback integrated over cosmic time from massive stars and black holes should be a strong function of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Let us quantify this for kinetic energy and then for momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' For kinetic energy, in a given galaxy (and assuming that global averages can be applied to individual galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' see below) the inventories above imply that the ratio KEBH/KEstar = 315 εstar-1 MBH/Mstar (where Mstar is the present-day stellar mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' For momentum, the corresponding ratio is pBH/pstar = 100 MBH/Mstar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' We can then plot these relations in Figure 2 to see the regimes in which feedback from supermassive black holes exceeds that from stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' For kinetic energy, we show this separately for values of εstar = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Figure 2 – A plot of the distribution of SDSS galaxies in the plane of galaxy stellar mass vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' supermassive black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The latter were estimated using the MBH vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' σ relation in McConnell & Ma (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The relative numbers of galaxies in each bin are indicated by the green contours (increasing by factors of 2) and the color-coding represents the fraction of galaxies that are quenched (SFR/Mstar < 10-11 yr-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The dark blue dashed line indicates where the momentum injected by black holes equals that from massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The two light blue dashed lines indicate where the kinetic energy from black holes equals that from massive stars for values of εstar = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0 (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The transition from predominantly star forming to predominantly quenched galaxies occurs near the relationship for εstar = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='8 Quenched fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1 KEBH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='2 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0 log1o(Stellar mass / Msun)In terms of momentum input, stars dominate over black holes in almost all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' However, considering kinetic energy, we find that the transition from galaxies that are mostly quenched to those that are mostly star-forming occurs very near the dividing line between jet-dominated feedback and stellar dominated feedback for a value of εstar ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This is suggestive evidence that quenching is driven by the feedback of kinetic energy from jets driven by supermassive black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' However, we caution that the transition from star forming to quiescent galaxies also occurs at the transition from disk dominated to bulge dominated galaxies, so the causal connections between galaxy structure, star formation, and black hole feedback are not entirely clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' We emphasize that the relations plotted in Figure 2 explicitly assume that the global relations can be applied to individual galaxies, namely that the amount of feedback from massive stars in a given galaxy is proportional to stellar mass and that the amount of feedback from jets is proportional to the black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The former seems like a safe assumption, but the dependence of the production of radio jets on black hole mass may be complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' We know that there are essentially two populations of radio galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Heckman & Best 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' In one case (“radiative mode”) the jets are launched by star-forming galaxies and are accompanied by strong nuclear radiation (QSO-like).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' In the other class (“jet-mode”) the jets are launched by quenched galaxies, with little accompanying nuclear radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The radiative mode becomes more important at higher luminosities and at higher redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' For the jet-mode galaxies, Sabater et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' (2019) find that the probability of producing a jet with a given luminosity depends on both the stellar and black hole mass (and more strongly on the former).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The situation for radiative-mode radio galaxies is less clear, although the indications are that any dependence of the ratio of KE/MBH on MBH or Mstar is weaker (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Janssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2012, Kondapally et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' In the context of Figure 2, it may be that the jet-mode is not the dominant population in terms of actively quenching, since the jet-mode galaxies are already quenched (instead, these may just ‘maintain’ a quenched state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' If quenching is due to jets in radiative-mode galaxies, the dividing line between quenched and star-forming galaxies in Figure 2 would imply that time-integrated amount of jet energy contributed by a radiative mode galaxy is proportional to its black hole mass (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' the integrated ratio of jet kinetic energy and energy carried by radiation is independent of black hole mass in these galaxies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Another way to consider this is to ask how the amount of energy supplied by stars and by black holes scales with the binding energy of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' We take Ebind ≈ Mstar vc2 where vc is the galaxy circular velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The Tully-Fischer relation for disk galaxies (McGaugh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2000) and the Faber-Jackson relation for ellipticals (Bernardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2003) both imply vc α Mstar1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Thus, we have Ebind α Mstar3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Given that KEstar α Mstar and KEBH α MBH α Mstar2, this implies that KEstar/Ebind α Mstar-1/2, while KEBH/Ebind α Mstar1/2 α MBH1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This again underscores the fundamental difference in the mass-dependence of feedback from massive stars and supermassive black holes: feedback from stars becomes increasingly impactful on the galaxy as the mass decreases, while feedback from black holes has greater impact as the mass increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='2 For the Intra-Group and Intra-Cluster Media It has long been known that the basic observed properties of the hot gas in groups and clusters of galaxies (Mhalo > 1013 Mʘ) are not consistent with simple models of purely gravitational processes operating during the formation of these systems (see Donahue & Voit 2022 and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' A particularly simple example of this is the observed relationship between the X-ray temperature (a proxy for halo mass) and X-ray luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' As the halo masses decrease, the observed X-ray luminosities fall further below the relationship expected simply from gravitational infall and heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' These lower luminosities arise because the hot gas in these less-massive halos is more spatially-extended than the dark matter, with the resulting drop in gas density leading to lower X-ray luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This could be due to the feedback of energy injected into the hot gas, which “lifts” the gas outward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' As described above, there is direct observational evidence in the local universe of radio jets delivering energy to the hot gas in groups and clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' As discussed in Donahue & Voit (2022), for this to be responsible for lifting the hot gas, an amount of kinetic energy equal to ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='5% MBH/c2 must be delivered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This is close to the value for jets and AGN winds that we estimated above of ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='34%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Note that this could be supplemented by the kinetic energy from massive stars (which would be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='25 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='5 the value for jets for εstar = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='0 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Summary Based on a global inventory of the amount of kinetic energy and momentum injected by massive stars (stellar winds and supernovae), and by winds and jets driven by supermassive black holes, we draw the following conclusions: i) The major sources of kinetic energy are massive stars and jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' Winds driven by supermassive black holes provide <10% of the total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' The global ratio of the kinetic energy injected by massive stars to that injected by jets is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='5 εstar (where εstar is the fraction of injected energy from stars that is not lost to radiative cooling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' ii) Massive stars are the dominant source of momentum injection (90% of the total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' AGN winds provide 10%, and radio jets are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' iii) The peak in the feedback from jets occurs at z ≈ 1, considerably later than the contributions of AGN-winds and massive stars (peaking at z ≈ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' iv) Since the ratio of the mass of the supermassive black hole to the galaxy stellar mass increases steeply with mass, there will be a mass-dependence in the relative importance of feedback from the two sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' v) For the assumptions that the total amount of kinetic energy from massive stars is proportional to the galaxy’s stellar mass, and that the total amount of kinetic energy from a supermassive black hole is proportional to its mass, we find that the populations of quenched and star-forming galaxies occur in the regimes where supermassive black hole feedback and massive star feedback dominate, respectively (for a value of εstar ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' vi) By comparing the amount of kinetic energy injected as a function of the binding energy of a galaxy, we show that feedback becomes more impactful as galaxy mass decreases for massive stars, but more impactful as galaxy mass increases for black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' vii) The global amount of kinetic energy injected by radio jets and AGN winds per unit volume, combined with the supermassive black hole mass function, yields an efficiency for producing kinetic energy in jets of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='34% c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' This is very close to the amount of energy needed to explain X-ray luminosity-temperature relation in groups and clusters (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='5% c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' REFERENCES Baade, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' & Minkowski, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 1954, ApJ, 119, 206 Bernardi, M 2003, AJ, 125, 1849 Bırzan L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Rafferty D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', McNamara B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Wise M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Nulsen P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2004, ApJ, 607, 800 Bırzan L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', McNamara B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Nulsen P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Carilli C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Wise, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' ApJ, 686, 859 Cavagnolo K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', McNamara B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Nulsen P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2010, ApJ, 720, 1066 Chabrier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2003, PASP, 115, 763 Chevalier, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', & Clegg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 1985, Nature, 317, 44 Churazov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Bruggen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Kaiser C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Bohringer H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Forman W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=" 2001, Ap J, 554,261 Dall'Agnol de Oliveira, B." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Storchi-Bergmann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Kraemer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2021, MNRAS, 504, 3890 Donahue, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', & Voit, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='M, 2022, Physics Reports, 973, 1 Dunn, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Fabian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Taylor, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2005, MNRAS, 364, 1343 Fielding, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' & Bryan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2022, ApJ, 984, 82 Fiore, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Feruglio, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Shankar, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2017, A&A, 601, 143 Heckman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' & Best, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2014, ARA&A, 52, 589 Hopkins, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Richards, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', & Hernquist, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2007, ApJ, 654, 731 Janssen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Best, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Röttgering, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Brinchmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2012, A&A, 541, 62 Kakkad, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=',Sani, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Rojas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2022, MNRAS, 511, 2105 Kennicutt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', & Evans, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2012, ARA&A, 50, 531 Kim, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Ostriker, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Somerville, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2020, ApJ, 900, 61 Kondapally, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Best, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Cochrane, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2022, MNRAS, 513, 3742 Lamperti, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Pereira-Santaella, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Perna, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2022, A&A, 668, 45 Leitherer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Schaerer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Goldader, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 1999, ApJS, 123, 3 Lutz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Sturm, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Janssen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2020, A&A, 633, 134 Madau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', & Dickinson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2014, ARA&A, 52, 415 McConnell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Ma, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', 2013, ApJ, 764, 184 McGaugh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2000, ApJ, 533, L99 McNamara, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', & Nulsen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2007, ARA&A, 45, 117 Miley, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 1980, ARA&A, 18, 165 Miller, T, Arav, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' & Kriss, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2020, MNRAS, 499, 1522 Naab, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' & Ostriker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2017, ARA&A, 55, 59 Rafferty, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', McNamara, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Nulsen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', & Wise M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2006, ApJ, 652, 216 Revalski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Meena, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Martinez, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2021, ApJ, 910, 139 Sabater, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Best, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Hardcastle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2019, A&A, 622, 17 Schneider, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Ostriker, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Robertson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', & Thompson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2020, ApJ, 895, 43 Somerville, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' & Davé, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2015, ARA&A, Tacconi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Genzel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', & Sternberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2020, ARA&A, 58, 157 Thompson, T, & Heckman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2023, ARA&A, in press Tombesi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Cappi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Reeves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2011, ApJ, 742, 44 Tombesi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Cappi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Reeves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', & Braito, 2012, MNRAS, 422, 1 Trindade Falcao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Kraemer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Fischer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2021, MNRAS, 500, 1491 Veilleux, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Maiolino, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Bolatto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' & Aalto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2020, A&Arv, 28, 2 Willott, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Rawlings, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Blundell, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', & Lacy M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 1999, MNRAS, 309, 1017 Yuan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=', Zhou, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} +page_content=' 2017, 846, 78' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf'} diff --git a/8NFLT4oBgHgl3EQfBC4o/content/tmp_files/2301.11968v1.pdf.txt b/8NFLT4oBgHgl3EQfBC4o/content/tmp_files/2301.11968v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..974cc012b3e343dfd74c3cc885367e197a635eff --- /dev/null +++ b/8NFLT4oBgHgl3EQfBC4o/content/tmp_files/2301.11968v1.pdf.txt @@ -0,0 +1,792 @@ +arXiv:2301.11968v1 [hep-th] 27 Jan 2023 +Strong Cosmic Censorship in light of Weak Gravity +Conjecture for Charged Black Holes +Jafar Sadeghi ⋆1, +Mohammad Reza Alipour ⋆2, +Saeed Noori Gashti⋆3 +⋆Department of Physics, Faculty of Basic Sciences, +University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran +Abstract +In this paper, we investigate the strong cosmic censorship conjecture (SCC) for charged +black holes in the de Sitter space by considering the weak gravity conjecture (WGC). Us- +ing analytical methods, we find that the SCC is preserved for dS-charged black holes with +respect to some restriction qQ ≫ 1 and r+ ≥ Q with the help of the WGC condition +viz +q +m ≥ 1 for scalar fields. Where q, m are the charge and mass of the scalar field, and +r+, Q determine the radius of the outer event horizon and the charge of the black hole, +respectively. In that case, when the (WGC) is valid, SCC will definitely be satisfied for +the dS-charged black holes. On the other hand, the SCC is violated when the WGC is +not satisfied. Also, we examined the RN-dS charged black hole in the extremality state +and found that SCC can be violated with the condition Λr2 ++ = 1. +Keywords: Strong cosmic censorship conjecture; Weak gravity conjecture; RN-dS charged +black hole +Contents +1 +Introduction +2 +2 +Weak Gravity Conjecture +4 +3 +Charged Black Holes in dS Space +6 +4 +The Quasinormal Resonant Frequency Spectrum +7 +5 +Discussion and Result +10 +1Email: +pouriya@ipm.ir +2Email: +mr.alipour@stu.umz.ac.ir +3Email: +saeed.noorigashti@stu.umz.ac.ir +1 + +1 +Introduction +One of the studies with a long history in general relativity is the study of the collapse of small +perturbations. We need more information on how these oscillations decay to understand better +the gravity concept, use gravitational wave data, and study and investigate the valuable features +of general relativity. One of the signs of the failure of determinism in general relativity can +be the emergence of an interesting phenomenon known as Cauchy horizons that appear in the +astrophysical solutions of Einstein’s equations. These horizons are such that it is impossible +to specify the history of the future of an observer that passes through such horizons using +Einstein’s equations and initial data. With these descriptions in the black holes’ space-time +background, it is an expected possibility that the perturbations of the outer region are infinitely +amplified by a mechanism known as the blue shift. They lead to a singularity boundary beyond +the Cauchy horizon in the interior of black holes, where field equations cease to make sense. The +Penrose strong cosmic censorship (SCC) confirms such an expectation. Of course, another point +is that astrophysical black holes are stable due to a special mechanism called the perturbation- +damping mechanism, which is applied in the outer region. +Therefore, considering whether +SCC retains real hinges or not depends on the very subtle competition between the collapse +of perturbations in the outer region and their amplification (blue shift) in the inner space- +time of black holes. In general, the fate of Cauchy horizons is related to the collapse of small +perturbations outside the event horizon. Hence, the validity of SCC is tied to the extent of +external damps fluctuation. In connection with various structures and conditions, SCC and +its satisfaction and violation have been investigated in various theories. The violation of this +conjecture near the extremal region studied in the investigation of higher curvature gravity [1]. +Also, this conjecture has been challenged in investigating many charged black holes. In [2,3], this +conjecture was checked for a charged AdS black hole. It was shown that for a specific interval +for the parameter (β), this conjecture is satisfied and violated in other areas as well. +The +strong cosmic censorship conjecture has also been investigated in two dimensions. There have +been interesting outcomes regarding the violation of this conjecture near the extremal region +at specific points [4]. The study of this conjecture in the structure of three-dimensional black +strings has also carried interesting results, which you can see [5] for a deeper study. Studying +the validity and violation of this conjecture in many recent studies in different conditions and +frameworks has led to exciting results that you can see [6–9] for further study. Therefore, in +this article, we want to study a different structure of this conjecture. According to the above +explanation, we consider the general configuration of charged black holes. Then, using the +weak gravity conjecture, we will prove that SCC is valid for specific values for all charged black +holes. We will use the weak gravity conjecture to prove a general relation for all charged black +holes about SCC. In connection with SCC, we need to pay attention to more concepts, which +we will mention here for further study. The effectiveness of mass-inflation systems, which are +involved in the transformations of the inner Cauchy horizon associated with the space-time of +2 + +black holes that are approximately flat, which is pathological in the estimation of SCC, into +a series of hypersurfaces which is singular non-extendable. Those that are in an indivisible +form are related to two different types of physical mechanisms [10–16]. First, the events in +the exterior space-time regions of dynamic black holes formed viz the collapse of the remnant +perturbation fields and second amplification of exponential blue shift related to the fields falling +into the inside of black holes. We can manage these two introduced different systems through +parameters such as (g) and (k−). It can be stated that the dimensionless physical ratio with +the help of these two parameters can determine the fate of the inner Cauchy horizons inside +such space-times of non-asymptotic flat black holes [8,17,18], +β ≡ g +k− +. +Of course, a certain range of parameters of black holes, such as mass and charge, etc., as +indicated in [8,17,18], +β > 1 +2. +So, space-time of the corresponding black holes can be physically expanded beyond their Cauchy +horizon which includes a pathological fact and a sign of algebraic failure or a violation of the +Penrose SCC in classical general relativity. For the dynamics of Einstein’s equations as well +as the destiny of the observers, the explosive structure of the curvature that is related to +(β < 1) does not have per se much physical significance: it indicates two theorems, not the +failure of the field equations mentioned in [19] and of course not the destruction of macroscopic +observers which is discussed in [13]. Therefore, the physical and mathematical formulation +of the conjecture of a SCC in such conditions leads to ignoring physical phenomena such as +impulsive gravitational waves or the formation of shocks in relativistic fluids. +Due to the +aforementioned reasons, the modern form of the CC conjecture was introduced that requires +a stronger constraint (β < 1 +2 ) and many works have been done to fit such constraints. For +example, by studying massless scalar fields in linear form and examining the entire parametric +space of a charged black hole, areas beyond mentioned range were obtained, which it seems +cannot be allowed. According to the above explanations, we organize the article in the following +form. +In section 2, we will give basic explanations about the weak gravity conjecture and also the +motivation to use it. In section 3, we will introduce charged black holes in dS space, and then +we will introduce the quasinormal resonant frequency spectrum in section 4. We will check the +conditions of compatibility and violation of (SCC) with respect to (WGC) for RNdS charged +black holes. Finally, we describe the results in section 5. +3 + +2 +Weak Gravity Conjecture +As it is known in the literature, a new idea has been put forward as a swampland program to +check theories coupled to gravity, to check the consistency of quantum gravity, and finally, a +proof for string theory. Recently, ones have done lots of work on this field [20–30]. Due to the +special conditions of string theory and the fact that its testing and experimental investigations +seem a bit difficult, this idea has been proposed to test and investigate various concepts of +cosmology. The swampland program is challenged from two sides. From an up-bottom view +for introducing principles and limitations to introduce conjectures, as well as mathematical +formulations to examine cosmological concepts. A second look from the bottom-up in order +to test each of these conjectures with various concepts of cosmology including inflation and +matching with observable data, which is both a proof for this new idea and a proof for string +theory. So far, many conjectures have been proposed from this theory, and now, according to +the structure and further investigations, new conjectures will be added to this program. Some +of these conjectures face challenges and as a result, corrections are made to the conjectures. +We face some limitations in quantum gravity (QG). At the point when gravity is considered at +the quantum level, the hypothesis will be incompatible. Generally having a reliable quantum +hypothesis of gravity isn’t really straightforward and can in any case hold many surprises and +can be interesting for physical science at low energies. The objective of the swampland program +is to decide the limitations that an effective field theory(EFT) should fulfill to be viable with the +consideration of ultraviolet completion(UV) in QG. They are called swampland limitations, and +different suggestions are figured out as far as swampland conjectures(SC). The objective is to +recognize these limitations, accumulate proof to demonstrate or refute them inside the structure +of QG, give reasoning to make sense of them in a model-free manner, and comprehend their +phenomenological suggestions for low-energy EFTs. Albeit the swampland idea isn’t restricted +to string theory on a fundamental level, SC are frequently examined by string theory backdrops. +Without a doubt, the string theory gives an ideal structure to thorough quantitative testing +of conjectures and works on how we might interpret potential compressions of string theory. +Strangely, it has as of late been uncovered that a large number of these conjectures are to be +sure related, recommending that they may essentially be various countenances of some yet-to- +be-found crucial standard of QG. As far as possible have significant ramifications for cosmology +and particle physics. They can give new core values to building conjectures past the standard +models in high-energy physics. They may likewise prompt UV/IR blending, which breaks the +assumption for scale detachment and possibly gives new bits of knowledge into the regular issues +seen in our universe. Consequently, the presence of swampland is extraordinary information +for phenomenology. For a total rundown of references connected with swampland that might +be valuable, we allude in [20] the swampland program (SP) has likewise been surveyed and +presented. The shortfall of global symmetry (GS) and the completeness of charge spectra are +at the center of the SP. Nonetheless, they need phenomenological suggestions except if we can +4 + +restrict the global symmetries [21,22] and whether there is any limitations point on the mass +of charged states. In any case, they just bound the complete hypothesis but not the low-energy +EFTs. Specifically, it is phenomenologically important whether all charged particles can be +really super heavy and even compare to black holes(BHs), or whether there is some thought of +completeness of the range that gets by at low energies. A large portion of the SCs examined +address exactly these inquiries. They want to profoundly explore these assertions and measure +them so we can draw nearer to the recuperation of a few global symmetries. For instance, we +can deduce recuperate a global symmetry (GS) U(1) by sending the gauge coupling(GC) to +nothing, which ought not to be permitted in QG. Attempting to comprehend string theory +for the study of this issue, it might turn out that if one somehow managed to try to do such +work, can give data about the imperatives that an EFT can fulfill to be viable with QG. +Likewise, WGC forbids this cycle by flagging the presence of new charged states that denies +the depiction of the EFTs. Thusly, it gives an upper bound on the mass of these charged states. +The WGC comprises of some parts: the electric and the magnetic electric-WGC: As indicated +by a quantum hypothesis, we have the following condition [20–30], +Q +m ≥ Q +M |ext = O(1), +(1) +and +Q = gq, +(2) +where, g and q are the gauge coupling and the quantized charge. The electric-WGC needs +the presence of an electrically charged condition of a higher charge-to-mass proportion than +extremal BH in that hypothesis, which is regularly a variable of the order one. One more +understanding of this conjecture is that the limitations region shows that scale force determines +stronger than the gravity on this mode — so subsequently is called WGC. This is an identical +equation since it expects that electromagnetic force is stronger gravitational force [20–30], +FGrav ≤ FEM +(3) +It implies that the charge is more prominent than the mass, so we get a similar condition as +above. This is as of now false within the sight of massless scalar fields. The motivations of +WGC are twofold. To begin with, it makes a QG boundary to reestablish the GS of U(1) by +sending g → 0. If a GC goes to zero as indicated by WGC, this conducts new light particles +and the cutoff the hypothesis arrives at nothing and nullifies the EFT. Because of the littleness +of the scale coupling, it relies upon how much energy the interaction with which you need to +portray the viable EFT. The smallness of the cycle energy leads to the smallness of the scale +coupling. On the other hand, if you need to keep the EFT substantial up to an extremely high +cut-off, the GC can’t be excessively small. This is an illustration of swampland limitations that +5 + +becomes more grounded for higher energies. Obviously, a hypothesis with disappearing measure +coupling i.e., GS is incompatible because the cutoff of the viable EFT is likewise zero. One more +fundamental inspiration for WGC is that a kinematic prerequisite permits extremality BH to +have decomposed. Charged BHs should fulfill an extremality breaking point to stay away from +the presence of exposed singularities, as shown by the weak cosmic censorship (WCC). For a +given charge Q, this super bound shows that the this super bound shows that the mass M of +the BHs should be more noteworthy than the charge [20–30], +M ≥ Q +(4) +For the BHs to have a regular horizon. +Here, we set the extremal factor O(1) to one for +simplicity. The primary condition for starting the decay to the small black hole and particle +is the existence of the extremal BHs (M = Q). So, one can consider the decay of an extremal +BHs which one of the rot items has a charge more modest than its mass as far as possible, so +M1 ≥ Q1. Then the rot item can never again have a charge more modest than the mass, that +is m2 ≤ Q2. It is just a kinematic necessity. Since the second rot item violates the WCC, it +can’t be a BH, so it should be a particle. The above kinematic necessity can be acquired by +applying preservation of mass/energy and protection of charge as follows. The initial mass of +the BH should be more prominent than the amount of the mass of the rot items Mi and the +charge of the initial BH. +3 +Charged Black Holes in dS Space +The metric of charged black hole in spherical symmetric space is defined as follows, +dS2 = f(r)dt2 − f −1(r)dr2 − r2dΩ2, +dΩ2 = (dθ2 + sin2(θ)dϕ2). +(5) +Here, we consider f(r) = H(M, Q) − Λr2 +3 +in general; where Q, M, Λ > 0 are electric charge, the +mass of the black hole and the cosmological constant respectively. In this case, we can obtain +its event horizons as follows, +f(r⋆) = 0 +→ +⋆ ∈ (−, +, ..., c). +(6) +Considering the metric in general terms, we have different event horizons, where (r−) is the +Cauchy horizon, (r+) is the outer event horizon, and (rc) is the cosmological horizons. Using +Klein-Gordon’s differential equation, we can determine the dynamics of a massive charged +particle near a charged black hole [31–34], +1 +√−g∂µ(gµν√−g∂νΦ) − 2iqgµνAµ∂νΦ − q2gµνAµAνΦ − m2Φ = 0, +(7) +6 + +where m and q are the mass and charge of the particle, respectively also, Aµ = +� Q +r , 0, 0, 0 +� +. We +can define the scalar field Φ according to relation (7) as follows [36], +Φ(t, r, θ, φ) = +� +m +� +ℓ +e−iωtYℓm(θ, ϕ)Φ(r). +(8) +The integer parameters ℓ and m are the spherical and the azimuthal harmonic indices of the +resonant eigenmodes which characterize the charged massive scalar fields in the charged black- +hole spacetime. By putting Eq.(8) in Eq.(7) and using dx = +dr +f(r), we get the Schr¨odinger-like +differential equation , +d2φ(r) +dx2 ++ V (r)φ(r) = 0. +(9) +The effective radial potential due to a massive charged particle near a charged black hole is +defined as [8], +V (r) = qm +r2 +� q +mα(r) − m +q β(r) +� +, +(10) +where +α(r) = Q2 +� +1 − ωr +qQ +�2 +, +β(r) = r2f(r)H(r), +H(r) = +�ℓ(ℓ + 1) +m2r2 ++ f ′(r) +m2r + 1 +� +. +(11) +Also, we can consider the boundary conditions for the special radial function near the outer +event horizon as an incoming wave and at the largest event horizon as an outgoing wave [34,35]: +φ(x) ∼ +� +e +−i(ω− qQ +r+ )x, +for +r → r+ (x → −∞); +e−i(ω− qQ +rc )x, +for +r → rc (x → ∞). +(12) +According to the above boundary conditions, we can obtain the discrete spectrum of ω, defined +as the resonance frequency of the imaginary quasi-normal state. +4 +The Quasinormal Resonant Frequency Spectrum +In this section, we need to obtain the imaginary part of the resonance frequency to investigate +the linear dynamics of a massive charged particle near a general charged black hole. Also, we +need to do this in a dimensionless regime to do this analytically. Since, the q2 +¯h ≃ +1 +137 relationship +exists in our universe, we can consider it for black holes, even slightly charged, and get qQ ≫ 1. +In addition, the mechanism of the Schwinger-type pair-production in space-time of charged +black hole creates a limit to the black hole electric field with the +Q +r2 ++ ≪ m2 +q relationship [37–40]. +7 + +Therefore, according to the above statement, we can consider SCC and define our constraint +regime following ansans, +m2r2 ++ ≫ ℓ(ℓ + 1) +and +m2r2 ++ ≫ 2k+r+, +(13) +where k+ = f ′(r+)/2 is the gravitational acceleration of the black hole at the outer event +horizon. In this area, we try to obtain the imaginary part of the resonance frequency in the +background of the general charged black hole near the event horizon. +Now, we use radial +potential (10) to determine the linear dynamics of the massive charged particle near the event +horizon of the black hole. We can consider this potential in region (13) as an effective potential +and obtain the quasinormal resonance modes analytically using standard WKB techniques +[41, 42]. In this region, we consider the maximum effective potential near the event horizon +of the black hole at point r = r0. In the following, we use the relationship (10), (11), and +V ′(r0) = 0 to obtain the point where the effective potential is maximum as follows, +r0 = +q2Q2 +qQω − m2r2 ++k+ +(14) +According to the Schr¨odinger-like differential equation (9) and [41–43], we use the WKB +method to obtain the quasinormal mode frequencies through the following, +iK − (n + 1 +2) − Λ(n) = Ω(n) +(15) +where +K = +V0 +� +2V (2) +0 +Λ(n) = +1 +� +2V (2) +0 + + +� +α2 + 1 +4 +� +8 +V (4) +0 +V (2) +0 +− (60α2 + 7) +288 +� +V (3) +0 +V (2) +0 +�2 + +Ω(n) = n + 1 +2 +2V (2) +0 + +5 (188α2 + 77) +6912 +� +V (3) +0 +V (2) +0 +�4 +− (100α2 + 51) +384 +� +V (3) +0 +�2 +V (4) +0 +� +V (2) +0 +�3 + + ++ n + 1 +2 +2V (2) +0 + +(68α2 + 67) +2304 +� +V (4) +0 +V (2) +0 +�2 ++ (28α2 + 19) +288 +� +V (3) +0 +V (5) +0 +� +� +V (2) +0 +�2 +− (4α2 + 5) +288 +V (6) +0 +V (2) +0 + + +(16) +Here, V (k) +0 +≡ dkV +dxk |r=r0 is the spatial derivative of the effective potential of equation (10), and +its scattering peak is evaluated at the point r = r0. Using relations (10), (11), (14) and (16), +8 + +we will have the following relation in the region of (13), +K ≃ +k2 ++m4r4 ++qQ +2f0 (k+m2r2 ++ − qQω)2 +Λ(n) ≃ +k2 ++m4 � +17 − 60 +� +n + 1 +2 +�2� +r4 ++ + 2k+m2 � +36 +� +n + 1 +2 +�2 − 7 +� +qQr2 ++ω +16qQ (qQω − 3k+m2r2 ++)2 +× f0 +A = 15k4 ++m8 +� +148(n + 1 +2)2 − 41 +� +r8 ++ + 12k3 ++m6 +� +121 − 420(n + 1 +2)2 +� +qQr6 ++ω +B = 64q5Q5 � +k+m2r2 ++ − qQω +�4 +Ω(n) ≃ −(n + 1 +2)q3Q3f 2 +0 × A +B +(17) +where f0 = f(r0). In the next step, to determine the study of SCC, we need to obtain the +minimum value of the fundamental imaginary resonance mode of the system. For this purpose, +using equations (15) and (17), we can calculate the Im(ω0), +ω ≃ qQ +r+ +− 2k+m2r2 ++ +qQ +� +1 − 14400 +11644 +�(n + 1/2)f0 +qQ +�4� +− i +� +4f0k+(n + 1 +2)m2r2 ++ +q2Q2 +� +1 − 34qQf 4 +0 +11664 +� ++ O(f 2 +0) +� +(18) +Since we consider r0 near the event horizon (r+), we have f0 ≪ 1. For investigation the SCC, +it is necessary to find the minimum value of the resonance mode and evaluate its ratio to the +surface gravity of the event horizon, +β = −Im(ω0) +k+ +≃ 2f0 +m2r2 ++ +q2Q2 +� +1 − 34qQf 4 +0 +11664 +� +. +(19) +Since it is f0 ≪ 1, it is sufficient to have the conditions q2Q2 > m2r2 ++ in the relation above +concepts so that −Im(ω0) +k+ +< 1 +2 is established. Therefore, we have the following condition for the +study of SCC, +q +m ≥ r+ +Q . +(20) +from equation (20) determine that when r+ ≥ Q, we have the weak gravity conjecture condition. +We know that k− > k+, so the relationship of (19) and (20) is also established for β = −Im(ω0) +k− +< +1 +2. Also, according to relation (19), when qQ < 2√f0mr+, SCC can be violated. Since qQ ≫ 1 +and f0 ≪ 1, the mass of the scalar field and the radius of the event horizon must be very +massive and very large respectively. In the following, we obtain the extremality state of the +9 + +RN-dS black hole. We will have the following relation for the RN-dS black hole with respect +to equation(5), +f(r) = 1 − 2M +r ++ Q2 +r2 − Λr2 +3 . +(21) +When k+ = k− = 0, we can obtain the black hole extremality state, +Qexe = r+ +� +1 − Λr2 ++, +Mexe = r+(1 − 2 +3Λr2 ++). +(22) +We substitute Eq.(22) in Eq.(19) to obtain β in the extremality state of the RN black hole, +β ≃ 2f0 +m2 +q2(1 − Λr2 ++) +� +1 − 34qf 4 +0 +� +1 − Λr2 ++r+ +11664 +� +(23) +According the above relationship, when the condition +q +m ≥ +1 +1−Λr2 ++ is satisfied, the SCC will +definitely be preserved, and since Λr2 ++ < 1, the weak gravity conjecture will also be satisfied. +On the other hand, when Λr2 ++ ≪ 1, we will have the SCC condition in light of the WGC, +q +m ≥ 1 + Λr2 ++, +(24) +from the above relation WGC is clearly obtained. In relation (23) when Λr2 ++ = 1, we have +β → ∞ and the SCC is violated. Also, these result and conditions are completely compatible +with [44,45]. +5 +Discussion and Result +One of the indications of the failure of determinism GR can be the rise of a fascinating pecu- +liarity known as the Cauchy horizon that shows up in the astrophysical solutions of Einstein’s +equations. These horizons are such that it is difficult to indicate the history of the future of +an observer that passes come of such horizons utilizing Einstein’s conditions and initial infor- +mation. With these descriptions in the black holes’ background space-time, it is a predicted +possibility that the perturbations of the external area are infinitely enhanced by a system +known as the blue shift. They lead to a singularity beyond the Cauchy horizon the inside +of BHs, where field conditions fail to seem good. The Penrose cosmic censorship conjecture +(SCC) affirms such an assumption. Obviously, another point is that astrophysical BHs are +stable because of an exceptional component called the perturbation-damping system, which is +applied in the outer region. Also, the SCC resolves the issue of the idea of the singularities +tracked down in many answers to Einstein’s gravitational field equations: Are such singular- +ities conventionally described by unbounded curvature? Is the presence of a Cauchy horizon, +10 + +an unsteady characteristic element of answers of Einstein’s equations? Recently researchers, +remarking on the historical backdrop of the SCC conjecture, overview a portion of the headway +made in research coordinated either toward satisfying SCC or toward revealing a portion of its +shortcomings. They specifically around model adaptations of SCC which have been demon- +strated for constrained groups of spacetimes viz the Gowdy spacetimes and the role played by +the conventional presence of Asymptotically speed term dominated conduct in these answers. +Also additionally note some work on spacetimes containing weak null singularities, and their +importance for the SCC [44, 45, 47]. SCC conjecture has been one of the main acts of pure +confidence with regard to GR, confirming the deterministic idea of the related field relations. +However, it holds well for asymptotically level spacetimes, an expected disappointment of the +SCC conjecture could emerge for spacetimes acquiring Cauchy horizon alongside a positive +cosmological constant viz its potential failure about this issue. Researchers have unequivocally +exhibited that infringement of the restriction SCC turns out as expected within the sight of +a Maxwell field even with the presence of higher spacetime aspects. Specifically, for higher +dimensions of the RN black holes, the infringement of SCC is at a bigger scope compared with +the 4D case, for specific of the cosmological constant. Then again, for a brane world BH, the +impact of an additional dimension is to make the infringement of cosmic censorship weaker. +For rotating BHs, intriguingly, the SCC is constantly holding even in the presence of higher +dimensions. A comparable situation is likewise noticed for rotating BHs on the brane [47]. +In this paper, we investigated dynamically formed charged black holes. Also, to satisfy the +SCC, the inner Cauchy horizons of the black hole must be unstable. Here, to check the SCC, +it is necessary to get two −Im(ω0) and k− parameters to demonstrate the decay rate of the +remaining perturbation fields in the outer regions of the black hole and the blue-shift growth +rate of the in-falling fields of the black hole, respectively. Therefore, if β = −Im(ω0) +k− +< 1/2, SCC +will be maintained. We found that for the dS charged black hole with respect to r+ ≥ Q in +light of the WGC, viz q/m ≥ 1, SCC will definitely be satisfied. We also found that there will +be a possibility of violation of SCC for the massive scalar field as well as when the radius of the +event horizon of the charged black hole is very large. We also found SCC will be violated in +the extremality state for the charged RN-dS black hole when Λr2 ++ = 1 which is also mentioned +in [44,45]. Also, these results and conditions are completely compatible with [44,45]. On the +other hand, in [8,46], when the scalar field is uncharged, the SCC is violated, which is consistent +with (19) in this paper. Because can be obtained β > 1/2 if assume the charge of the scalar +field is zero viz q = 0. The above study also raises some questions as follows. +Is the relationship researched in this article also valid for black holes in higher dimensions? Do +other black holes in different frames satisfy the SCC and WGC simultaneously? Is it possible to +consider the SCC relation with WGC monitoring for all black holes? Is it may such a structure +also be established for black holes on the brane? We leave these questions for future work. +11 + +References +[1] A. K. Mishra and S. Chakraborty, strong cosmic censorship conjecture in higher curvature +gravity, Phys. Rev. D 101, 064041 (2020). +[2] C. Singha, S. Chakraborty and N. Dadhich, Strong cosmic censorship conjecture for a +charged BTZ black hole, J. High Energ. Phys. 2022, 28 (2022). +[3] C. Singha and N. Dadhich, Strong cosmic censorship conjecture for a charged AdS black +hole, arXiv:2209.10153 (2022). +[4] U. Moitra, Strong cosmic censorship in two dimensions, Phys. Rev. D 103, L081502 (2021). +[5] J. Ho, W. Kim, W. Kim and B. H. Lee, Investigations of strong cosmic censorship in +3-dimensional black strings, J. High Energ. Phys. 2022, 18 (2022). +[6] C. Y. Shaoa, L. J. Xina, W. Zhangb, C. G. Shaoa, Strong cosmic censorship for a charged +black hole surrounded by quintessence, Physics Letters B 835, 137512 (2022). +[7] M. Casals and C. I. S. Marinho, Glimpses of violation of strong cosmic censorship in +rotating black holes, Phys. Rev. D 106, 044060 (2022). +[8] V. Cardoso, J. L. Costa, K. Destounis, P. Hintz and A. Jansen, Quasinormal Modes and +Strong Cosmic Censorship, Phys. Rev. Lett. 120, 031103 (2018). +[9] R. A. Konoplya and A. Zhidenko, How general is the strong cosmic censorship bound for +quasinormal modes?, Journal of Cosmology and Astroparticle Physics 2022, (2022). +[10] P.R. Brady, I.G. Moss and R.C. Myers, Cosmic Censorship: As Strong As Ever, Phys. +Rev. Lett. 80, 3432 (1998). +[11] W.A. Hiscock, Evolution of the interior of a charged black hole, Phys. Lett. A 83, 110 +(1981). +[12] E. Poisson and W. Israel, Internal structure of black holes, Phys. Rev. D 41, 1796 (1990). +[13] A. Ori, Inner structure of a charged black hole: An exact mass-inflation solution, Phys. +Rev. Lett. 67, 789 (1991). +[14] M. L. Gnedin and N. Y. Gnedin, Destruction of the Cauchy horizon in the Reissner- +Nordstrom black hole, Class. Quantum Gravity 10, 1083 (1993). +[15] P.R. Brady and J.D. Smith, Black Hole Singularities: A Numerical Approach, Phys. Rev. +Lett. 75, 1256 (1995). +12 + +[16] S. Hod and T. Piran, Mass Inflation in Dynamical Gravitational Collapse of a Charged +Scalar Field, Phys. Rev. Lett. 81, 1554 (1998). +[17] M. Dafermos, Black Holes Without Spacelike Singularities, Commun. Math. Phys. 332, +729 (2014). +[18] P. Hintz and A. Vasy, Analysis of linear waves near the Cauchy horizon of cosmological +black holes, J. Math. Phys. 58, 081509 (2017). +[19] S. Klainerman, I. Rodnianski, and J. Szeftel, The bounded L2 curvature conjecture, Invent. +Math. 202, 91 (2015). +[20] E. Palti, The Swampland: Introduction and Review, Fortsch. Phys., 67, 1900037 (2019). +[21] S. Fichet and P. Saraswat, Approximate Symmetries and Gravity, JHEP. 01, 088 (2020). +[22] T. Daus, A. Hebecker, S. Leonhardt and J. March-Russell, Towards a Swampland Global +Symmetry conjcture using weak gravity, Nucl. Phys. B. 960, 115167 (2020). +[23] M. van Beest, J. Calder´on-Infante, D. Mirfendereski and I. Valenzuela, Lectures on the +swampland program in string compactifications. Physics Reports 989, 1-50 (2022). +[24] E. Palti, A Brief Introduction to the Weak Gravity Conjecture, LHEP 2020, 176 (2020). +[25] C. Vafa, The String landscape and the swampland, hep-th/0509212. +[26] N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The String landscape, black holes and +gravity as the weakest force, JHEP 06, 060 (2007). +[27] J. Sadeghi, M. Shokri, S. N. Gashti and M. R. Alipour, RPS thermodynamics of Taub–NUT +AdS black holes in the presence of central charge and the weak gravity conjecture. Gen +Relativ Gravit 54, 129 (2022). +[28] J. Sadeghi, M. Shokri, M. R. Alipour and S. N. Gashti, Weak gravity conjecture from +conformal field theory: a challenge from hyperscaling violating and Kerr-Newman-AdS +black holes, Chinese Physics C 47, 015103 (2023). +[29] J. Sadeghi, S. N. Gashti and E N. Mezerji, The investigation of universal relation between +corrections to entropy and extremality bounds with verification WGC, Physics of the Dark +Universe 30, 100626 (2020). +[30] J. Sadeghi, B. Pourhassan, S. N. Gashti and S. Upadhyay, Weak Gravity Conjecture, +Black Branes and Violations of Universal Thermodynamic Relation, Annals of Physics +447, 169168 (2022). +13 + +[31] R. A. Konoplya and A. Zhidenko, Massive charged scalar field in the Kerr-Newman back- +ground: Quasinormal modes, late-time tails and stability, Phys. Rev. D 88, 024054 (2013). +[32] S. Hod, Mode-Coupling in Rotating Gravitational Collapse of a Scalar Field, Phys. Rev. +D 61, 024033 (2000). +[33] E. N. Mezerji, J. Sadeghi, The correlation of WGC and hydrodynamics bound with R4 +correction in the charged AdSd+ 2 black brane, Nuclear Physics B 981, 115858 (2022). +[34] S. Hod, Strong cosmic censorship in charged black-hole spacetimes: As strong as ever, +Nuclear PhysicsB 941, 636 (2019). +[35] R.A. Konoplya, A. Zhidenko, Charged scalar field instability between the event and cos- +mological horizons, Phys. Rev. D 90, 064048 (2014). +[36] R. A. Konoplya and A. Zhidenko, Quasinormal modes of black holes: From astrophysics +to string theory, Rev. Mod. Phys. 83, 793 (2011). +[37] J. Schwinger, On Gauge Invariance and Vacuum Polarization, Phys. Rev. 82, 664 (1951). +[38] B. Carter, Charge and Particle Conservation in Black-Hole Decay, Phys. Rev. Lett. 33, +558 (1974). +[39] S. Hod, Best approximation to a reversible process in black-hole physics and the area +spectrum of spherical black holes, Phys. Rev. D 59, 024014 (1999). +[40] W. T. Zaumen, Upper bound on the electric charge of a black hole, Nature 247, 531 (1974). +[41] S. Iyer and C.M. Will, Black-hole normal modes: A WKB approach. I. Foundations and +application of a higher-order WKB analysis of potential-barrier scattering, Phys. Rev. D +35, 3621 (1987). +[42] S. Iyer, Black-hole normal modes: A WKB approach. II. Schwarzschild black holes, Phys. +Rev. D 35, 3632 (1987). +[43] B. F. Schutz and C. M. Will, Black hole normal modes - A semianalytic approach, Astro- +phys. J. 291, L33 (1985). +[44] O. J. C. Dias, H. S. Reall, and J. E. Santos, Strong cosmic censorship for charged de Sitter +black holes with a charged scalar field, Class. Quantum Grav. 36, 045005 (2019). +[45] O. J. C. Dias, H. S. Reall, and J. E. Santos, Strong cosmic censorship: taking the rough +with the smooth, J. High Energ. Phys. 2018, 1 (2018). +14 + +[46] V. Cardoso, J. L. Costa, K. Destounis, P. Hintz and A. Jansen, Strong cosmic censorship +in charged black-hole spacetimes: Still subtle, Phys. Rev. D 98, 104007 (2018). +[47] M. Rahman, S. Chakraborty, S. S. Guptab and A. A. Sen, Fate of strong cosmic censorship +conjecture in presence of higher spacetime dimensions, JHEP 03, 178 (2019). +15 + diff --git a/8NFLT4oBgHgl3EQfBC4o/content/tmp_files/load_file.txt b/8NFLT4oBgHgl3EQfBC4o/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b188915becedb936f024e39511094140f1a2094a --- /dev/null +++ b/8NFLT4oBgHgl3EQfBC4o/content/tmp_files/load_file.txt @@ -0,0 +1,588 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf,len=587 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='11968v1 [hep-th] 27 Jan 2023 Strong Cosmic Censorship in light of Weak Gravity Conjecture for Charged Black Holes Jafar Sadeghi ⋆1, Mohammad Reza Alipour ⋆2, Saeed Noori Gashti⋆3 ⋆Department of Physics, Faculty of Basic Sciences, University of Mazandaran P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Box 47416-95447, Babolsar, Iran Abstract In this paper, we investigate the strong cosmic censorship conjecture (SCC) for charged black holes in the de Sitter space by considering the weak gravity conjecture (WGC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Us- ing analytical methods, we find that the SCC is preserved for dS-charged black holes with respect to some restriction qQ ≫ 1 and r+ ≥ Q with the help of the WGC condition viz q m ≥ 1 for scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Where q, m are the charge and mass of the scalar field, and r+, Q determine the radius of the outer event horizon and the charge of the black hole, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' In that case, when the (WGC) is valid, SCC will definitely be satisfied for the dS-charged black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' On the other hand, the SCC is violated when the WGC is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Also, we examined the RN-dS charged black hole in the extremality state and found that SCC can be violated with the condition Λr2 + = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Keywords: Strong cosmic censorship conjecture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Weak gravity conjecture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' RN-dS charged black hole Contents 1 Introduction 2 2 Weak Gravity Conjecture 4 3 Charged Black Holes in dS Space 6 4 The Quasinormal Resonant Frequency Spectrum 7 5 Discussion and Result 10 1Email: pouriya@ipm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='ir 2Email: mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='alipour@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='umz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='ir 3Email: saeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='noorigashti@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='umz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='ir 1 1 Introduction One of the studies with a long history in general relativity is the study of the collapse of small perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' We need more information on how these oscillations decay to understand better the gravity concept, use gravitational wave data, and study and investigate the valuable features of general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' One of the signs of the failure of determinism in general relativity can be the emergence of an interesting phenomenon known as Cauchy horizons that appear in the astrophysical solutions of Einstein’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' These horizons are such that it is impossible to specify the history of the future of an observer that passes through such horizons using Einstein’s equations and initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' With these descriptions in the black holes’ space-time background, it is an expected possibility that the perturbations of the outer region are infinitely amplified by a mechanism known as the blue shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' They lead to a singularity boundary beyond the Cauchy horizon in the interior of black holes, where field equations cease to make sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The Penrose strong cosmic censorship (SCC) confirms such an expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Of course, another point is that astrophysical black holes are stable due to a special mechanism called the perturbation- damping mechanism, which is applied in the outer region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Therefore, considering whether SCC retains real hinges or not depends on the very subtle competition between the collapse of perturbations in the outer region and their amplification (blue shift) in the inner space- time of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' In general, the fate of Cauchy horizons is related to the collapse of small perturbations outside the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Hence, the validity of SCC is tied to the extent of external damps fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' In connection with various structures and conditions, SCC and its satisfaction and violation have been investigated in various theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The violation of this conjecture near the extremal region studied in the investigation of higher curvature gravity [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Also, this conjecture has been challenged in investigating many charged black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' In [2,3], this conjecture was checked for a charged AdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' It was shown that for a specific interval for the parameter (β), this conjecture is satisfied and violated in other areas as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The strong cosmic censorship conjecture has also been investigated in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' There have been interesting outcomes regarding the violation of this conjecture near the extremal region at specific points [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The study of this conjecture in the structure of three-dimensional black strings has also carried interesting results, which you can see [5] for a deeper study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Studying the validity and violation of this conjecture in many recent studies in different conditions and frameworks has led to exciting results that you can see [6–9] for further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Therefore, in this article, we want to study a different structure of this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' According to the above explanation, we consider the general configuration of charged black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Then, using the weak gravity conjecture, we will prove that SCC is valid for specific values for all charged black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' We will use the weak gravity conjecture to prove a general relation for all charged black holes about SCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' In connection with SCC, we need to pay attention to more concepts, which we will mention here for further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The effectiveness of mass-inflation systems, which are involved in the transformations of the inner Cauchy horizon associated with the space-time of 2 black holes that are approximately flat, which is pathological in the estimation of SCC, into a series of hypersurfaces which is singular non-extendable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Those that are in an indivisible form are related to two different types of physical mechanisms [10–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' First, the events in the exterior space-time regions of dynamic black holes formed viz the collapse of the remnant perturbation fields and second amplification of exponential blue shift related to the fields falling into the inside of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' We can manage these two introduced different systems through parameters such as (g) and (k−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' It can be stated that the dimensionless physical ratio with the help of these two parameters can determine the fate of the inner Cauchy horizons inside such space-times of non-asymptotic flat black holes [8,17,18], β ≡ g k− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Of course, a certain range of parameters of black holes, such as mass and charge, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=', as indicated in [8,17,18], β > 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' So, space-time of the corresponding black holes can be physically expanded beyond their Cauchy horizon which includes a pathological fact and a sign of algebraic failure or a violation of the Penrose SCC in classical general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' For the dynamics of Einstein’s equations as well as the destiny of the observers, the explosive structure of the curvature that is related to (β < 1) does not have per se much physical significance: it indicates two theorems, not the failure of the field equations mentioned in [19] and of course not the destruction of macroscopic observers which is discussed in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Therefore, the physical and mathematical formulation of the conjecture of a SCC in such conditions leads to ignoring physical phenomena such as impulsive gravitational waves or the formation of shocks in relativistic fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Due to the aforementioned reasons, the modern form of the CC conjecture was introduced that requires a stronger constraint (β < 1 2 ) and many works have been done to fit such constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' For example, by studying massless scalar fields in linear form and examining the entire parametric space of a charged black hole, areas beyond mentioned range were obtained, which it seems cannot be allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' According to the above explanations, we organize the article in the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' In section 2, we will give basic explanations about the weak gravity conjecture and also the motivation to use it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' In section 3, we will introduce charged black holes in dS space, and then we will introduce the quasinormal resonant frequency spectrum in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' We will check the conditions of compatibility and violation of (SCC) with respect to (WGC) for RNdS charged black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Finally, we describe the results in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 3 2 Weak Gravity Conjecture As it is known in the literature, a new idea has been put forward as a swampland program to check theories coupled to gravity, to check the consistency of quantum gravity, and finally, a proof for string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Recently, ones have done lots of work on this field [20–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Due to the special conditions of string theory and the fact that its testing and experimental investigations seem a bit difficult, this idea has been proposed to test and investigate various concepts of cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The swampland program is challenged from two sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' From an up-bottom view for introducing principles and limitations to introduce conjectures, as well as mathematical formulations to examine cosmological concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' A second look from the bottom-up in order to test each of these conjectures with various concepts of cosmology including inflation and matching with observable data, which is both a proof for this new idea and a proof for string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' So far, many conjectures have been proposed from this theory, and now, according to the structure and further investigations, new conjectures will be added to this program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Some of these conjectures face challenges and as a result, corrections are made to the conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' We face some limitations in quantum gravity (QG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' At the point when gravity is considered at the quantum level, the hypothesis will be incompatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Generally having a reliable quantum hypothesis of gravity isn’t really straightforward and can in any case hold many surprises and can be interesting for physical science at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The objective of the swampland program is to decide the limitations that an effective field theory(EFT) should fulfill to be viable with the consideration of ultraviolet completion(UV) in QG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' They are called swampland limitations, and different suggestions are figured out as far as swampland conjectures(SC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The objective is to recognize these limitations, accumulate proof to demonstrate or refute them inside the structure of QG, give reasoning to make sense of them in a model-free manner, and comprehend their phenomenological suggestions for low-energy EFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Albeit the swampland idea isn’t restricted to string theory on a fundamental level, SC are frequently examined by string theory backdrops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Without a doubt, the string theory gives an ideal structure to thorough quantitative testing of conjectures and works on how we might interpret potential compressions of string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Strangely, it has as of late been uncovered that a large number of these conjectures are to be sure related, recommending that they may essentially be various countenances of some yet-to- be-found crucial standard of QG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' As far as possible have significant ramifications for cosmology and particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' They can give new core values to building conjectures past the standard models in high-energy physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' They may likewise prompt UV/IR blending, which breaks the assumption for scale detachment and possibly gives new bits of knowledge into the regular issues seen in our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Consequently, the presence of swampland is extraordinary information for phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' For a total rundown of references connected with swampland that might be valuable, we allude in [20] the swampland program (SP) has likewise been surveyed and presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The shortfall of global symmetry (GS) and the completeness of charge spectra are at the center of the SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Nonetheless, they need phenomenological suggestions except if we can 4 restrict the global symmetries [21,22] and whether there is any limitations point on the mass of charged states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' In any case, they just bound the complete hypothesis but not the low-energy EFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Specifically, it is phenomenologically important whether all charged particles can be really super heavy and even compare to black holes(BHs), or whether there is some thought of completeness of the range that gets by at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' A large portion of the SCs examined address exactly these inquiries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' They want to profoundly explore these assertions and measure them so we can draw nearer to the recuperation of a few global symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' For instance, we can deduce recuperate a global symmetry (GS) U(1) by sending the gauge coupling(GC) to nothing, which ought not to be permitted in QG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Attempting to comprehend string theory for the study of this issue, it might turn out that if one somehow managed to try to do such work, can give data about the imperatives that an EFT can fulfill to be viable with QG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Likewise, WGC forbids this cycle by flagging the presence of new charged states that denies the depiction of the EFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Thusly, it gives an upper bound on the mass of these charged states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The WGC comprises of some parts: the electric and the magnetic electric-WGC: As indicated by a quantum hypothesis, we have the following condition [20–30], Q m ≥ Q M |ext = O(1), (1) and Q = gq, (2) where, g and q are the gauge coupling and the quantized charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The electric-WGC needs the presence of an electrically charged condition of a higher charge-to-mass proportion than extremal BH in that hypothesis, which is regularly a variable of the order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' One more understanding of this conjecture is that the limitations region shows that scale force determines stronger than the gravity on this mode — so subsequently is called WGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' This is an identical equation since it expects that electromagnetic force is stronger gravitational force [20–30], FGrav ≤ FEM (3) It implies that the charge is more prominent than the mass, so we get a similar condition as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' This is as of now false within the sight of massless scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The motivations of WGC are twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' To begin with, it makes a QG boundary to reestablish the GS of U(1) by sending g → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' If a GC goes to zero as indicated by WGC, this conducts new light particles and the cutoff the hypothesis arrives at nothing and nullifies the EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Because of the littleness of the scale coupling, it relies upon how much energy the interaction with which you need to portray the viable EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The smallness of the cycle energy leads to the smallness of the scale coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' On the other hand, if you need to keep the EFT substantial up to an extremely high cut-off, the GC can’t be excessively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' This is an illustration of swampland limitations that 5 becomes more grounded for higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Obviously, a hypothesis with disappearing measure coupling i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=', GS is incompatible because the cutoff of the viable EFT is likewise zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' One more fundamental inspiration for WGC is that a kinematic prerequisite permits extremality BH to have decomposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Charged BHs should fulfill an extremality breaking point to stay away from the presence of exposed singularities, as shown by the weak cosmic censorship (WCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' For a given charge Q, this super bound shows that the this super bound shows that the mass M of the BHs should be more noteworthy than the charge [20–30], M ≥ Q (4) For the BHs to have a regular horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Here, we set the extremal factor O(1) to one for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The primary condition for starting the decay to the small black hole and particle is the existence of the extremal BHs (M = Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' So, one can consider the decay of an extremal BHs which one of the rot items has a charge more modest than its mass as far as possible, so M1 ≥ Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Then the rot item can never again have a charge more modest than the mass, that is m2 ≤ Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' It is just a kinematic necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Since the second rot item violates the WCC, it can’t be a BH, so it should be a particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The above kinematic necessity can be acquired by applying preservation of mass/energy and protection of charge as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The initial mass of the BH should be more prominent than the amount of the mass of the rot items Mi and the charge of the initial BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 3 Charged Black Holes in dS Space The metric of charged black hole in spherical symmetric space is defined as follows, dS2 = f(r)dt2 − f −1(r)dr2 − r2dΩ2, dΩ2 = (dθ2 + sin2(θ)dϕ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' (5) Here, we consider f(r) = H(M, Q) − Λr2 3 in general;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' where Q, M, Λ > 0 are electric charge, the mass of the black hole and the cosmological constant respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' In this case, we can obtain its event horizons as follows, f(r⋆) = 0 → ⋆ ∈ (−, +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=', c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' (6) Considering the metric in general terms, we have different event horizons, where (r−) is the Cauchy horizon, (r+) is the outer event horizon, and (rc) is the cosmological horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Using Klein-Gordon’s differential equation, we can determine the dynamics of a massive charged particle near a charged black hole [31–34], 1 √−g∂µ(gµν√−g∂νΦ) − 2iqgµνAµ∂νΦ − q2gµνAµAνΦ − m2Φ = 0, (7) 6 where m and q are the mass and charge of the particle, respectively also, Aµ = � Q r , 0, 0, 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' We can define the scalar field Φ according to relation (7) as follows [36], Φ(t, r, θ, φ) = � m � ℓ e−iωtYℓm(θ, ϕ)Φ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' (8) The integer parameters ℓ and m are the spherical and the azimuthal harmonic indices of the resonant eigenmodes which characterize the charged massive scalar fields in the charged black- hole spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' By putting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' (8) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' (7) and using dx = dr f(r), we get the Schr¨odinger-like differential equation , d2φ(r) dx2 + V (r)φ(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' (9) The effective radial potential due to a massive charged particle near a charged black hole is defined as [8], V (r) = qm r2 � q mα(r) − m q β(r) � , (10) where α(r) = Q2 � 1 − ωr qQ �2 , β(r) = r2f(r)H(r), H(r) = �ℓ(ℓ + 1) m2r2 + f ′(r) m2r + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' (11) Also, we can consider the boundary conditions for the special radial function near the outer event horizon as an incoming wave and at the largest event horizon as an outgoing wave [34,35]: φ(x) ∼ � e −i(ω− qQ r+ )x, for r → r+ (x → −∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' e−i(ω− qQ rc )x, for r → rc (x → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' (12) According to the above boundary conditions, we can obtain the discrete spectrum of ω, defined as the resonance frequency of the imaginary quasi-normal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 4 The Quasinormal Resonant Frequency Spectrum In this section, we need to obtain the imaginary part of the resonance frequency to investigate the linear dynamics of a massive charged particle near a general charged black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Also, we need to do this in a dimensionless regime to do this analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Since, the q2 ¯h ≃ 1 137 relationship exists in our universe, we can consider it for black holes, even slightly charged, and get qQ ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' In addition, the mechanism of the Schwinger-type pair-production in space-time of charged black hole creates a limit to the black hole electric field with the Q r2 + ≪ m2 q relationship [37–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 7 Therefore, according to the above statement, we can consider SCC and define our constraint regime following ansans, m2r2 + ≫ ℓ(ℓ + 1) and m2r2 + ≫ 2k+r+, (13) where k+ = f ′(r+)/2 is the gravitational acceleration of the black hole at the outer event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' In this area, we try to obtain the imaginary part of the resonance frequency in the background of the general charged black hole near the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Now, we use radial potential (10) to determine the linear dynamics of the massive charged particle near the event horizon of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' We can consider this potential in region (13) as an effective potential and obtain the quasinormal resonance modes analytically using standard WKB techniques [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' In this region, we consider the maximum effective potential near the event horizon of the black hole at point r = r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' In the following,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' we use the relationship (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' (11),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' and V ′(r0) = 0 to obtain the point where the effective potential is maximum as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' r0 = q2Q2 qQω − m2r2 +k+ (14) According to the Schr¨odinger-like differential equation (9) and [41–43],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' we use the WKB method to obtain the quasinormal mode frequencies through the following,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='iK − (n + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='2) − Λ(n) = Ω(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='(15) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='K = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='V0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='2V (2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='Λ(n) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='2V (2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='\uf8ee ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='\uf8f9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='\uf8fa\uf8fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='(16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' V (k) 0 ≡ dkV dxk |r=r0 is the spatial derivative of the effective potential of equation (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' and its scattering peak is evaluated at the point r = r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Using relations (10), (11), (14) and (16), 8 we will have the following relation in the region of (13), K ≃ k2 +m4r4 +qQ 2f0 (k+m2r2 + − qQω)2 Λ(n) ≃ k2 +m4 � 17 − 60 � n + 1 2 �2� r4 + + 2k+m2 � 36 � n + 1 2 �2 − 7 � qQr2 +ω 16qQ (qQω − 3k+m2r2 +)2 × f0 A = 15k4 +m8 � 148(n + 1 2)2 − 41 � r8 + + 12k3 +m6 � 121 − 420(n + 1 2)2 � qQr6 +ω B = 64q5Q5 � k+m2r2 + − qQω �4 Ω(n) ≃ −(n + 1 2)q3Q3f 2 0 × A B (17) where f0 = f(r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' In the next step, to determine the study of SCC, we need to obtain the minimum value of the fundamental imaginary resonance mode of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' For this purpose, using equations (15) and (17), we can calculate the Im(ω0), ω ≃ qQ r+ − 2k+m2r2 + qQ � 1 − 14400 11644 �(n + 1/2)f0 qQ �4� − i � 4f0k+(n + 1 2)m2r2 + q2Q2 � 1 − 34qQf 4 0 11664 � + O(f 2 0) � (18) Since we consider r0 near the event horizon (r+), we have f0 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' For investigation the SCC, it is necessary to find the minimum value of the resonance mode and evaluate its ratio to the surface gravity of the event horizon, β = −Im(ω0) k+ ≃ 2f0 m2r2 + q2Q2 � 1 − 34qQf 4 0 11664 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' (19) Since it is f0 ≪ 1, it is sufficient to have the conditions q2Q2 > m2r2 + in the relation above concepts so that −Im(ω0) k+ < 1 2 is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Therefore, we have the following condition for the study of SCC, q m ≥ r+ Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' (20) from equation (20) determine that when r+ ≥ Q, we have the weak gravity conjecture condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' We know that k− > k+, so the relationship of (19) and (20) is also established for β = −Im(ω0) k− < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Also, according to relation (19), when qQ < 2√f0mr+, SCC can be violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Since qQ ≫ 1 and f0 ≪ 1, the mass of the scalar field and the radius of the event horizon must be very massive and very large respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' In the following, we obtain the extremality state of the 9 RN-dS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' We will have the following relation for the RN-dS black hole with respect to equation(5), f(r) = 1 − 2M r + Q2 r2 − Λr2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' (21) When k+ = k− = 0, we can obtain the black hole extremality state, Qexe = r+ � 1 − Λr2 +, Mexe = r+(1 − 2 3Λr2 +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' (22) We substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' (22) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' (19) to obtain β in the extremality state of the RN black hole, β ≃ 2f0 m2 q2(1 − Λr2 +) � 1 − 34qf 4 0 � 1 − Λr2 +r+ 11664 � (23) According the above relationship, when the condition q m ≥ 1 1−Λr2 + is satisfied, the SCC will definitely be preserved, and since Λr2 + < 1, the weak gravity conjecture will also be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' On the other hand, when Λr2 + ≪ 1, we will have the SCC condition in light of the WGC, q m ≥ 1 + Λr2 +, (24) from the above relation WGC is clearly obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' In relation (23) when Λr2 + = 1, we have β → ∞ and the SCC is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Also, these result and conditions are completely compatible with [44,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 5 Discussion and Result One of the indications of the failure of determinism GR can be the rise of a fascinating pecu- liarity known as the Cauchy horizon that shows up in the astrophysical solutions of Einstein’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' These horizons are such that it is difficult to indicate the history of the future of an observer that passes come of such horizons utilizing Einstein’s conditions and initial infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' With these descriptions in the black holes’ background space-time, it is a predicted possibility that the perturbations of the external area are infinitely enhanced by a system known as the blue shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' They lead to a singularity beyond the Cauchy horizon the inside of BHs, where field conditions fail to seem good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The Penrose cosmic censorship conjecture (SCC) affirms such an assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Obviously, another point is that astrophysical BHs are stable because of an exceptional component called the perturbation-damping system, which is applied in the outer region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Also, the SCC resolves the issue of the idea of the singularities tracked down in many answers to Einstein’s gravitational field equations: Are such singular- ities conventionally described by unbounded curvature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Is the presence of a Cauchy horizon, 10 an unsteady characteristic element of answers of Einstein’s equations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Recently researchers, remarking on the historical backdrop of the SCC conjecture, overview a portion of the headway made in research coordinated either toward satisfying SCC or toward revealing a portion of its shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' They specifically around model adaptations of SCC which have been demon- strated for constrained groups of spacetimes viz the Gowdy spacetimes and the role played by the conventional presence of Asymptotically speed term dominated conduct in these answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Also additionally note some work on spacetimes containing weak null singularities, and their importance for the SCC [44, 45, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' SCC conjecture has been one of the main acts of pure confidence with regard to GR, confirming the deterministic idea of the related field relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' However, it holds well for asymptotically level spacetimes, an expected disappointment of the SCC conjecture could emerge for spacetimes acquiring Cauchy horizon alongside a positive cosmological constant viz its potential failure about this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Researchers have unequivocally exhibited that infringement of the restriction SCC turns out as expected within the sight of a Maxwell field even with the presence of higher spacetime aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Specifically, for higher dimensions of the RN black holes, the infringement of SCC is at a bigger scope compared with the 4D case, for specific of the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Then again, for a brane world BH, the impact of an additional dimension is to make the infringement of cosmic censorship weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' For rotating BHs, intriguingly, the SCC is constantly holding even in the presence of higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' A comparable situation is likewise noticed for rotating BHs on the brane [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' In this paper, we investigated dynamically formed charged black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Also, to satisfy the SCC, the inner Cauchy horizons of the black hole must be unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Here, to check the SCC, it is necessary to get two −Im(ω0) and k− parameters to demonstrate the decay rate of the remaining perturbation fields in the outer regions of the black hole and the blue-shift growth rate of the in-falling fields of the black hole, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Therefore, if β = −Im(ω0) k− < 1/2, SCC will be maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' We found that for the dS charged black hole with respect to r+ ≥ Q in light of the WGC, viz q/m ≥ 1, SCC will definitely be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' We also found that there will be a possibility of violation of SCC for the massive scalar field as well as when the radius of the event horizon of the charged black hole is very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' We also found SCC will be violated in the extremality state for the charged RN-dS black hole when Λr2 + = 1 which is also mentioned in [44,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Also, these results and conditions are completely compatible with [44,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' On the other hand, in [8,46], when the scalar field is uncharged, the SCC is violated, which is consistent with (19) in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Because can be obtained β > 1/2 if assume the charge of the scalar field is zero viz q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' The above study also raises some questions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Is the relationship researched in this article also valid for black holes in higher dimensions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Do other black holes in different frames satisfy the SCC and WGC simultaneously?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Is it possible to consider the SCC relation with WGC monitoring for all black holes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Is it may such a structure also be established for black holes on the brane?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' We leave these questions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 11 References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Mishra and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Chakraborty, strong cosmic censorship conjecture in higher curvature gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' D 101, 064041 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Singha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Chakraborty and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Dadhich, Strong cosmic censorship conjecture for a charged BTZ black hole, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' High Energ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 2022, 28 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Singha and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Dadhich, Strong cosmic censorship conjecture for a charged AdS black hole, arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='10153 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [4] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Moitra, Strong cosmic censorship in two dimensions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' D 103, L081502 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Ho, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Kim, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Kim and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Lee, Investigations of strong cosmic censorship in 3-dimensional black strings, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' High Energ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 2022, 18 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Shaoa, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Xina, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Zhangb, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Shaoa, Strong cosmic censorship for a charged black hole surrounded by quintessence, Physics Letters B 835, 137512 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Casals and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Marinho, Glimpses of violation of strong cosmic censorship in rotating black holes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' D 106, 044060 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [8] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Cardoso, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Costa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Destounis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Hintz and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Jansen, Quasinormal Modes and Strong Cosmic Censorship, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 120, 031103 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Konoplya and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Zhidenko, How general is the strong cosmic censorship bound for quasinormal modes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=', Journal of Cosmology and Astroparticle Physics 2022, (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [10] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Brady, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Moss and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Myers, Cosmic Censorship: As Strong As Ever, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 80, 3432 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [11] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Hiscock, Evolution of the interior of a charged black hole, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' A 83, 110 (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [12] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Poisson and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Israel, Internal structure of black holes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' D 41, 1796 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Ori, Inner structure of a charged black hole: An exact mass-inflation solution, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 67, 789 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Gnedin and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Gnedin, Destruction of the Cauchy horizon in the Reissner- Nordstrom black hole, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Quantum Gravity 10, 1083 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [15] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Brady and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Smith, Black Hole Singularities: A Numerical Approach, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 75, 1256 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 12 [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Hod and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Piran, Mass Inflation in Dynamical Gravitational Collapse of a Charged Scalar Field, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 81, 1554 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Dafermos, Black Holes Without Spacelike Singularities, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 332, 729 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [18] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Hintz and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Vasy, Analysis of linear waves near the Cauchy horizon of cosmological black holes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 58, 081509 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Klainerman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rodnianski, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Szeftel, The bounded L2 curvature conjecture, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 202, 91 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [20] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Palti, The Swampland: Introduction and Review, Fortsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=', 67, 1900037 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [21] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Fichet and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Saraswat, Approximate Symmetries and Gravity, JHEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 01, 088 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [22] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Daus, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Hebecker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Leonhardt and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' March-Russell, Towards a Swampland Global Symmetry conjcture using weak gravity, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 960, 115167 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' van Beest, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Calder´on-Infante, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Mirfendereski and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Valenzuela, Lectures on the swampland program in string compactifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Physics Reports 989, 1-50 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [24] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Palti, A Brief Introduction to the Weak Gravity Conjecture, LHEP 2020, 176 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [25] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Vafa, The String landscape and the swampland, hep-th/0509212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [26] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Arkani-Hamed, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Motl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Nicolis and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Vafa, The String landscape, black holes and gravity as the weakest force, JHEP 06, 060 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Sadeghi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Shokri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Gashti and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Alipour, RPS thermodynamics of Taub–NUT AdS black holes in the presence of central charge and the weak gravity conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Gen Relativ Gravit 54, 129 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Sadeghi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Shokri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Alipour and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Gashti, Weak gravity conjecture from conformal field theory: a challenge from hyperscaling violating and Kerr-Newman-AdS black holes, Chinese Physics C 47, 015103 (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Sadeghi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Gashti and E N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Mezerji, The investigation of universal relation between corrections to entropy and extremality bounds with verification WGC, Physics of the Dark Universe 30, 100626 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Sadeghi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Pourhassan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Gashti and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Upadhyay, Weak Gravity Conjecture, Black Branes and Violations of Universal Thermodynamic Relation, Annals of Physics 447, 169168 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 13 [31] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Konoplya and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Zhidenko, Massive charged scalar field in the Kerr-Newman back- ground: Quasinormal modes, late-time tails and stability, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' D 88, 024054 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [32] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Hod, Mode-Coupling in Rotating Gravitational Collapse of a Scalar Field, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' D 61, 024033 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [33] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Mezerji, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Sadeghi, The correlation of WGC and hydrodynamics bound with R4 correction in the charged AdSd+ 2 black brane, Nuclear Physics B 981, 115858 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [34] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Hod, Strong cosmic censorship in charged black-hole spacetimes: As strong as ever, Nuclear PhysicsB 941, 636 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [35] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Konoplya, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Zhidenko, Charged scalar field instability between the event and cos- mological horizons, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' D 90, 064048 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [36] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Konoplya and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Zhidenko, Quasinormal modes of black holes: From astrophysics to string theory, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 83, 793 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [37] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Schwinger, On Gauge Invariance and Vacuum Polarization, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 82, 664 (1951).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [38] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Carter, Charge and Particle Conservation in Black-Hole Decay, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 33, 558 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [39] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Hod, Best approximation to a reversible process in black-hole physics and the area spectrum of spherical black holes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' D 59, 024014 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [40] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Zaumen, Upper bound on the electric charge of a black hole, Nature 247, 531 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [41] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Iyer and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Will, Black-hole normal modes: A WKB approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Foundations and application of a higher-order WKB analysis of potential-barrier scattering, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' D 35, 3621 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [42] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Iyer, Black-hole normal modes: A WKB approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Schwarzschild black holes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' D 35, 3632 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [43] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Schutz and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Will, Black hole normal modes - A semianalytic approach, Astro- phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 291, L33 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [44] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Dias, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Reall, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Santos, Strong cosmic censorship for charged de Sitter black holes with a charged scalar field, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Quantum Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 36, 045005 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [45] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Dias, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Reall, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Santos, Strong cosmic censorship: taking the rough with the smooth, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' High Energ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 2018, 1 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 14 [46] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Cardoso, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Costa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Destounis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Hintz and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Jansen, Strong cosmic censorship in charged black-hole spacetimes: Still subtle, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' D 98, 104007 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' [47] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Rahman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Chakraborty, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Guptab and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' Sen, Fate of strong cosmic censorship conjecture in presence of higher spacetime dimensions, JHEP 03, 178 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} +page_content=' 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfBC4o/content/2301.11968v1.pdf'} diff --git a/9dE3T4oBgHgl3EQfrApq/content/tmp_files/2301.04656v1.pdf.txt b/9dE3T4oBgHgl3EQfrApq/content/tmp_files/2301.04656v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..676207a8ce7f720bff4339f4fcfe99e8ec84b7f4 --- /dev/null +++ b/9dE3T4oBgHgl3EQfrApq/content/tmp_files/2301.04656v1.pdf.txt @@ -0,0 +1,2367 @@ +Astronomy & Astrophysics manuscript no. disc +©ESO 2023 +January 13, 2023 +Towards a population synthesis of discs and planets +II. Confronting disc models and observations at the population level⋆ +Alexandre Emsenhuber1 , Remo Burn2 , Jesse Weder3 , Kristina Monsch4, 1 , Giovanni Picogna1 , Barbara +Ercolano1, 5 , and Thomas Preibisch1 +1 Universitäts-Sternwarte, Ludwig-Maximilians-Universität München, Scheinerstraße 1, 81679 München, Germany +e-mail: emsenhuber@usm.lmu.de +2 Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany +3 Physikalisches Institut, Universität Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland +4 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA +5 Excellence Cluster ‘Origins’, Boltzmannstraße 2, 85748 Garching, Germany +Received 18 August 2022 / Accepted 9 December 2022 +ABSTRACT +Aims. We want to find the distribution of initial conditions that best reproduces disc observations at the population level. +Methods. We first ran a parameter study using a 1D model that includes the viscous evolution of a gas disc, dust, and pebbles, coupled +with an emission model to compute the millimetre flux observable with ALMA. This was used to train a machine learning surrogate +model that can compute the relevant quantity for comparison with observations in seconds. This surrogate model was used to perform +parameter studies and synthetic disc populations. +Results. Performing a parameter study, we find that internal photoevaporation leads to a lower dependency of disc lifetime on stellar +mass than external photoevaporation. This dependence should be investigated in the future. Performing population synthesis, we find +that under the combined losses of internal and external photoevaporation, discs are too short lived. +Conclusions. To match observational constraints, future models of disc evolution need to include one or a combination of the follow- +ing processes: infall of material to replenish the discs, shielding of the disc from internal photoevaporation due to magnetically driven +disc winds, and extinction of external high-energy radiation. Nevertheless, disc properties in low-external-photoevaporation regions +can be reproduced by having more massive and compact discs. Here, the optimum values of the α viscosity parameter lie between +3 × 10−4 and 10−3 and with internal photoevaporation being the main mode of disc dispersal. +Key words. Protoplanetary disk — Methods: numerical +1. Introduction +Protoplanetary discs are the birthplace of planets (the ‘nebular +hypothesis’ of Kant and Laplace). Discs serve as a source of gas +and solids from which the planets accrete. Planet–disc interac- +tions lead to planetary migration. To model planetary formation, +it is therefore essential to have disc characteristics that are as +close as possible to those observed to provide the highest possi- +ble fidelity. +Disc observations are not an entirely new subject of research. +Disc masses (e.g. Beckwith & Sargent 1996; Andrews et al. +2009) and lifetimes (e.g. Haisch et al. 2001; Mamajek 2009; +Fedele et al. 2010; Kraus et al. 2012; Ribas et al. 2014) have +been observed for over two decades. However, there have been +many new results concerning protoplanetary discs in the last sev- +eral years, including the mass and physical extent of early discs +(Tychoniec et al. 2018, 2020; Tobin et al. 2020) and at later times +(Hendler et al. 2020). +Nevertheless, some aspects of disc evolution are not cap- +tured by observations, such as the process that leads to trans- +port of material. These are usually taken to be turbulent viscosity +⋆ Tables 3 and 4 are only available in electronic form at the +CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via +https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/ +generated by the magnetorotational instability and magnetically +driven disc winds (Suzuki & Inutsuka 2009; Suzuki et al. 2016). +The strength of the turbulent viscosity has not yet been properly +determined and is usually parametrised using a factor α (Shakura +& Sunyaev 1973). +There are indirect methods to estimate the value of α. +Ultraviolet(UV)-excess measurements of the accretion luminos- +ity were used to derive the accretion rate onto the star for the +Chamaeleon I (Manara et al. 2016a, 2017) and Lupus (Alcalá +et al. 2014, 2017) star forming regions. These measurements +coupled to the disc masses for the same regions of Cha I (Pas- +cucci et al. 2016) and Lupus (Ansdell et al. 2016) provide a rela- +tionship between mass and accretion rate (Manara et al. 2016b). +Together, these can be used to calibrate numerical models (Ma- +nara et al. 2019) and to provide an estimate of the mass flux +onto the disc (Mulders et al. 2017; Sellek et al. 2020). A second +method to estimate α is to compare dust and gas emission, either +using spatially resolved observations of disc substructures from +ALMA (Andrews et al. 2018a) such as pressure bumps (Dulle- +mond et al. 2018) or from the overall disc sizes (Toci et al. 2021). +Mass loss does not occur only due to accretion onto the star. +For instance, observations also point towards protoplanetary disc +dispersal occurring from the inside out and on relatively short +timescales (Ercolano et al. 2011, 2015; Koepferl et al. 2013). +Article number, page 1 of 18 +arXiv:2301.04656v1 [astro-ph.EP] 11 Jan 2023 + +A&A proofs: manuscript no. disc +This suggests there is an additional mechanism removing gas +close to the star, with one possibility being internal photoevapo- +ration. Coupled with the findings that young stars emit a larger +fraction of their flux in UV (e.g. Gómez de Castro 2009) and +X-rays (e.g. Preibisch et al. 1996, 2005; Feigelson & Montmerle +1999; Favata & Micela 2003), it is proposed that extreme UV +(EUV; Hollenbach et al. 1994; Clarke et al. 2001) and/or X- +rays (e.g. Alexander et al. 2004; Ercolano et al. 2008, 2009) are +responsible for this mass loss. Using hydrodynamical simula- +tions, it is possible to predict the mass-loss rate as a function of +disc properties and stellar luminosity (Owen et al. 2012; Picogna +et al. 2019, 2021; Ercolano et al. 2021). +Nevertheless, irradiation from the host star is not the only +mass-loss mechanism: most stars are born in clusters where +many stars form concurrently. Consequently, protoplanetary +discs are exposed to a larger ambient radiation field than mature +stars. This leaves an additional mechanism for mass removal by +external photoevaporation (e.g. Matsuyama et al. 2003; Winter & +Haworth 2022). This is supported by observational findings that +discs near massive stars have lower masses than others (Ansdell +et al. 2017; van Terwisga et al. 2020) and that clusters with a low +ambient radiation field have longer disc lifetimes (Michel et al. +2021). As for external photoevaporation, hydrodynamical simu- +lations were performed to predict mass-loss rate (Haworth et al. +2018) as function of disc properties and ambient flux. Together +with simulations of cluster evolution (e.g. Adams et al. 2006; +Qiao et al. 2022), this enables us to determine the mass-loss rate +over an entire disc population. +All these observations and theoretical predictions put a lot +of constraints on protoplanetary disc evolution, as the number +of free parameters is limited. Whether or not the combination +of initial disc properties and predicted accretion and mass-loss +rates can be used to reproduce the distribution of, for instance, +disc lifetimes remains to be determined. Previous studies in this +direction usually consider only one type of photoevaporation, +either internal (Gorti et al. 2009; Owen et al. 2011; Kunitomo +et al. 2021) or external (Kunitomo et al. 2020). +Burn et al. (2022, hereafter Paper I) introduced a relatively +simple 1D radial disc model that is capable of consistently evolv- +ing gas, dust, pebbles, and planetesimals. In addition, this model +is capable of predicting how the modelled disc would be ob- +served by current instrumentation, such as ALMA (see Birnstiel +et al. 2018). Further, the light computational requirements of that +model make it possible to perform many such evolutions in order +to study the effects of initial disc properties. +Our goals are twofold: first, we aim to determine whether we +can understand the general picture of protoplanetary discs set by +the observations and predictions highlighted above. Second, we +want to find the combinations of disc properties that best repro- +duce the various observations. This should then serve as initial +conditions for future planetary population syntheses, such as in +Mordasini et al. (2009) or Emsenhuber et al. (2021b). +To fit the best parameters, many calculations need to be per- +formed, each involving the evolution of a population of proto- +planetary discs. To alleviate the computational requirements of +this procedure, we use machine learning to fit neural networks +that can reproduce the result of the underlying model with lim- +ited resources (Cambioni et al. 2019a, 2021; Emsenhuber et al. +2020). This ‘surrogate model’ can then be used as the forward +model in the fitting procedure (Cambioni et al. 2019b). +In this work, we aim to find initial conditions for the disc +evolution calculations that best match observations. For this pur- +pose, we first compute two series of calculations using the model +presented in Paper I (Sect. 4.1). These data are then used to fit +several surrogate models that hold the necessary outcomes for +comparison with observations (Sect. 4.2). Using these surrogate +models, we study the effect of the photoevaporation prescriptions +(Sect. 4.3) and find initial conditions that best match the obser- +vational constraints discussed in Sect. 3 as a whole (Sect. 5). A +study dedicated to this last aspect using a Bayesian approach in- +stead will be presented in Burn et al. (in prep., hereafter Paper +III). +2. Methods +The disc evolution model is based on the Bern global model +of planetary formation and evolution (e.g. Alibert et al. 2004, +2005; Mordasini et al. 2009; Fortier et al. 2013; Voelkel et al. +2020; Emsenhuber et al. 2021a) where planet formation has +been turned off to retain only the disc part. Paper I presented +an updated version of the coupled gas and solids model that in- +cludes proper modelling of the disc dispersal stage. As the model +was extensively described in Paper I, we only provide a brief +overview of the physical processes included in the model. +2.1. Gas disc +The gas disc is modelled by an azimuthally averaged 1D ra- +dial structure. Its evolution is obtained by solving the advection– +diffusion equation (Lüst 1952; Lynden-Bell & Pringle 1974) +∂ΣG +∂t += 3 +r +∂ +∂r +� +r +1 +2 ∂ +∂r +� +νΣGr +1 +2 �� +− ˙Σint − ˙Σext, +(1) +where ΣG = +� ∞ +−∞ ρGdz is the surface density and ν = αcsH the +viscosity (parametrised using the α prescription of Shakura & +Sunyaev 1973), with cs and H being the sound speed and scale +height of the disc, and ˙Σint and ˙Σext the sink terms due to in- +ternal and external photoevaporation, respectively. To compute +the vertical structure of the disc (and with this ρG, H, and cs), +we proceed as in Paper I and use the vertically integrated ap- +proach of Nakamoto & Nakagawa (1994), including stellar irra- +diation (Hueso & Guillot 2005) from an evolving stellar lumi- +nosity computed from Baraffe et al. (2015). +2.1.1. Internal photoevaporation +Internal photoevaporation is modelled assuming X-ray-driven +mass loss. This prescription requires one parameter that is not +obtained from elsewhere in the model, the stellar X-ray lumi- +nosity LX. This luminosity is converted into a total mass-loss +rate ˙MX, and then into a profile ˙Σint using fits to hydrodynamical +simulations performed by Picogna et al. (2019), Ercolano et al. +(2021), and Picogna et al. (2021), as described in Paper I. +2.1.2. External photoevaporation +The mass-loss rate due to external photoevaporation is obtained +from the FRIED grid (Haworth et al. 2018). Interpolation in the +grid requires the stellar mass M⋆, the current disc mass MG, +its outer radius rout, and the ambient far-UV (FUV) field F. All +but the latter parameter can be computed consistently from the +disc structure. The grid spans values of the ambient field F be- +tween 10 and 104 G0, where G0 = 1.6 × 10−3 erg s−1 cm−2 ap- +proximately represents the interstellar value (Habing 1968). The +total mass-loss rate is converted into a profile ˙Σext assuming mass +is lost in the outermost 10 % where the gas disc is present at a +given time (Paper I). +Article number, page 2 of 18 + +A. Emsenhuber et al.: Towards a population synthesis of discs and planets. II. +The FRIED grid, in its current state, presents two shortcom- +ings that we adapt here: (1) the lack of data for ambient fluxes +below 10 G0 and (2) a floor evaporation rate of 10−10 M⊙ yr−1. +Both items lead to a significant external photoevaporation rate +under any circumstances, which makes it very difficult to disen- +tangle the effects of external photoevaporation from the rest (in- +cluding internal photoevaporation). To remedy these problems, +we make one addition and one change to the FRIED prescrip- +tion. The change is to take the lower boundary of the external +photoevaporation rate down to 1 × 10−15 M⊙ yr−1, which repre- +sents a negligible mass-loss rate. For this, we remove the floor +value of 10−10 M⊙ yr−1 from the value returned from the interpo- +lation in the grid and ensure that the resulting value is at least +1 × 10−15 M⊙ yr−1. The change we make is to extend the domain +down to 1 G0 to be able to study low-ambient-field cases. In the +region below 10 G0, we perform a linear interpolation between +the value returned from the grid at that boundary and a fixed +value of 1 × 10−15 M⊙ yr−1 at 1 G0. +2.2. Solids disc model +The solid component of the disc is modelled using the two- +population model of Birnstiel et al. (2012). This approximates +the full size distribution using only its two extremes: the smaller +a0 well coupled to the gas (which can be seen as dust) and the +larger, rapidly drifting a1 (which can be seen as pebbles). The +smaller size a0 = 0.1 µm is fixed while the larger size is con- +strained by various limits. The fragmentation limit is given by +a1 = fF +2 +3π +ΣG +ρsα +v2 +frag +c2s +, +(2) +where fF = 0.37 is a factor fitted to hydrodynamical simula- +tions of Birnstiel et al. (2010) for the typical size of pebbles, +ρs = 1.675 g cm−3 is the bulk density, and vfrag the fragmentation +velocity. In the drift limit, the large size is given by +a1 = fD +2Σdustv2 +K +πρsc2sζ +���∂ ln P +∂ ln r +���−1, +(3) +where vK is Keplerian velocity, Σ0 the surface density of dust +only, and ζ is an efficiency parameter of the drift (to account for +the fact that drift is more limited in discs with features such gaps +created by planets; e.g. Zormpas et al. 2022, while we only study +smooth discs). +The surface density of solids ΣD is divided into the two com- +ponents with Σ0 = ΣD (1 − fm) and Σ1 = ΣD fm. The factor +fm = 0.75 when growth is fragmentation-limited and fm = 0.97 +when drift-limited. Gas drag onto both dust and pebbles is as- +sumed to be in the Epstein regime. The radial velocity of solids +is made of two components, coupling to the radial gas flow and +headwind (Nakagawa et al. 1986; Paper I), +u0/1 = +uG,red +1 + St2 +0/1 +− +2udr +St0/1 + (St0/1ϱ2)−1 , +(4) +where uG,red is the reduced radial gas velocity according to +Gárate et al. (2020) and Paper I, udr = − +r +2vKρG ζ ∂P +∂r , St is the Stokes +number, ϱ = ρG/(ρG + ρD), and ρD is the midplane dust density. +Here, we introduce a drift efficiency ζ to parametrise mecha- +nisms that reduce the headwind-induced drift velocity of dust. +In particular, it is possible to use this approach to represent the +effect that radial substructures have on the drift of solids without +modelling them in full detail. The mass-averaged radial velocity +is then given by ¯u = (1 − fm) u0 + fmu1. As in the gas disc, time +evolution is provided by an advection–diffusion equation, +∂ΣD +∂t += 1 +r +∂ +∂r +� +r +� +ΣD¯u − DGΣG +∂ +∂r +�ΣD +ΣG +��� +− ˙Σphoto − ˙Σrad − ˙Σpts, (5) +where DG is the gas diffusion coefficient. +The terms ˙Σphoto and ˙Σrad are sink terms due to dust being +entrained by photoevaporative winds (e.g. Facchini et al. 2016; +Franz et al. 2020) and ejected due to radiation pressure, respec- +tively; they are both described in Paper I. In contrast to Paper I, +we allow planetesimals to form. This is parametrised using the +term +˙Σpts = ε +d +˙MD +2πr = ε +d|¯udr|ΣD, +(6) +which follows the prescription of Lenz et al. (2019) as imple- +mented and described in detail in Voelkel et al. (2020). Here ε +is a parameter that specifies the conversion efficiency into plan- +etesimals over a length scale of d = 5H (Dittrich et al. 2013), ¯udr +is the drift component of the mass-averaged radial velocity, and +˙MD the relative mass flux of dust and pebbles through the gas. +2.3. Conversion into observed disc masses +For consistency with disc mass observations, millimetre (mm) +emission from dust and pebbles is computed from the disc sur- +face density and temperature profiles. The method is similar to +that of Birnstiel et al. (2018) and will be discussed in more de- +tail in Paper III. The calculation is performed for a wavelength +of λ = 0.89 mm to reproduce ALMA observations. The flux is +converted back into a mass using a simple prescription assuming +T = 20 K and the corresponding opacity. +For comparison, we also provide the unbiased disc masses of +gas and solids. To be presented alongside disc gas masses, solid +masses are multiplied by a factor 100, which is typically used as +a gas-to-dust ratio in this context. +2.4. Model parameters and initial conditions +The evolution model requires several initial conditions and pa- +rameters for evolution. These are: the mass of the central star +M⋆; the mass of the gas disc MG; the initial dust-to-gas ratio +fD/G; the power-law index for the initial profile β; the inner edge +of the disc rin; the characteristic radius of the disc r1; the tur- +bulent viscosity parameter α; the planetesimals formation effi- +ciency ε; the fragmentation velocity vfrag; the efficiency of drift +ζ; the stellar X-ray luminosity LX; and the ambient UV field +strength F. +The initial surface density profile of the gas disc is set as +(Veras & Armitage 2004) +ΣG(t = 0) = Σini +� r +r0 +�−β +exp +�������− +� r +r1 +�2−β������� +� +1 − +� +rin +r +� +, +(7) +where Σini is the surface density at r0 = 5.2 au, the reference dis- +tance. The conversion between that and the total mass is obtained +with +MG = 2πΣini +2 − β (r0)β (r1)2−β . +(8) +The initial solid profile of the disc ΣD(t = 0) is set by multiplying +the initial gas profile ΣG(t = 0) by the dust-to-gas ratio fD/G. +Article number, page 3 of 18 + +A&A proofs: manuscript no. disc +Table 1. Parameter range for the main simulation grids. +Variable +Sampling +Min. +Max. +M⋆/M⊙ +linear +0.1 +1.4 +MG/M⋆ +logarithmic +10−3 +10−0.5 +β +linear +0.8 +1.2 +Pin/d +logarithmic +10−0.15 +3 × 101 +r1/au +logarithmic +3 +3 × 102 +α +logarithmic +10−5 +10−2 +fD/G +logarithmic +10−2.5 +10−1.3 +vfrag/cm s−1 +logarithmic +2 × 101 +2 × 103 +ε +logarithmic +10−3 +10−1 +ζ +logarithmic +10−2 +1 +LX/1030 erg s−1 +logarithmic +10−2 +102 +F/G0 +logarithmic +1 +104 +In the remainder of this work, we do not provide all param- +eters as such. For instance, the inner edge is parametrised by its +period Pin, which we convert into distance by means of Kepler’s +third law. Also, we generally set the initial disc mass by its solid +content MD. The ratio between the initial solids and gas masses is +readily given by the dust-to-gas ratio, such that fD/G = MD/MG. +2.5. Simulation list +To generate the list of the simulations to be performed, we se- +lected the Latin hypercube sampling (LHS) method (e.g. McKay +et al. 1979). By dividing each dimension into n intervals and +then selecting one random sample from each interval, LHS en- +sures that the entire range of possible values for each parameter +is sampled with a uniform probability. Additional criteria are re- +quired to avoid correlation between selected values of different +parameters (to disentangle their effects) and to ensure that the +entire space is well sampled (to avoid locations with no results). +To build the grid, we use the pyDOE2 Python package with the +minmax setting. Each generated grid contains values in the [0,1] +range with uniform probability. +These values have to be mapped into the range to be stud- +ied. For our main grids, we outline these in Table 1. The selec- +tion was made to encompass the needs of this and future works, +as well as the limitations of the model. For instance, the stellar +mass M⋆ is taken in steps of 0.1 M⊙ to lie on the stellar evolu- +tion tracks of Baraffe et al. (2015), and the limits of the ambient +UV field strength F match those of the FRIED grid (Haworth +et al. 2018) with the extrapolation for low field values from Pa- +per I. The gas mass of the disc is given in terms of the stellar +mass to roughly follow the scaling of Burn et al. (2021). The +dust-to-gas ratio was selected to span the possible stellar metal- +licities, with a reference stellar metallicity fD/G,⊙ = 0.0149 (Lod- +ders 2003). The range power-law index was selected to cover the +possible values of Andrews et al. (2009). The lower boundary +of the period at the inner edge Pin corresponds to 0.7 d, which +is nearly the maximum value of the stellar radii in the models +of Baraffe et al. (2015). The fragmentation velocity was cho- +sen to encompass the previously assumed value of ∼10 m s−1 +(e.g. Dr˛a˙zkowska & Alibert 2017, and references therein), and +more current values of ∼1 m s−1, as ice was not found to be more +sticky than silicates in recent experiments (Gundlach et al. 2018; +Musiolik & Wurm 2019; Steinpilz et al. 2019). The range of +planetesimal-formation efficiencies was selected to be able to +study low efficiencies where only a small fraction of the mass +of solids is converted into planetesimals and to cover the case +ε = 0.05, which forms a sufficient amount of planetesimals. +2.6. Machine learning +Surrogate models of disc evolution are obtained by means of +a neural network. These neural networks are trained, validated, +and tested using the scikit-learn Python package (Pedregosa +et al. 2011). scikit-learn uses cross-validation to train and +validate the neural network with five passes. This means that the +combined training and validation set is divided into five equal- +sized batches, and five successive training and validation steps +are performed, each using four of the five batches for training +and one batch for validation. The neural networks are fitted us- +ing either the L-BFGS-B algorithm (Byrd et al. 1995), which is +part of the SciPy package (Virtanen et al. 2020) or the ADAM +method (Kingma & Ba 2014). +3. Observational constraints +To compute disc populations that are comparable to observa- +tions, we must first describe the constraints on their initial prop- +erties and their outcomes. These are then used to set the initial +conditions and the comparison point for the outcomes. +3.1. Stellar mass +The stellar initial mass function (IMF) has been determined +(e.g. Chabrier 2003), and so it could in principle be used to re- +produce the stellar population. However, the stellar mass func- +tions for different star-forming regions deviate from the IMF. In +the case of Taurus, Luhman (2000) found a peak around 0.6 +to 1 M⊙, while for the Orion Nebula Cluster (ONC), Da Rio +et al. (2012) found that the best log-normal fit has a mean at +log10(M⋆/M⊙) = −0.45 (corresponding to M⋆ = 0.35 M⊙), +using the stellar evolution model of Baraffe et al. (1998). Cor- +roborating this, the sample of Flaischlen et al. (2021), which is +based on that of Manara et al. (2012), has a stellar mass distri- +bution peaking around 0.4 M⊙: a simple log-normal fit to that +data gives log10(µ/M⊙) = −0.481 and a narrower standard devi- +ation of log10(σ) = 0.2383. As the main aim of this work is to +compare our model with disc lifetimes, their mass, and the stellar +accretion rate of nearby star-forming regions, we chose to follow +the stellar mass function of Da Rio et al. (2012), with a mean of +log10(µ/M⊙) = −0.45 and standard deviation log10(σ) = 0.44. +This should offer a distribution that is representative of both +nearby clusters in general and of stars for which observations +of disc masses and stellar accretion rates are available. +Our model uses the stellar evolution tracks of Baraffe et al. +(2015) to obtain the luminosity for disc irradiation. These are +only defined for mass increments of 0.1 M⊙ from 0.1 to 1.4 M⊙. +To properly track stellar luminosities, we restricted ourselves to +stellar masses that match these values. +3.2. Initial dust mass +Initial dust masses can be obtained from works targeting the +youngest stars known to date, such as Tychoniec et al. (2018), +Williams et al. (2019), or Tobin et al. (2020). Emsenhuber et al. +(2021b) fitted the masses of the Class 0 discs of Tychoniec et al. +(2018), which gave log10(µ/M⊙) = −3.49 and σ = 0.35 dex, +taking out the conversion from dust mass to gas mass using the +standard factor of 100 that was used there. These disc masses +Article number, page 4 of 18 + +A. Emsenhuber et al.: Towards a population synthesis of discs and planets. II. +Table 2. Best-fit parameters for stellar X-ray luminosity. +Work +a +b +Sca. +Preibisch et al. (2005) +1.44 ± 0.10 +30.37 ± 0.06 +0.65 +Güdel et al. (2007) +1.52 ± 0.12 +30.31 ± 0.06 +0.54 +Parameters a and b are those of the fit +log +� +LX/erg s−1� += a × log (M⋆/M⊙) + b. The ‘Sca.’ column +provides the scatter of the residuals from the fit. +were used for a population of stars with masses of 1 M⊙ while +the populations around lower-mass stars of Burn et al. (2021) +scaled the disc masses proportionally to the stellar masses. +However, a complication arises from the fact that the mass +of the central body is not known for the objects observed by +Tychoniec et al. (2018) and Tobin et al. (2020). To properly con- +vert the absolute masses into disc-to-star mass ratios, as we do +in this work, we must assume a reference stellar mass Mref +⋆ . The +method of Emsenhuber et al. (2021b) and Burn et al. (2021) was +equivalent to setting Mref +⋆ += 1 M⊙. We use this as our default +conversion factor, although consistency with the stellar mass dis- +tribution discussed in the previous section would call for a lower +value of Mref +⋆ . We explore different values of this factor later in +this work. +3.3. Sizes +Protoplanetary disc sizes have been found to be correlated with +their mass. Andrews et al. (2010) found that discs in the Ophi- +uchus star-forming region have MD ∝ r(1.6±0.3) +1 +and β = 0.9±0.2. +More recent studies, such as those of Tripathi et al. (2017) and +Andrews et al. (2018b), found that MD ∝ r2 +1, while Hendler et al. +(2020) obtained different scalings across various star-forming re- +gions. For young and non-multiple discs, Tobin et al. (2020) ob- +tained r1 ∝ M(0.25±0.03) +D +. Adding a normalisation from the same +work, we get +r1 +70 au = +� +MD +100 M⊕ +�0.25 +, +(9) +plus a residual scatter of the order of 0.1 dex. +3.4. Dust-to-gas ratio +The ratio between the initial masses of the gas and dust discs +is given by fD/G. We select this parameter as in Emsenhuber +et al. (2021b), that is, we assume it is the same as the stellar +metallicity (Gáspár et al. 2016). Thus, we can use the relation +fD/G +fD/G,⊙ = 10[Fe/H] (Murray et al. 2001), where fD/G,⊙ = 0.0149 +is the primordial solar value (Lodders 2003). The distribution +of metallicity is chosen to be that of the CORALIE RV search +sample (Santos et al. 2005), which was modelled as a Gaus- +sian with a mean of −0.02 and a standard deviation of 0.22. +To avoid extreme values, we restrict the parameter to within +−0.6 < [Fe/H] < 0.5. +3.5. Stellar X-ray luminosity +A couple of surveys have been performed to determine the X- +ray luminosities of young stars, the relevant results of which +are provided in Table 2. One is the Chandra Orion Ultradeep +Project (COUP; Getman et al. 2005; Preibisch et al. 2005), which +covers stellar masses M⋆ between 0.5 and 0.9 M⊙. The survey +found a stellar-mass dependency of LX ∝ M(1.44±0.10) +⋆ +. Another +survey, the XMM-Newton Extended Survey of Taurus (XEST; +Güdel et al. 2007), found that LX varies with stellar mass as +LX ∝ M(1.54±0.12) +⋆ +, which we used to correct for the stellar-mass +effect and recompute the inherent scatter. The two surveys have +similar stellar mass dependence, meaning that using one or the +other to set the stellar X-ray luminosities should not affect the +outcomes in any significant manner. For this work, we compute +LX using a log-normal distribution with the parameters selected +following XEST (Güdel et al. 2007), as the stellar mass depen- +dence is consistent with the prescription used to compute the +X-ray photoevaporation profiles in Picogna et al. (2021). +3.6. Ambient FUV field strength +The external photoevaporation prescription of Haworth et al. +(2018) requires the stellar mass, disc mass, outer radius, and am- +bient FUV field strength. The first three can be readily obtained +consistently from the simulation, but the latter, F, needs to be +specified. +Most stars are formed in stellar clusters (e.g. Lada & Lada +2003), which result in high stellar densities. To retrieve the am- +bient FUV relevant during the lifetime of protoplanetary discs, +we use the simulation of Adams et al. (2006). The authors deter- +mined that F is well described by a log-normal distribution with +a median close to 103.25G0, where G0 = 1.6 × 10−3 erg s−1 cm−2 +is nearly the interstellar FUV field (Habing 1968). +3.7. Inner edge of the gas disc +The location of the inner edge of the gas disc is most relevant for +the location of the close-in planets (such as hot Jupiters). As we +are mostly interested in warm giants further away than the inner +edge, this parameter is of less importance in this work. We chose +this parameter in the same way as in Emsenhuber et al. (2021b), +that is, by assuming that the disc is truncated at the corotation ra- +dius of the star. For the distribution of stellar rotation periods, we +follow the results of Venuti et al. (2017). This gives a log-normal +distribution with a median period of log10(µ/d) = 0.67617866 +and deviation σ = 0.305 673 3 dex. +For comparison, the distribution of initial rotation periods +used by Johnstone et al. (2021) has a median of log10(µ/d) = +0.5181 and a standard deviation of σ = 0.3236 dex. The median +rotation period here is smaller here (3.3 d) than the 4.7 d value of +Venuti et al. (2017) but not by a large amount, while the devia- +tions are similar. The exact choice should therefore not affect the +results significantly. +4. Results +4.1. Full model +To generate the training, validation, and testing data for the sur- +rogate models, we generated two sets of simulations. The first set +contains 100 000 models that are used for the combined training +and validation steps, while the second set contains 20 000 mod- +els and is used for the testing step. The values of the first set are +provided in Table 3 while the values of the second set are pro- +vided in Table 4. Both tables are available at the CDS and have +the same format; they contain the following columns: columns 1 +to 12 are the initial conditions in the same order and units that +are given in Table 1. Column 13 gives the disc lifetime accord- +ing to when the mass becomes lower than 10−6M⋆ or when the +surface density is lower than 1 × 10−3 g cm−2 inside 100 au (or +Article number, page 5 of 18 + +A&A proofs: manuscript no. disc +30 au for M⋆ = 0.1 M⊙ or 0.2 M⊙) and 1 × 10−2 g cm−2 outside +that (this second criterion on the surface density is to avoid ex- +cessively long-lived discs when photoevaporation rates, particu- +larly external ones, are low). Column 14 gives the lifetime us- +ing the minimum value of the criterion of column 13 and the +observability criterion in the near-infrared (NIR) from Kimura +et al. (2016). Columns 15-19 give the following outcomes at +1 × 105 yr: stellar accretion rate log10 +� ˙M⋆/M⊙ yr−1� +, the true gas +mass log10 (MG/M⊙), the true solids mass log10 (MD/M⊙), the +observed mass (Sect. 2.3) log10 (Mobs/M⊙) and the radius en- +compassing 68 % of the flux log10 (r68/au). Columns 20-24 re- +peat the same information, but at 2 × 106 yr. +Two epochs (1 × 105 yr and 2 × 106 yr) were selected to be +compatible with the observations we are comparing to. The first +epoch is for comparison with early discs, such as their initial +masses. Its selection is a trade-off between two items: on the one +hand, we would like to have the data as early as possible, while +on the other hand, we need to wait until the initial dust growth +has taken place. From the analysis of individual discs, we found +that 1 × 105 yr represents a good compromise in that sense. The +second epoch is for comparison with the star-forming regions of +Lupus and Cha I. As the stars in these regions are between 1 +and 3 Myr old, we take the results at 2 Myr, as in Manara et al. +(2019). +Our results indicate that the two criteria for disc dispersal +produce nearly identical results. In only about 10 % of the cases, +the NIR criterion predicts a lower disc lifetime than the crite- +rion based on the mass, and the difference remains small when +this occurs (we do not check for the reverse, as calculations stop +when the mass criterion is reached). These results are consis- +tent with the findings of Kunitomo et al. (2021). As a conse- +quence, hereafter we only report the disc lifetimes based on the +NIR criterion of Kimura et al. (2016). Also, we stop the calcu- +lation at 100 Myr in any case. This affects some long-lived discs +with minimal photoevaporation and accretion. In such cases, the +lifetime based on the mass criterion is not reported while that +based on the NIR emission is. +4.2. Performance of the surrogate model +We asses the performance of the surrogate models in terms of +the best regression (obtained using ordinary least squares), the +Pearson correlation coefficient R2, and the RMS of the differ- +ences between the predicted and target lifetimes (the square root +of the mean square error). These were computed on the testing +set (Table 4) that the surrogate model has not seen before. The +hyper-parameters and results for all surrogate models that are +part of this work are presented in Table 5. For three of them, we +also show correlation plots in Fig. 1. In all cases, the fitting pro- +cedure was performed on the logarithm (base 10) of the values, +and so all the reported performances are given in these units. +Concerning the different surrogate models, the ones for the +disc lifetimes and for the stellar accretion rates provide the best +performance. The ones that are based on the dust disc, namely +masses and radii, show a lower performance, especially at 2 Myr. +We note that these values are for each single prediction; they rep- +resent the level of additional uncertainty for the parameter stud- +ies (Sect. 4.3 and Appendix A) while for the population studies +(Sect. 5) these errors can average out and result in an even better +global accuracy. +The neural networks predicting the disc masses, their radii, +and stellar accretion rates were fitted only on the discs that had +not vanished at the time. This means that they are supported +by a lower number of points than the ones predicting the life- +times. This also implies that these surrogate models are only +constrained in the region of the parameter space where lifetimes +are larger than the time of the analysis. Thus, in the remainder of +this work we only provide disc masses and stellar accretion rates +for discs that have not yet dispersed. +4.3. Effects of photoevaporation +We began our investigations using the surrogate model, perform- +ing a parameter study of the effects of the photoevaporation pre- +scriptions on disc lifetimes. For this purpose, we generated two +maps, one for internal photoevaporation and one for external +photoevaporation, which vary the stellar mass and the control- +ling parameter of each photoevaporation prescription. In each +case, the value of the parameter controlling the other photoevap- +oration prescription was set at the minimum of the studied range +in order to avoid cross effects. We assumed typical values of the +remaining parameters: disc-to-star gas mass ratio MG/M⋆ = 0.1, +dust-to-gas ratio fD/G = 0.0149, power-law index β = 0.9, period +at the inner edge Pin = 10 d, characteristic radius r1 computed +according to Eq. (9), viscosity parameter α = 1 × 10−3, fragmen- +tation velocity vfrag = 2 m s−1, planetesimal formation efficiency +ε = 1 × 10−3, and drift efficiency ζ = 1. +4.3.1. Internal photoevaporation +The resulting map for internal photoevaporation is provided in +Fig. 2. Here we observe that the surrogate model predicts sev- +eral sharp transitions of disc lifetime with stellar mass. The most +evident are those between 0.2 and 0.3 M⊙ and between 0.8 and +0.9 M⊙ where disc lifetime increases. There are other transitions +between 0.4 and 0.5 M⊙ and between 0.6 and 0.7 M⊙ where +disc lifetime decreases, but only for large X-ray luminosities +(LX > 1030 erg s−1). These transitions match the switch from +one photoevaporative profile to another, which are marked by +the dashed white lines. This indicates that the profile of surface- +density loss has a strong effect on disc lifetime and not only the +total mass-loss rate, which gradually changes between each stel- +lar mass. Also, the further out the location of the peak of internal +photoevaporation (which is for the profiles of 0.3 M⊙ and 1.0 M⊙ +stars; see top panel of Figure 7 of Picogna et al. 2021), the longer +the disc lifetimes are in general. We find that this effect is due to +a larger inner region where material is not evaporated at all and +can only be dispersed by viscous accretion. In this case, the ob- +served disc lifetime is set by the dispersal timescale of the inner +disc, which is given by the viscous timescale at the outer radius +of the inner disc. +As discussed in Sect. 3.5, the X-ray luminosity is correlated +with stellar mass. To highlight this, we show in addition the me- +dian stellar X-ray luminosity as a function of stellar mass from +Güdel et al. (2007) with the green dashed curve. To determine +the expected relationship between disc lifetime and stellar mass, +one needs to follow this curve rather than a horizontal line on +the plot. We see that internal photoevaporation leads to a lim- +ited change in disc lifetime with stellar mass. This is because +more massive stars lead to stronger mass-loss rates (owing to a +corresponding increase in stellar X-ray luminosity), which com- +pensates for the increase in disc mass (as we assume disc mass +to be proportional to stellar mass). This is shown by the black +line that traces disc lifetimes of 3 Myr (a typical value in obser- +vations), which is consistently lower than the median LX by a +factor of a few. +Article number, page 6 of 18 + +A. Emsenhuber et al.: Towards a population synthesis of discs and planets. II. +��� +��� +��������������������� +��� +��� +��� +������������������������ +� � ���� � � ���� +�� � ����� +����������� +�������������� +�� +�� +�� +�� +������ ���� ���� �� +���� +�� +�� +�� +�� +�� +� +��������� ���� ���� �� +���� +� � ���� � +���� +�� � ����� +����������� +����������������������� +�� +�� +�� +� +������ ���� �� +� +�� +�� +�� +�� +�� +� +��������� ���� �� +� +� � ���� � +���� +�� � ����� +����������� +����������������������� +Fig. 1. Performance of three surrogate models based on the comparison of the predicted and actual values of the testing set. The insert values show +the best regression (ordinary least squares), the Pearson correlation coefficient R2, and the RMS of the differences between each predicted and +actual value. +Table 5. Hyper parameters and performance of the surrogate models. +Age +Model +Solver +Activ. +HLS +alpha +Val. R2 +Test. R2 +Val. MSE +Test. MSE +Lifetime +L-BFGS +logistic +25, 50, 45 +2.592 × 10−3 +0.99465 +0.99436 +0.00368 +0.00389 +100 kyr +Accretion +L-BFGS +tahn +15, 30, 50 +1.098 × 10−3 +0.99909 +0.99829 +0.00163 +0.00300 +Mass +ADAM +tahn +55, 40, 65 +2.659 × 10−3 +0.96330 +0.95040 +0.05690 +0.07677 +Radius +ADAM +tahn +60, 55, 45 +1.131 × 10−3 +0.89222 +0.88045 +0.04460 +0.04853 +2 Myr +Accretion +L-BFGS +tahn +60, 45, 70 +9.407 × 10−3 +0.99865 +0.99506 +0.00349 +0.01297 +Mass +ADAM +tahn +65, 25, 55 +1.915 × 10−3 +0.91565 +0.89798 +0.25667 +0.32089 +Radius +ADAM +tahn +70, 65, 35 +6.901 × 10−3 +0.85665 +0.80786 +0.11881 +0.15925 +4.3.2. External photoevaporation +The resulting map for external photoevaporation is shown in +Fig. 3. Unlike internal photoevaporation, the prescription for ex- +ternal photoevaporation provides for gradual changes of lifetime +with stellar mass. However, these changes lead to a larger depen- +dency of disc lifetime on stellar mass than what is expected from +internal photoevaporation. This is illustrated by the black line, +which tracks a 3 Myr lifetime, as in the map for internal photo- +evaporation. Its position in terms of ambient UV field strength +varies across the entire parameter range studied here, from less +that 2 G0 for M⋆ = 0.1 M⊙ to about 4 × 103 G0 for M⋆ = 1 M⊙; it +becomes independent of stellar mass for M⋆ > 1 M⊙ and fluxes +above ∼103 G0. +While the trend of reduced disc lifetimes in regions with +strong ambient UV fields (e.g. Michel et al. 2021) is reproduced, +the general behaviour of correlated disc lifetimes with stellar +mass for a given ambient UV field strength is problematic for +several reasons. First, this general behaviour is inconsistent with +observations that suggest disc lifetimes are independent of, or +slightly decreasing with, increasing stellar mass (Carpenter et al. +2006; Kennedy & Kenyon 2009; Bayo et al. 2012; Ribas et al. +2015). There are several possibilities to remedy this, although +they are unlikely. To obtain a behaviour similar to observations, +the mass loss would need to be correlated with stellar mass (Ko- +maki et al. 2021), which in turn would require that the ambi- +ent field be correlated with stellar mass. However, the ambient +field is usually dominated by the few most massive stars (Adams +et al. 2006), which means that it depends more on the cluster as +a whole than on the mass of the star in question. Another av- +enue is that disc masses scale to a lesser extent with stellar mass +than assumed here. However, this would not yield the expected +behaviour of stellar accretion rates with stellar masses (e.g. Hart- +mann et al. 2016; Alcalá et al. 2017; Flaischlen et al. 2021). We +find that the FRIED grid prescription that we use in this work +produces incompatible results that show at most a dependence +of the disc lifetime on stellar mass. The second concern is that +further lifetime analyses will be strongly affected by the selec- +tion of the stellar masses, in contrast to internal photoevaporation +where this dependency is weaker. +4.4. Parameter sensitivity +The sensitivity of disc lifetimes, disc masses, and stellar accre- +tion rates at 2 Myr is studied in detail in Appendix A. These +results can be summarised as follows: all the outcomes are in- +sensitive to the power-law index β and the inner edge of the gas +disc rin. The characteristic radius r1 and viscosity α control the +viscous timescale of the disc, and therefore the stellar accretion +rate. Disc lifetimes and observed dust masses are more strongly +affected by the viscosity α than by the characteristic radius r1. +The twopop model parameters only affect the observed dust +masses. Observed dust masses are less affected by the dust-to- +gas ratio fD/G than by the initial mass of the gas disc MG, except +for discs close to dispersal. The fragmentation velocity vfrag and +the drift efficiency strongly affect the observed dust masses, but +only for vfrag ≳ 200 cm s−1, while the planetesimal formation +efficiency ε only has a limited effect for values close to the max- +imum we study, namely of 0.1. +These results narrow down the parameter space that we ex- +plore in the remainder of this work. First, we keep the values of +the power-law index β and the inner edge of the gas disc rin as +described in Sect. 3, because they are of negligible importance. +We then only use the initial mass of the gas disc MG to con- +trol disc masses, not the dust-to-gas ratio fD/G; as the latter is in +Article number, page 7 of 18 + +A&A proofs: manuscript no. disc +��� +��� +��� +��� +��� +��� +��� +��������������� +� +�� +� +�� +� +��� +��� +��� +������������������� ����� ������ +��������� +���� +����� +���� +����� +���� +����� +���� +����� +���� +������� +��� +��� +��� +��� +��� +��� +������������������������� +Fig. 2. Map of disc lifetimes as a function of stellar mass and X-ray +luminosity, which is the main driver of internal photoevaporation, ac- +cording to the surrogate model described in Sect. 4.2. External photoe- +vaporation was set to its minimum value (F = 1 G0). Other parameters +were selected as MG/M⋆ = 0.1, fD/G = 0.0149, β = 0.9, Pin = 10 d, r1 +according to Eq. (9), α = 1 × 10−3, vfrag = 2 m s−1, ε = 1 × 10−3, and +ζ = 1. The green dashed line represents the dependency of LX on M⋆ +from Güdel et al. (2007) and the solid black line shows the location of a +3 Myr lifetime (a typical value). The results are discussed in Sect. 4.3.1. +most cases of lower importance and well constrained by obser- +vations. Also, we keep the planetesimal formation efficiency to +the minimum value of ε = 1 × 10−3, the fragmentation velocity +to vfrag = 200 cm s−1, and the drift efficiency to ζ = 1. +5. Disc populations +We now compare synthetic disc populations with observations. +For this, we proceed as follows: we draw 10 000 random discs +whose initial conditions follow given distributions. The out- +comes of each disc are obtained by means of the different sur- +rogate models. For the analysis, we first compare the cumulative +distribution of disc lifetimes so that it can be compared to the +fraction of stars that have a protoplanetary disc for stellar clus- +ters with different ages, as in Haisch et al. (2001). The second +analysis is to compare observed disc masses and stellar accre- +tion rates with the data of Manara et al. (2019). Here, we use +the data at 2 Myr. Further, we only use discs whose lifetime, as +determined by the surrogate model from the previous analysis, +is larger than the time of analysis in order to avoid being in the +region where the surrogate model is not supported by any under- +lying data. +5.1. Canonical +To determine if all the processes that are predicted from theory +are able to reproduce disc observations, we compute a population +of discs whose properties are as close as possible to observations +��� +��� +��� +��� +��� +��� +��� +��������������� +� +��� +��� +��� +��� +��� +����������������������� +��� +��� +��� +��� +��� +��� +������������������������� +Fig. 3. Map of disc lifetime as functions of stellar mass and ambient UV +field strength, this latter being the main driver of external photoevapo- +ration according to the surrogate model described in Sect. 4.2. Internal +photoevaporation is set to its minimum value (LX = 1 × 1028 erg s−1). +Other parameters were selected as MG/M⋆ = 0.1, fD/G = 0.0149, β = +0.9, Pin = 10 d, r1 according to Eq. (9), α = 1 × 10−3, vfrag = 2 m s−1, +ε = 1 × 10−3, and ζ = 1. The solid black line shows the location of a +3 Myr lifetime (a typical value). The results are discussed in Sect. 4.3.2. +Table 6. Random distributions for the canonical population +Variable +Distribution +log10(M⋆/M⊙) +N(−0.45, 0.442) +MG +MD/fD/G +β +0.9 +log10(Pin/d) +N(0.67617866, 0.30567332) +r1/au +70(MD/100 M⊕)0.25 × 10N(0,0.12) +log10(α) +U(−3.5, −3.0) +log10( fD/G) +N(−1.85, 0.222) +log10(MD/M⋆) +N(−3.49, 0.352) +vfrag/cm s−1 +200 +ε +10−3 +ζ +1 +LX/1030 erg s−1 +10N(0.31,0.542) × (M⋆/1 M⊙)1.52 +log10(F/G0) +N(3.25, 0.932) +from early discs. The only parameter that has some freedom is +α. Here, we selected to draw log10 (α) with a uniform probabil- +ity of between −3.5 and −3. This was decided as a compromise +between disc lifetime and stellar accretion rate, as we discuss be- +low. For the other parameters, their distributions were selected as +described in the discussion of Sect. 3; for convenience, these are +summarised in Table 6. +The resulting distribution of disc lifetimes is shown with +the blue curve in Fig. 4. It becomes immediately apparent that +the synthetic lifetimes are too short overall in comparison with +observed discs. The median lifetime of the synthetic discs is +Article number, page 8 of 18 + +A. Emsenhuber et al.: Towards a population synthesis of discs and planets. II. +�� +� +��� +��� +��������� +��� +��� +��� +��� +��� +��� +���������������������������� +������� +��������� +��������� +���������� +����������������� +�������������������� +����������� +Fig. 4. Cumulative distribution of disc lifetimes for a population with +canonical parameter distribution (see text). Two exponential decays fol- +lowing Fedele et al. (2010) with a characteristic time of 2.3 Myr (accre- +tion) and 3 Myr (infrared excess) and the results of Ribas et al. (2014) +are shown as well. +0.42 Myr. Disc lifetime depends on the assumed distribution +of α, which we chose such that it results in the largest stellar +accretion rates at 2 Myr. A histogram of stellar accretion rate +versus observed dust mass for the observed discs in the Lupus +and Chamaeleon I star-forming regions is shown in the top-left +panel of Fig. 5. Only the synthetic discs that live beyond 2 Myr +contribute to this histogram. The few remaining discs have low +masses, as the discs are close to being dissipated. Using larger +values of α, for instance between roughly 10−3 and 10−2 as was +proposed by Mulders et al. (2017), would have resulted in even +shorter disc lifetimes. This means that there would have been no +discs that would live long enough to produce stellar accretion at +2 Myr. Conversely, selecting a distribution of α with even lower +values would allow disc lifetimes to be matched by observations, +but this would result in even lower stellar accretion rates, which +would be in tension with the results of Dullemond et al. (2018), +who concluded that α ≥ 1 × 10−4 from the sizes of disc substruc- +tures. +The discrepancy between our modelled lifetimes and obser- +vations arises from the strong mass-loss rates predicted for in- +ternal and external photoevaporation, as we discuss in Sect. 6. +As disc lifetime depends on stellar mass (especially for exter- +nal photoevaporation; Sect. 4.3), this analysis is affected by the +assumed stellar mass distribution. Had we selected larger stel- +lar masses, the lifetimes would better match observations. How- +ever, selecting stellar masses of around 1 M⊙, which would lead +to a fairly good match including both photoevaporation prescrip- +tions, is not representative of the star-forming regions that we are +comparing to (Sect. 3.1). +5.2. Effect of photoevaporation on disc populations +The mismatch in disc lifetimes and disc masses that we obtained +is in contrast with other studies, such as that of Mulders et al. +(2017), who did not include photoevaporation, Kunitomo et al. +(2020), who included only internal photoevaporation, and Weder +et al. (in prep.), who used an EUV-only internal photoevapora- +tion prescription with much lower mass-loss rates. Further, as +already discussed, photoevaporation leads to considerable short- +ening of disc lifetimes (Sect. 4.3). Therefore, here we investi- +gate how the photoevaporation prescriptions affect the evolution +of disc populations and determine whether they are responsible +for the mismatch. To this end, we created two additional popula- +tions, each time neglecting one of the photoevaporation process +by taken the corresponding controlling parameter to the mini- +mum value. For the population with internal photoevaporation +only, the ambient flux has been set to F = 1G0, which corre- +sponds to ˙M = 1 × 10−15 M⊙ yr−1 (Sect. 2.1.2). The results are +shown as ‘Int. only’ in Figs. 4 and 5. For the population with +external photoevaporation only, we set LX = 1028 erg s−1, the +results of which are shown as ‘Ext. only’ in the same figures. +In each population, the disc lifetimes strongly increase com- +pared to the canonical case. However, the distributions are differ- +ent: internal photoevaporation leads to a relatively narrow distri- +bution around 2 to 3 Myr, while with external photoevaporation +disc lifetimes are more spread out, including very short-lived +discs. This leads to more discs being shown in the accretion ver- +sus mass diagram. +The population with only external photoevaporation has low +disc masses combined with a narrow range of stellar accretion +between 10−10 and 10−9 M⊙ yr−1. This is because the mass loss +occurs in the outer disc, which reduces its size, limiting the area +from which the dust emits (as dust is also lost where there is no +longer gas present). At the same time, the mass-accretion rate is +weakly affected by the mass loss in the outer disc; again because +external photoevaporation affects only the outer disc. +Conversely, using internal photoevaporation leads to a larger +spread in stellar accretion rate. As internal photoevaporation re- +moves material relatively close to the star, it competes with stel- +lar accretion to some extent. This, coupled with the spread of X- +ray luminosities, leads to a spread in accretion rate (Owen et al. +2011). Thus, internal photoevaporation is needed to reproduce +the observed spread in stellar accretion rate. We further note that +neither population is able to reproduce the discs with an accre- +tion rate larger than 10−8 M⊙ yr−1. +In addition, internal photoevaporation leaves discs with an +inner cavity that have low accretion rates but where the outer disc +is out of reach of internal photoevaporation and therefore takes a +long time to dissipate; these are what Owen et al. (2011) refer to +as relic discs. It is possible to avoid this situation to a large extent +by having more compact discs initially, which leave nearly all of +their mass within reach of internal photoevaporation. +We conclude that previous studies managed to reproduce +disc lifetimes because they used only one photoevaporation +mechanism as the main loss mechanism. However, when both +are accounted for, the combined mass-loss rate is so large that +discs are very short lived. We discuss the implications of this +and possible remedies in Sect. 6. +5.3. Towards a best match +Despite all the differences between models and observations +highlighted so far, we now try to find a set of initial conditions +that is able to better match disc evolution characteristics. To this +Article number, page 9 of 18 + +A&A proofs: manuscript no. disc +�� +� +�� +� +�� +� +�� +� +�� +� +�� +� +��� +���� ���� ���� � ��� �� +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +� +�� +� +�� +� +��������� ���� ���� ���� �� +���� +������������� +�������� +������������������� +����� +����� +�� +� +�� +� +�� +� +�� +� +�� +� +�� +� +��� +���� ���� ���� � ��� �� +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +� +�� +� +�� +� +��������� ���� ���� ���� �� +���� +���������������� +�������� +��������������������� +����� +����� +�� +� +�� +� +�� +� +�� +� +�� +� +�� +� +��� +���� ���� ���� � ��� �� +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +� +�� +� +�� +� +��������� ���� ���� ���� �� +���� +����������������� +�������������� +��������������������� +����� +����� +�� +� +�� +� +�� +� +�� +� +�� +� +�� +� +��� +���� ���� ���� � ��� �� +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +� +�� +� +�� +� +��������� ���� ���� ���� �� +���� +����������������� +������������������� +��������������� +���������������������� +����� +����� +��� +��� +������������������������������������ +Fig. 5. Histogram for stellar accretion rate vs. disc mass at 2 Myr and the same synthetic disc populations shown in Fig. 4. The observed data +from the Lupus (in red) and Chamaeleon I (orange) star forming regions from Manara et al. (2019) are shown for comparison. Two disc dispersal +timescales τ = ˙MG/MG of 1 Myr and 10 Myr (assuming that the gas disc mass MG is 100 times the observed dust mass) and the best fit to the data +from Manara et al. (2016b) are shown as well. +end, we investigate how the initial conditions can be modified +from our canonical values provided in Table 6. +The initial disc-to-star mass ratios were taken from Ty- +choniec et al. (2018) and Tobin et al. (2020) assuming they were +measured on stars of Mref +⋆ = 1 M⊙. However, this assumption is +inconsistent with the stellar mass distribution we selected, which +has a median value of M⋆ = 0.35 M⊙ (Sect. 3.1). Were we to se- +lect a lower reference stellar mass, such as Mref +⋆ = 0.33 M⊙, this +would increase the disc masses. At the same time, selecting a +reference stellar mass that is similar to the median value from +our initial conditions results in an agreement between the disc +masses in observations and our initial conditions, as shown with +the dashed black and red lines in Fig. 6, respectively. In addition, +to account for observational biases, we also want to check the +observed dust masses after a short evolution time of 1 × 105 yr, +which we provide with the solid lines in Fig. 6. The results show +that the discs in the nominal populations have low masses com- +pared to the non-multiple discs measured by Tobin et al. (2020), +Article number, page 10 of 18 + +A. Emsenhuber et al.: Towards a population synthesis of discs and planets. II. +�� +� +�� +� +�� +� +�� +� +�� +� +���� �� +� +��� +��� +��� +��� +��� +��� +����������������� +���������������� +������������������� +��������������� +�������� � � ����� +���� ������ � � ����� +����������� +Fig. 6. Kernel density estimate for two populations from Figs. 4 and 5, +both for their initial conditions (dashed lines) and retrieved disc masses +at 100 kyr (solid lines). The initial mass distribution of the best-match +population with Mref +⋆ += 0.33 M⊙ (in red) is consistent with the fit by +Emsenhuber et al. (2021b) to the data of Tychoniec et al. (2018), which +is shown as the dashed black line, while the retrieved masses at 100 kyr +are compatible with the non-multiple discs of Tobin et al. (2020). +while a population with more massive initial discs has slightly +larger masses. Thus, the larger initial disc masses lead to a rea- +sonable match with observations as a whole. +We point out that the disc-to-star mass ratio in the population +with Mref +⋆ += 0.33 M⊙ is about twice (2.1 times) that of Emsen- +huber et al. (2021b) and Burn et al. (2021), rather than a factor +three as one might assume from the change of Mref +⋆ from 1 M⊙ +to 0.33 M⊙. This is due to an inconsistency in the selection of +the disc masses in the previous work: there, the gas masses were +taken as a Monte Carlo variable that were converted from dust +observations using a dust-to-gas ratio of 1 %. However, the ini- +tial mass of solids in the model was recomputed from the gas +mass using the same distribution as in this work (Sect. 3.4), +which has a median value of 1.42 % (fD/G,⊙ = 1.49 % with +[Fe/H] = −0.02). In contrast, we use the solid disc mass as a +Monte Carlo variable here. +Another possibility is that early protoplanetary discs are not +as extended as what is suggested by the findings of Tobin et al. +(2020). More compact discs are less susceptible to external pho- +toevaporation, as there is less surface exposed to ambient radia- +tion and they are more tightly bound to the star. At the same time, +a more compact disc means that more material is concentrated +in the region where internal photoevaporation is most efficient, +which allows the stellar accretion rate to remain larger. Mag- +netohydrodynamics models of protoplanetary disc formation by +Hennebelle et al. (2016), Lebreuilly et al. (2021), and Lee et al. +(2021) could favour such a possibility. +Finally, nearby star-forming regions have low masses, which +results in a low ambient UV field strength F; for instance, the +value in Lupus is F ≈ 4G0 (Cleeves et al. 2016). Our nomi- +nal distribution of ambient UV fields overestimates mass losses +due to external photoevaporation. Therefore, we set F = 1G0, +which results in negligible mass losses due to external photoe- +vaporation ( ˙M = 1 × 10−15 M⊙ yr−1) and only internal photoe- +vaporation remains. Using internal photoevaporation confers the +further advantage that a wider range of stellar accretion rates is +obtained. +Below, we investigate whether more massive and compact +discs are able to improve the match with observations. The ef- +fects of the parameters mentioned above are discussed in Ap- +pendix B. From these, we find that the following modifications +to the initial conditions given in Table 6 are best able to repro- +duce disc lifetimes and the accretion rate–mass relationship: +– A decrease in the reference stellar mass to Mref +⋆ = 0.33 M⊙, +which corresponds to an increase in the disc mass by a factor +of three compared to the nominal population; +– r1 = 2/3 × 70 au (MD/100 M⊕)0.25, a factor 2/3 compared to +Eq. (9); and +– only using internal photoevaporation (F = 1G0). +The population using these distributions is shown as ‘Best +match’ in Figs. 4 and 5. +While the overall stellar mass distribution we chose is repre- +sentative of the observed stars, our visual optimisation approach +to reproducing the observed disc masses and accretion rates is +independent of the exact stellar mass dependency of the observ- +ables. We will improve on this with a Bayesian framework in +Paper III to optimise the initial parameters when reproducing a +set of observations in four-dimensional space made up of stellar +mass, disc mass, disc radius, and accretion rate. +Compared to the population with only internal photoevap- +oration (the other population that is closest in terms of initial +conditions), we can see several differences. First, the larger disc +masses result in an increased median lifetime. About 2.2 % of +the discs have a lifetime of greater than 10 Myr, representing a +100-fold increase. Then, the combination of larger initial disc +mass and smaller extent results in a certain number of cases with +a stellar accretion rate of higher than 10−8 M⊙ yr−1, which was +not previously seen. The smaller extent of the disc also causes +less discs to be relics (towards the bottom of Fig. 5). Here, the +highest concentration of discs is found near the best-fit value +of Manara et al. (2016b, the pink dashed line in Fig. 5) with a +similar number on either side. We are still failing to reproduce +the discs with the large accretion rates. However, part of these +discs with large accretion rates could be due to binaries (Zagaria +et al. 2022), which we do not model. To corroborate this, the +largest stellar accretion rates are biased to larger-mass stars (the +mean stellar mass for systems with a stellar accretion rate higher +than 3 × 10−9 M⊙ yr−1 is 0.74 M⊙ versus 0.43 M⊙ for the gen- +eral population), which at the same time are more likely to be +in binary systems (e.g. Duchêne & Kraus 2013). The mismatch +should therefore not be the source of significant concern. We find +that this combination of parameters is able to reproduce the disc +mass–accretion rate relationship and provide a reasonable match +to most observations. +6. Discussion +It is difficult to reconcile the results of our synthetic disc popu- +lations with observations. We find that it is particularly hard to +Article number, page 11 of 18 + +A&A proofs: manuscript no. disc +���� +���� +���� +���� +���� +���� +���������� +��� +��� +��� +��� +��� +��� +������������� +������������ +������������ +������������ +������������ +Fig. 7. Evolution of the relative gas mass: in the disc (blue), accreted +onto the star (green), and lost by internal (orange) or external (red) pho- +toevaporation until disc dispersal at 1.38 Myr. This represents a typi- +cal disc, with M⋆ = 0.5 M⊙, MG/M⋆ = 0.1, β = 0.9, Rin = 0.1 au, +r1 = 87.8 au, α = 1 × 10−3, LX = 7.02 × 1029 erg s−1, and F = 10 G0. +The parameters for the solid disc are irrelevant, as this shows only the +gas component, except that we used fD/G = 0.0149 to compute the solid +mass needed to set the characteristic radius r1. +obtain discs with characteristic lifetimes of 2 to 3 Myr accord- +ing to Mamajek (2009) or Fedele et al. (2010), and even less +lifetimes of 5 to 10 Myr following Pfalzner et al. (2022). Here, +we assumed that the initial mass is constrained by the dust-mass +measurements of Tychoniec et al. (2018, MD/M⋆ ∼ 10−3) , and +dust-to-gas ratios similar to the solar initial abundance, namely +fD/G = 1.49 % (Lodders 2003), combined with the predictions +of internal and external photoevaporation models. To illustrate +this issue, in Fig. 7 we provide the evolution of the gas mass +still present in the disc and removed by the processes modelled +here. Here we choose typical values for the initial conditions, +with a star of mass M⋆ = 0.5 M⊙, a gas disc with a mass of +MG/M⋆ = 0.1, a power-law index of β = 0.9, an inner ra- +dius of rin = 0.1 au, and a characteristic radius of r1 = 87.8 au, +which was computed from Eq. (9) assuming a dust-to-gas ratio +of fD/G = 0.0149. The ambient UV field strength F = 10 G0 was +set at the lower boundary of the computed grid while the X-ray +luminosity LX = 7.02 × 1029 erg s−1 follows the best fit of the +XEST survey (Güdel et al. 2007, Table 2) for the given stellar +mass. +Figure 7 shows that photoevaporation (both internal and ex- +ternal) is responsible for the loss of nearly all the gas; only 6 % +of the gas is accreted onto the star. The final disc lifetime of +1.38 Myr is short compared to the characteristic lifetimes from +observations. We therefore have a problem with the mass bud- +get, which could be resolved by (i) larger initial disc masses, (ii) +mass replenishment after disc formation, or (iii) lower mass-loss +rates by photoevaporation. +Stellar accretion already plays a minor role in the mass bud- +get and cannot be reduced further, in order to remain consis- +tent with observed stellar accretion rates. However, radial mass +transport could be the result of magnetically driven disc winds +(Suzuki & Inutsuka 2009; Suzuki et al. 2010) rather than pure +viscous dissipation as we assumed here. This would add another +mass-loss channel, which would further exacerbate the problem +(though it could shield stellar radiation, as discussed below). +We experiment with larger initial disc masses in this work. +However, even with gas masses on the order of 10 % of the stel- +lar mass, disc lifetimes are not sufficiently long. Therefore, in- +creasing the masses even more would be required, but this in- +crease would bring another series of problems. For one, such +large discs are likely gravitationally unstable and produce spi- +ral density waves. Second, such large discs would lead to strong +gas-driven migration, which hinders planet formation (Nayak- +shin et al. 2022). Also, discs around 10 Myr-old stars HD163296 +and TW Hya have at least 10 % of the stellar mass (Powell et al. +2019) and it is unclear what their initial mass would have been +for them to remain so large at their age. We therefore do not find +that massive initial discs would be able to solve the conundrum. +Discs do not form instantly. Rather, they grow from gas +falling from the envelope. This allows for disc masses to in- +crease during the early stages of disc evolution, which is at +odds with the models described here. Modelling the infall stage +would allow us to have longer-lived discs without them being +very massive early on. Accretion can persist for several million +years (Throop & Bally 2008), providing replenishment even at +late times. The complex morphology of the gas disc around RU +Lup (a ∼0.5 Myr old star) could be an outcome of this process +(Huang et al. 2020). The long-lived discs could also be second- +generation discs, formed following accretion from the molecu- +lar cloud (Kuffmeier et al. 2020) or the disruption of a planet +(Nayakshin et al. 2020). While none of these individual pro- +cesses are sufficient to explain the presence of massive discs, +they should still be explored if they can explain certain charac- +teristics of the overall disc population. +Finally, the photoevaporation rates used here could be over- +estimated. Lower mass-loss rates would increase the disc life- +times and masses at later times. An argument in favour of this +hypothesis is that young discs can be shielded from the radiation +of both their host and nearby stars. The launching of magneti- +cally driven disc winds occurs inside the location where internal +photoevaporation is effective and would thus prevent EUV and +X-ray photoevaporation during the early stages of disc evolution +(Pascucci et al. 2022). Similarly, early discs are likely embed- +ded, preventing the radiation of nearby stars from reaching the +disc. Both effects would reduce the photoevaporation rates dur- +ing the early stage of disc evolution compared to what we model +here. +7. Summary and conclusion +In this work, we investigate whether protoplanetary disc obser- +vations can be reconciled with theoretical predictions of pro- +cesses such as viscous accretion and photoevaporation (both in- +ternal and external). We first compute two sets of simulations +that we use to fit neural networks (Fig. 1). With these neural net- +works, we can perform parameter studies and compute the out- +comes of synthetic disc populations with limited computational +resources. +We first compare how internal and external photoevaporation +affect disc lifetime as a function of stellar mass. We find that be- +cause of a direct link between stellar mass and X-ray luminos- +ity (e.g. Preibisch et al. 2005; Güdel et al. 2007), which means +mass-loss rate due to internal photoevaporation, discs around +more massive stars are not significantly longer-lived than those +Article number, page 12 of 18 + +A. Emsenhuber et al.: Towards a population synthesis of discs and planets. II. +around low-mass stars (Fig. 2). Conversely, external photoevap- +oration leads to a strong positive correlation between disc life- +time and stellar mass (Fig. 3), because gas is more bound for +higher-mass stars. This positive correlation is at odds with obser- +vations that find that disc lifetimes are either independent of or +anticorrelated with stellar mass (Carpenter et al. 2006; Kennedy +& Kenyon 2009; Bayo et al. 2012; Ribas et al. 2015; Pfalzner +et al. 2022) and should be investigated in the future. +Turning to protoplanetary disc populations, we find that ac- +counting for both internal and external photoevaporation accord- +ing to theoretical predictions leads to disc lifetimes (Fig. 4) that +are much too short. Discs whose initial mass is 10 % of the stellar +mass are dispersed in roughly 1 Myr under the combined effects +of internal and external photoevaporation (Fig. 7). +Despite the dissimilarities, a reasonably good match to the +disc properties of the Lupus and Chaemeleon I low-mass star- +forming regions is obtained starting with more massive discs of +smaller sizes, and with only internal photoevaporation. This is +valid for clusters with low ambient field strengths, such as Lu- +pus (4 G0; Cleeves et al. 2016), where losses due to external pho- +toevaporation are low. The larger masses and smaller sizes are +needed to improve the match in stellar accretion rates and ob- +served masses. The corresponding initial conditions and model +parameters can be used to study planetary formation in similar +environments. A more robust comparison with observations is +performed in Paper III. +However, initial disc masses cannot be arbitrarily increased +or discs would become gravitationally unstable. Instead, we sug- +gest that future studies should include the modelling of the initial +stages of disc formation, including the presence of an envelope. +This envelope would allow discs to be replenished after their +initial formation and provide shielding from UV radiation from +nearby stars. Also, magnetically driven disc winds would shield +UV and X-ray radiation from the central star. This would provide +a reduction of losses by both internal and external photoevapora- +tion. Both effects allow for longer lifetimes and larger masses at +later times without the need for extremely large masses at earlier +times. +We decided here to use observed dust masses as the main +comparison point. However, this is not the only possible avenue. +For instance, disc radii could be less susceptible to the degen- +eracy caused by regions that are optically thick (Pascucci et al. +2022). One likely difficulty would be in accounting for the large +disc radii and sustained stellar accretion rate. Already with the +comparatively small discs that we find to best match the disc +mass–stellar accretion rate relationship, we are not able to re- +produce the largest observed accretion rates. Having larger discs +would lead to a reduction of the stellar accretion rates for a given +disc mass (Fig. B.1); which would lead to a mismatch with ob- +servations of stellar accretion rates. +In this work, we assume that the gas discs evolve viscously. +However, simulations of disc evolution that account for non- +ideal magnetohydrodynamical (MHD) effects find that the mag- +netorotational instability (MRI), which is the likely mechanism +generating the turbulence, is largely suppressed (e.g. Bai & +Stone 2013; Lesur et al. 2014). Instead, it has also been pro- +posed that the evolution is driven by magnetically driven disc +winds (Suzuki & Inutsuka 2009). Magnetically driven disc wind +prescriptions, such as those of Suzuki et al. (2016) or Tabone +et al. (2022), include several model possibilities, which can be +narrowed down by performing a similar comparison to that pre- +sented here (Weder et al. in prep.). Once such a model is properly +coupled with internal photoevaporation to account for shielding, +a similar study to that presented here is possible. +Acknowledgements. The authors thank Christian Rab, Ilaria Pascucci, Susanne +Pfalzner, and Aashish Gupta for fruitful discussions. We also thank the anony- +mous reviewer, whose comments and suggestions greatly helped improve the +manuscript’s quality. This work was funded by the Deutsche Forschungsge- +meinschaft (DFG, German Research Foundation) - 362051796. This research +was supported by the Excellence Cluster ORIGINS which is funded by the +Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under +Germany’s Excelence Strategy – EXC-2094-390783311. The plots shown in this +work were generated using matplotlib (Hunter 2007). +References +Adams, F. C., Proszkow, E. M., Fatuzzo, M., & Myers, P. C. 2006, ApJ, 641, 504 +Alcalá, J. M., Manara, C. F., Natta, A., et al. 2017, A&A, 600, A20 +Alcalá, J. M., Natta, A., Manara, C. F., et al. 2014, A&A, 561, A2 +Alexander, R. D., Clarke, C. J., & Pringle, J. E. 2004, MNRAS, 354, 71 +Alibert, Y., Mordasini, C., & Benz, W. 2004, A&A, 417, L25 +Alibert, Y., Mordasini, C., Benz, W., & Winisdoerffer, C. 2005, A&A, 434, 343 +Andrews, S. M., Huang, J., Pérez, L. M., et al. 2018a, ApJ, 869, L41 +Andrews, S. M., Terrell, M., Tripathi, A., et al. 2018b, ApJ, 865, 157 +Andrews, S. M., Wilner, D. J., Hughes, A. M., Qi, C., & Dullemond, C. P. 2009, +ApJ, 700, 1502 +Andrews, S. M., Wilner, D. J., Hughes, A. M., Qi, C., & Dullemond, C. P. 2010, +ApJ, 723, 1241 +Ansdell, M., Williams, J. P., Manara, C. F., et al. 2017, AJ, 153, 240 +Ansdell, M., Williams, J. P., van der Marel, N., et al. 2016, ApJ, 828, 46 +Bai, X.-N. & Stone, J. M. 2013, ApJ, 769, 76 +Baraffe, I., Chabrier, G., Allard, F., & Hauschildt, P. H. 1998, A&A, 337, 403 +Baraffe, I., Homeier, D., Allard, F., & Chabrier, G. 2015, A&A, 577, A42 +Bayo, A., Barrado, D., Huélamo, N., et al. 2012, A&A, 547, A80 +Beckwith, S. V. W. & Sargent, A. I. 1996, Nature, 383, 139 +Birnstiel, T., Dullemond, C. P., & Brauer, F. 2010, A&A, 513, A79 +Birnstiel, T., Dullemond, C. P., Zhu, Z., et al. 2018, ApJ, 869, L45 +Birnstiel, T., Klahr, H., & Ercolano, B. 2012, A&A, 539, A148 +Burn, R., Emsenhuber, A., Melon Fuksman, J. D., et al. in prep., A&A +Burn, R., Emsenhuber, A., Weder, J., et al. 2022, A&A, 666, A73 +Burn, R., Schlecker, M., Mordasini, C., et al. 2021, A&A, 656, A72 +Byrd, R. H., Lu, P., Nocedal, J., & Zhu, C. 1995, SIAM Journal on Scientific +Computing, 16, 1190 +Cambioni, S., Asphaug, E., Emsenhuber, A., et al. 2019a, ApJ, 875, 40 +Cambioni, S., Delbo, M., Ryan, A. J., Furfaro, R., & Asphaug, E. 2019b, Icarus, +325, 16 +Cambioni, S., Jacobson, S. A., Emsenhuber, A., et al. 2021, PSJ, 2, 93 +Carpenter, J. M., Mamajek, E. E., Hillenbrand, L. A., & Meyer, M. R. 2006, ApJ, +651, L49 +Chabrier, G. 2003, PASP, 115, 763 +Clarke, C. J., Gendrin, A., & Sotomayor, M. 2001, MNRAS, 328, 485 +Cleeves, L. I., Öberg, K. I., Wilner, D. J., et al. 2016, ApJ, 832, 110 +Da Rio, N., Robberto, M., Hillenbrand, L. A., Henning, T., & Stassun, K. G. +2012, ApJ, 748, 14 +Dittrich, K., Klahr, H., & Johansen, A. 2013, ApJ, 763, 117 +Dr˛a˙zkowska, J. & Alibert, Y. 2017, A&A, 608, A92 +Duchêne, G. & Kraus, A. 2013, ARA&A, 51, 269 +Dullemond, C. P., Birnstiel, T., Huang, J., et al. 2018, ApJ, 869, L46 +Emsenhuber, A., Cambioni, S., Asphaug, E., et al. 2020, ApJ, 891, 6 +Emsenhuber, A., Mordasini, C., Burn, R., et al. 2021a, A&A, 656, A69 +Emsenhuber, A., Mordasini, C., Burn, R., et al. 2021b, A&A, 656, A70 +Ercolano, B., Clarke, C. J., & Drake, J. J. 2009, ApJ, 699, 1639 +Ercolano, B., Clarke, C. J., & Hall, A. C. 2011, MNRAS, 410, 671 +Ercolano, B., Drake, J. J., Raymond, J. C., & Clarke, C. C. 2008, ApJ, 688, 398 +Ercolano, B., Koepferl, C., Owen, J., & Robitaille, T. 2015, MNRAS, 452, 3689 +Ercolano, B., Picogna, G., Monsch, K., Drake, J. J., & Preibisch, T. 2021, MN- +RAS, 508, 1675 +Facchini, S., Clarke, C. J., & Bisbas, T. G. 2016, MNRAS, 457, 3593 +Favata, F. & Micela, G. 2003, Space Sci. Rev., 108, 577 +Fedele, D., van den Ancker, M. E., Henning, T., Jayawardhana, R., & Oliveira, +J. M. 2010, A&A, 510, A72 +Feigelson, E. D. & Montmerle, T. 1999, ARA&A, 37, 363 +Flaischlen, S., Preibisch, T., Manara, C. F., & Ercolano, B. 2021, A&A, 648, +A121 +Fortier, A., Alibert, Y., Carron, F., Benz, W., & Dittkrist, K. M. 2013, A&A, 549, +A44 +Franz, R., Picogna, G., Ercolano, B., & Birnstiel, T. 2020, A&A, 635, A53 +Gárate, M., Birnstiel, T., Dr˛a˙zkowska, J., & Stammler, S. M. 2020, A&A, 635, +A149 +Gáspár, A., Rieke, G. H., & Ballering, N. 2016, ApJ, 826, 171 +Getman, K. V., Flaccomio, E., Broos, P. S., et al. 2005, ApJS, 160, 319 +Gómez de Castro, A. I. 2009, Ap&SS, 320, 97 +Article number, page 13 of 18 + +A&A proofs: manuscript no. disc +Gorti, U., Dullemond, C. P., & Hollenbach, D. 2009, ApJ, 705, 1237 +Güdel, M., Briggs, K. R., Arzner, K., et al. 2007, A&A, 468, 353 +Gundlach, B., Schmidt, K. P., Kreuzig, C., et al. 2018, MNRAS, 479, 1273 +Habing, H. J. 1968, Bull. Astron. Inst. Netherlands, 19, 421 +Haisch, Karl E., J., Lada, E. A., & Lada, C. J. 2001, ApJ, 553, L153 +Hartmann, L., Herczeg, G., & Calvet, N. 2016, ARA&A, 54, 135 +Haworth, T. J., Clarke, C. J., Rahman, W., Winter, A. J., & Facchini, S. 2018, +MNRAS, 481, 452 +Hendler, N., Pascucci, I., Pinilla, P., et al. 2020, ApJ, 895, 126 +Hennebelle, P., Commerçon, B., Chabrier, G., & Marchand, P. 2016, ApJ, 830, +L8 +Hollenbach, D., Johnstone, D., Lizano, S., & Shu, F. 1994, ApJ, 428, 654 +Huang, J., Andrews, S. M., Öberg, K. I., et al. 2020, ApJ, 898, 140 +Hueso, R. & Guillot, T. 2005, A&A, 442, 703 +Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90 +Johnstone, C. P., Bartel, M., & Güdel, M. 2021, A&A, 649, A96 +Kennedy, G. M. & Kenyon, S. J. 2009, ApJ, 695, 1210 +Kimura, S. S., Kunitomo, M., & Takahashi, S. Z. 2016, MNRAS, 461, 2257 +Kingma, D. P. & Ba, J. 2014, arXiv e-prints, arXiv:1412.6980 +Koepferl, C. M., Ercolano, B., Dale, J., et al. 2013, MNRAS, 428, 3327 +Komaki, A., Nakatani, R., & Yoshida, N. 2021, ApJ, 910, 51 +Kraus, A. L., Ireland, M. J., Hillenbrand, L. A., & Martinache, F. 2012, ApJ, 745, +19 +Kuffmeier, M., Goicovic, F. G., & Dullemond, C. P. 2020, A&A, 633, A3 +Kunitomo, M., Ida, S., Takeuchi, T., et al. 2021, ApJ, 909, 109 +Kunitomo, M., Suzuki, T. K., & Inutsuka, S.-i. 2020, MNRAS, 492, 3849 +Lada, C. J. & Lada, E. A. 2003, ARA&A, 41, 57 +Lebreuilly, U., Hennebelle, P., Colman, T., et al. 2021, ApJ, 917, L10 +Lee, Y.-N., Marchand, P., Liu, Y.-H., & Hennebelle, P. 2021, ApJ, 922, 36 +Lenz, C. T., Klahr, H., & Birnstiel, T. 2019, ApJ, 874, 36 +Lesur, G., Kunz, M. W., & Fromang, S. 2014, A&A, 566, A56 +Lodders, K. 2003, ApJ, 591, 1220 +Luhman, K. L. 2000, ApJ, 544, 1044 +Lüst, R. 1952, Zeitschrift Naturforschung Teil A, 7, 87 +Lynden-Bell, D. & Pringle, J. E. 1974, MNRAS, 168, 603 +Mamajek, E. E. 2009, in American Institute of Physics Conference Series, +Vol. 1158, American Institute of Physics Conference Series, ed. T. Usuda, +M. Tamura, & M. Ishii, 3–10 +Manara, C. F., Fedele, D., Herczeg, G. J., & Teixeira, P. S. 2016a, A&A, 585, +A136 +Manara, C. F., Mordasini, C., Testi, L., et al. 2019, A&A, 631, L2 +Manara, C. F., Robberto, M., Da Rio, N., et al. 2012, ApJ, 755, 154 +Manara, C. F., Rosotti, G., Testi, L., et al. 2016b, A&A, 591, L3 +Manara, C. F., Testi, L., Herczeg, G. J., et al. 2017, A&A, 604, A127 +Matsuyama, I., Johnstone, D., & Hartmann, L. 2003, ApJ, 582, 893 +McKay, M. D., Beckman, R. J., & Conover, W. J. 1979, Technometrics, 21, 239 +Michel, A., van der Marel, N., & Matthews, B. C. 2021, ApJ, 921, 72 +Mordasini, C., Alibert, Y., & Benz, W. 2009, A&A, 501, 1139 +Mulders, G. D., Pascucci, I., Manara, C. F., et al. 2017, ApJ, 847, 31 +Murray, N., Chaboyer, B., Arras, P., Hansen, B., & Noyes, R. W. 2001, ApJ, 555, +801 +Musiolik, G. & Wurm, G. 2019, ApJ, 873, 58 +Nakagawa, Y., Sekiya, M., & Hayashi, C. 1986, Icarus, 67, 375 +Nakamoto, T. & Nakagawa, Y. 1994, ApJ, 421, 640 +Nayakshin, S., Elbakyan, V., & Rosotti, G. 2022, MNRAS, 512, 6038 +Nayakshin, S., Tsukagoshi, T., Hall, C., et al. 2020, MNRAS, 495, 285 +Owen, J. E., Clarke, C. J., & Ercolano, B. 2012, MNRAS, 422, 1880 +Owen, J. E., Ercolano, B., & Clarke, C. J. 2011, MNRAS, 412, 13 +Pascucci, +I., +Cabrit, +S., +Edwards, +S., +et +al. +2022, +arXiv +e-prints, +arXiv:2203.10068 +Pascucci, I., Testi, L., Herczeg, G. J., et al. 2016, ApJ, 831, 125 +Pedregosa, F., Varoquaux, G., Gramfort, A., et al. 2011, Journal of Machine +Learning Research, 12, 2825 +Pfalzner, S., Dehghani, S., & Michel, A. 2022, ApJ, 939, L10 +Picogna, G., Ercolano, B., & Espaillat, C. C. 2021, MNRAS, 508, 3611 +Picogna, G., Ercolano, B., Owen, J. E., & Weber, M. L. 2019, MNRAS, 487, 691 +Powell, D., Murray-Clay, R., Pérez, L. M., Schlichting, H. E., & Rosenthal, M. +2019, ApJ, 878, 116 +Preibisch, T., Kim, Y.-C., Favata, F., et al. 2005, ApJS, 160, 401 +Preibisch, T., Zinnecker, H., & Herbig, G. H. 1996, A&A, 310, 456 +Qiao, L., Haworth, T. J., Sellek, A. D., & Ali, A. A. 2022, MNRAS, 512, 3788 +Ribas, Á., Bouy, H., & Merín, B. 2015, A&A, 576, A52 +Ribas, Á., Merín, B., Bouy, H., & Maud, L. T. 2014, A&A, 561, A54 +Santos, N. C., Israelian, G., Mayor, M., et al. 2005, A&A, 437, 1127 +Sellek, A. D., Booth, R. A., & Clarke, C. J. 2020, MNRAS, 498, 2845 +Shakura, N. I. & Sunyaev, R. A. 1973, A&A, 500, 33 +Steinpilz, T., Teiser, J., & Wurm, G. 2019, ApJ, 874, 60 +Suzuki, T. K. & Inutsuka, S.-i. 2009, ApJ, 691, L49 +Suzuki, T. K., Muto, T., & Inutsuka, S.-i. 2010, ApJ, 718, 1289 +Suzuki, T. K., Ogihara, M., Morbidelli, A., Crida, A., & Guillot, T. 2016, A&A, +596, A74 +Tabone, B., Rosotti, G. P., Cridland, A. J., Armitage, P. J., & Lodato, G. 2022, +MNRAS, 512, 2290 +Throop, H. B. & Bally, J. 2008, AJ, 135, 2380 +Tobin, J. J., Sheehan, P. D., Megeath, S. T., et al. 2020, ApJ, 890, 130 +Toci, C., Rosotti, G., Lodato, G., Testi, L., & Trapman, L. 2021, MNRAS, 507, +818 +Tripathi, A., Andrews, S. M., Birnstiel, T., & Wilner, D. J. 2017, ApJ, 845, 44 +Tychoniec, Ł., Manara, C. F., Rosotti, G. P., et al. 2020, A&A, 640, A19 +Tychoniec, Ł., Tobin, J. J., Karska, A., et al. 2018, ApJS, 238, 19 +van Terwisga, S. E., van Dishoeck, E. F., Mann, R. K., et al. 2020, A&A, 640, +A27 +Venuti, L., Bouvier, J., Cody, A. M., et al. 2017, A&A, 599, A23 +Veras, D. & Armitage, P. J. 2004, MNRAS, 347, 613 +Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261 +Voelkel, O., Klahr, H., Mordasini, C., Emsenhuber, A., & Lenz, C. 2020, A&A, +642, A75 +Weder, J., Mordasini, C., & Emsenhuber, A. in prep., A&A +Williams, J. P., Cieza, L., Hales, A., et al. 2019, ApJ, 875, L9 +Winter, A. J. & Haworth, T. J. 2022, European Physical Journal Plus, 137, 1132 +Zagaria, F., Clarke, C. J., Rosotti, G. P., & Manara, C. F. 2022, MNRAS, 512, +3538 +Zormpas, A., Birnstiel, T., Rosotti, G. P., & Andrews, S. M. 2022, A&A, 661, +A66 +Article number, page 14 of 18 + +A. Emsenhuber et al.: Towards a population synthesis of discs and planets. II. +Appendix A: Parameter study +The surrogate models allow us to study the effects of the differ- +ent parameters, as shown for stellar mass and photoevaporation +in Sect. 4.3. Here, we expand the study to other parameters. The +parameters that are not varied are selected as in Sect. 4.3, ex- +cept for the photoevaporation-related parameters, which are set +as LX = 1 × 1029 erg s−1 and F = 10 G0 to provide lifetimes that +are globally in line with the observations. +We study in Fig. A.1 the effects of the parameters of the gas +disc model. The top row shows the outcomes as functions of +the power-law index of the initial profile β and the inner edge +rin. These two parameters have very little effect on the final life- +times, as all values lie within about 0.1 dex, corresponding to a +maximum relative difference of 26 %. The same applies to disc +masses and stellar accretion rates. Thus, the choice of these pa- +rameters has negligible effects on the final properties and we do +not discuss these parameters further in this work. +The bottom row of Fig. A.1 shows the effect of the viscosity +parameter α and the disc’s characteristic radius r1. The charac- +teristic radius has limited effect on the disc lifetimes while α, +which affects the whole viscous evolution, has an important ef- +fect. However, both parameters affect the stellar accretion rate, +making it possible to combine these two parameters to set the +behaviour of stellar accretion versus lifetime. +Figure A.2 shows the same analysis but for the parameters +of the solid disc. As such, only the observed dust masses are +shown for each parameter combination. The left panel features +the dust-to-gas ratio fD/G and the disc’s gas mass MG/M⋆. The +results show that the dust-to-gas ratio is only important to con- +trol the observed disc masses for discs that are close to disper- +sal (towards the left of the panel) while it has a lower effect on +more massive discs. The centre panel shows two of the twopop +model parameters, planetesimal formation efficiency and dust- +to-gas ratio. The panel shows that the planetesimal formation +efficiency parameter ε is of lower importance than the initial +dust mass (which is controlled by the dust-to-gas ratio fD/G) +for the observed disc masses. The right panels show two other +twopop model parameters, the drift efficiency ζ and the fragmen- +tation velocity vfrag. Here we see that fragmentation velocities +vfrag ≳ 300 cm s−1 lead to lower disc masses because drift be- +comes efficient. This effect can be counterbalanced by reducing +the drift efficiency ζ. However, for a value of vfrag = 200 cm s−1, +the drift efficiency has a small effect on the observed disc masses, +which indicate that drift is already inefficient. +One might expect observed dust masses to be affected +equally by the two parameters shown in the left panel (as the +initial solid mass is the product of the dust-to-gas ratio and the +gas mass MG). However, our results indicate that the initial gas +mass has a greater effect than the dust-to-gas ratio. +Article number, page 15 of 18 + +A&A proofs: manuscript no. disc +��� +��� +��� +���������������� +��� +��� +�������������������� +���� ���� ���� ���� +�������������������� +��� +��� +��� +���������������� +��� +��� +�������������������� +���� +���� +���� +���������� �� +�� +��� +��� +��� +���������������� +��� +��� +�������������������� +���� ���� ���� ���� +������� +�� +����� +�� +� +�� +� +���������� +��� +��� +�������������������� +��� +��� +��� +��� +�������������������� +�� +� +�� +� +���������� +��� +��� +�������������������� +� +� +� +���������� �� +�� +�� +� +�� +� +���������� +��� +��� +�������������������� +���� ��� +��� +������� +�� +����� +Fig. A.1. Gas disc lifetime (left), observed dust masses at 2 Myr (centre), and stellar accretion rates at 2 Myr (right) as functions of the power-law +index of the initial profile β and the inner edge rin (top) or viscosity parameter α and characteristic radius r1 (bottom) In the bottom row, the white +region is where disc lifetimes are less than 2 Myr and the values cannot be constrained by the model. +�� +� +�� +� +�� +� +�������������� +�� +� +���������������������� +��� +��� +��� +��� +���������� �� +�� +�� +� +���������������������� +�� +� +�� +� +�� +� +����������������� +��� +��� +��� +��� +���������� �� +�� +��� +��� +����������������������� +�� +� +�� +� +��� +����������� +��� +��� +��� +���������� �� +�� +Fig. A.2. Observed mass at 2 Myr as function of the parameters of the twopop model: gas-disc mass and dust-to-gas ratio (left), planetesimal +formation efficiency and fragmentation velocity (centre), and drift efficiency and fragmentation velocity (right). +Article number, page 16 of 18 + +A. Emsenhuber et al.: Towards a population synthesis of discs and planets. II. +Appendix B: Best parameters for population +To show how the different model parameters affect the mass– +accretion rate relationship and how we came to select the best +model parameters, in Fig. B.1 we provide several 2D histograms +similar to those shown in Fig. 5, but varying one parameter at +a time. These were generated with the parameters of the ‘best +match’ population, except for the parameter being varied. +The first row of the figure shows the effect of varying the disc +mass (both gas and solids). We see that an increase in the initial +mass is recovered in the observed mass after 2 Myr of evolution. +In addition, the stellar accretion rate is correlated with the disc +mass similarly to the best fit of Manara et al. (2016b). The initial +disc mass can therefore be used to set observed masses and an in- +crease from the canonical value is required to obtain disc masses +consistent with observations. However, this parameter cannot be +used to control the behaviour of the stellar accretion rates for +given disc masses. +The second row of the same figure shows the effect of reduc- +ing the disc’s characteristic radius by a certain factor. Smaller +discs will result in increased stellar accretion rates for a given +disc mass while also reducing the occurrence of non-accreting +discs (the ones at the bottom of the plot). In addition, smaller +discs will also tend to have greater disc lifetimes because they +are less affected by photoevaporation. Discs that follow the re- +lationship of Tobin et al. (2020) are found to have overly low +accretion rates for their masses, while when reduced by a factor +two, stellar accretion rates are now too high. The best-fit param- +eter is therefore between these two. +Then, the third row shows the effect of the α viscosity pa- +rameter. The synthetic populations shown here show less spread +due to the use of a fixed value of α, whereas the ones above +have the individual values selected from a distribution that goes +from 10−3.5 to 10−3.0. The results are in line with the discus- +sion in Sect. 5.1: low values of α result in low accretion rates +with a large amount of remaining discs, while large values re- +sult in a small number of discs with excessive stellar accretion +rates. Therefore, while large values of α could be used to have +large accretion rates, they would also require a corresponding in- +crease in the initial disc masses to maintain the expected lifetime +distribution. We find that values around α = 1 × 10−3 provide a +reasonable match to disc lifetimes and stellar accretion rates. In +addition, we also note that there is an increase in the observed +dust masses along with α, which is related to the extent of the +solid disc. With large α, the grains do not grow as much as with +lower α, and as a consequence do not drift as fast. Hence, the +dust emits from a larger area, which is then reflected in the emit- +ted flux and the observed mass. However, this effect only lasts +until the dispersal of the gas disc. +Article number, page 17 of 18 + +A&A proofs: manuscript no. disc +�� +� +�� +� +��� +���� ���� � ��� �� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +���� +�� +�� +� +�� +� +��� +���� ���� � ��� �� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +���� +�� +� +�� +� +�� +� +��� +���� ���� � ��� �� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +���� +�� +� +�� +� +�� +� +��� +���� ���� � ��� �� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +���� +�� +�� +� +�� +� +��� +���� ���� � ��� �� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +���� +���� +�� +� +�� +� +��� +���� ���� � ��� �� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +���� +���� +�� +� +�� +� +��� +���� ���� � ��� �� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +���� +� �� +� +�� +� +�� +� +��� +���� ���� � ��� �� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +���� +� �� +� +�� +� +�� +� +��� +���� ���� � ��� �� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +���� +� �� +� +��� +��� +������������������������������������ +Fig. B.1. 2D histograms for stellar accretion rate versus disc mass at 2 Myr, showing the effects of the different parameters, as given above each +panel. The top row shows the effect of increasing the initial disc mass MD. The middle row shows the effect of reducing the disc characteristic +radius r1. The bottom row shows the effect of the disc viscosity parameter α. The observed data from the Lupus (in red) and Chamaeleon I (orange) +star-forming regions from Manara et al. (2019) are shown for comparison. +Article number, page 18 of 18 + diff --git a/9dE3T4oBgHgl3EQfrApq/content/tmp_files/load_file.txt b/9dE3T4oBgHgl3EQfrApq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..71256411c2f441e6f095ebe6bb1fdf493c02660e --- /dev/null +++ b/9dE3T4oBgHgl3EQfrApq/content/tmp_files/load_file.txt @@ -0,0 +1,2297 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf,len=2296 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' disc ©ESO 2023 January 13, 2023 Towards a population synthesis of discs and planets II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Confronting disc models and observations at the population level⋆ Alexandre Emsenhuber1 , Remo Burn2 , Jesse Weder3 , Kristina Monsch4, 1 , Giovanni Picogna1 , Barbara Ercolano1, 5 , and Thomas Preibisch1 1 Universitäts-Sternwarte, Ludwig-Maximilians-Universität München, Scheinerstraße 1, 81679 München, Germany e-mail: emsenhuber@usm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='de 2 Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany 3 Physikalisches Institut, Universität Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland 4 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 5 Excellence Cluster ‘Origins’, Boltzmannstraße 2, 85748 Garching, Germany Received 18 August 2022 / Accepted 9 December 2022 ABSTRACT Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We want to find the distribution of initial conditions that best reproduces disc observations at the population level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We first ran a parameter study using a 1D model that includes the viscous evolution of a gas disc, dust, and pebbles, coupled with an emission model to compute the millimetre flux observable with ALMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This was used to train a machine learning surrogate model that can compute the relevant quantity for comparison with observations in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This surrogate model was used to perform parameter studies and synthetic disc populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Performing a parameter study, we find that internal photoevaporation leads to a lower dependency of disc lifetime on stellar mass than external photoevaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This dependence should be investigated in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Performing population synthesis, we find that under the combined losses of internal and external photoevaporation, discs are too short lived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To match observational constraints, future models of disc evolution need to include one or a combination of the follow- ing processes: infall of material to replenish the discs, shielding of the disc from internal photoevaporation due to magnetically driven disc winds, and extinction of external high-energy radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Nevertheless, disc properties in low-external-photoevaporation regions can be reproduced by having more massive and compact discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Here, the optimum values of the α viscosity parameter lie between 3 × 10−4 and 10−3 and with internal photoevaporation being the main mode of disc dispersal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Protoplanetary disk — Methods: numerical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Introduction Protoplanetary discs are the birthplace of planets (the ‘nebular hypothesis’ of Kant and Laplace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Discs serve as a source of gas and solids from which the planets accrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Planet–disc interac- tions lead to planetary migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To model planetary formation, it is therefore essential to have disc characteristics that are as close as possible to those observed to provide the highest possi- ble fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Disc observations are not an entirely new subject of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Disc masses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Beckwith & Sargent 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2009) and lifetimes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Haisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Mamajek 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Fedele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Kraus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Ribas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2014) have been observed for over two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, there have been many new results concerning protoplanetary discs in the last sev- eral years, including the mass and physical extent of early discs (Tychoniec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2018, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Tobin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020) and at later times (Hendler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Nevertheless, some aspects of disc evolution are not cap- tured by observations, such as the process that leads to trans- port of material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' These are usually taken to be turbulent viscosity ⋆ Tables 3 and 4 are only available in electronic form at the CDS via anonymous ftp to cdsarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='fr (130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='5) or via https://cdsarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='fr/cgi-bin/qcat?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='J/A+A/ generated by the magnetorotational instability and magnetically driven disc winds (Suzuki & Inutsuka 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Suzuki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The strength of the turbulent viscosity has not yet been properly determined and is usually parametrised using a factor α (Shakura & Sunyaev 1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' There are indirect methods to estimate the value of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Ultraviolet(UV)-excess measurements of the accretion luminos- ity were used to derive the accretion rate onto the star for the Chamaeleon I (Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016a, 2017) and Lupus (Alcalá et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2014, 2017) star forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' These measurements coupled to the disc masses for the same regions of Cha I (Pas- cucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016) and Lupus (Ansdell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016) provide a rela- tionship between mass and accretion rate (Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Together, these can be used to calibrate numerical models (Ma- nara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2019) and to provide an estimate of the mass flux onto the disc (Mulders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Sellek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' A second method to estimate α is to compare dust and gas emission, either using spatially resolved observations of disc substructures from ALMA (Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2018a) such as pressure bumps (Dulle- mond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2018) or from the overall disc sizes (Toci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Mass loss does not occur only due to accretion onto the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For instance, observations also point towards protoplanetary disc dispersal occurring from the inside out and on relatively short timescales (Ercolano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2011, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Koepferl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Article number, page 1 of 18 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='04656v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='EP] 11 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' disc This suggests there is an additional mechanism removing gas close to the star, with one possibility being internal photoevapo- ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Coupled with the findings that young stars emit a larger fraction of their flux in UV (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Gómez de Castro 2009) and X-rays (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Preibisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1996, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Feigelson & Montmerle 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Favata & Micela 2003), it is proposed that extreme UV (EUV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Hollenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Clarke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2001) and/or X- rays (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Ercolano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2008, 2009) are responsible for this mass loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Using hydrodynamical simula- tions, it is possible to predict the mass-loss rate as a function of disc properties and stellar luminosity (Owen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Picogna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2019, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Ercolano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Nevertheless, irradiation from the host star is not the only mass-loss mechanism: most stars are born in clusters where many stars form concurrently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Consequently, protoplanetary discs are exposed to a larger ambient radiation field than mature stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This leaves an additional mechanism for mass removal by external photoevaporation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Matsuyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Winter & Haworth 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This is supported by observational findings that discs near massive stars have lower masses than others (Ansdell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' van Terwisga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020) and that clusters with a low ambient radiation field have longer disc lifetimes (Michel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' As for external photoevaporation, hydrodynamical simu- lations were performed to predict mass-loss rate (Haworth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2018) as function of disc properties and ambient flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Together with simulations of cluster evolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2022), this enables us to determine the mass-loss rate over an entire disc population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' All these observations and theoretical predictions put a lot of constraints on protoplanetary disc evolution, as the number of free parameters is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Whether or not the combination of initial disc properties and predicted accretion and mass-loss rates can be used to reproduce the distribution of, for instance, disc lifetimes remains to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Previous studies in this direction usually consider only one type of photoevaporation, either internal (Gorti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Owen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Kunitomo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021) or external (Kunitomo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Burn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2022, hereafter Paper I) introduced a relatively simple 1D radial disc model that is capable of consistently evolv- ing gas, dust, pebbles, and planetesimals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In addition, this model is capable of predicting how the modelled disc would be ob- served by current instrumentation, such as ALMA (see Birnstiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Further, the light computational requirements of that model make it possible to perform many such evolutions in order to study the effects of initial disc properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Our goals are twofold: first, we aim to determine whether we can understand the general picture of protoplanetary discs set by the observations and predictions highlighted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Second, we want to find the combinations of disc properties that best repro- duce the various observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This should then serve as initial conditions for future planetary population syntheses, such as in Mordasini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2009) or Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To fit the best parameters, many calculations need to be per- formed, each involving the evolution of a population of proto- planetary discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To alleviate the computational requirements of this procedure, we use machine learning to fit neural networks that can reproduce the result of the underlying model with lim- ited resources (Cambioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2019a, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This ‘surrogate model’ can then be used as the forward model in the fitting procedure (Cambioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In this work, we aim to find initial conditions for the disc evolution calculations that best match observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For this pur- pose, we first compute two series of calculations using the model presented in Paper I (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' These data are then used to fit several surrogate models that hold the necessary outcomes for comparison with observations (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Using these surrogate models, we study the effect of the photoevaporation prescriptions (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3) and find initial conditions that best match the obser- vational constraints discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3 as a whole (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' A study dedicated to this last aspect using a Bayesian approach in- stead will be presented in Burn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', hereafter Paper III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Methods The disc evolution model is based on the Bern global model of planetary formation and evolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Alibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2004, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Mordasini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Fortier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Voelkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021a) where planet formation has been turned off to retain only the disc part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Paper I presented an updated version of the coupled gas and solids model that in- cludes proper modelling of the disc dispersal stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' As the model was extensively described in Paper I, we only provide a brief overview of the physical processes included in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Gas disc The gas disc is modelled by an azimuthally averaged 1D ra- dial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Its evolution is obtained by solving the advection– diffusion equation (Lüst 1952;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Lynden-Bell & Pringle 1974) ∂ΣG ∂t = 3 r ∂ ∂r � r 1 2 ∂ ∂r � νΣGr 1 2 �� − ˙Σint − ˙Σext, (1) where ΣG = � ∞ −∞ ρGdz is the surface density and ν = αcsH the viscosity (parametrised using the α prescription of Shakura & Sunyaev 1973), with cs and H being the sound speed and scale height of the disc, and ˙Σint and ˙Σext the sink terms due to in- ternal and external photoevaporation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To compute the vertical structure of the disc (and with this ρG, H, and cs), we proceed as in Paper I and use the vertically integrated ap- proach of Nakamoto & Nakagawa (1994), including stellar irra- diation (Hueso & Guillot 2005) from an evolving stellar lumi- nosity computed from Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Internal photoevaporation Internal photoevaporation is modelled assuming X-ray-driven mass loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This prescription requires one parameter that is not obtained from elsewhere in the model, the stellar X-ray lumi- nosity LX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This luminosity is converted into a total mass-loss rate ˙MX, and then into a profile ˙Σint using fits to hydrodynamical simulations performed by Picogna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2019), Ercolano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021), and Picogna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021), as described in Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' External photoevaporation The mass-loss rate due to external photoevaporation is obtained from the FRIED grid (Haworth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Interpolation in the grid requires the stellar mass M⋆, the current disc mass MG, its outer radius rout, and the ambient far-UV (FUV) field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' All but the latter parameter can be computed consistently from the disc structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The grid spans values of the ambient field F be- tween 10 and 104 G0, where G0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='6 × 10−3 erg s−1 cm−2 ap- proximately represents the interstellar value (Habing 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The total mass-loss rate is converted into a profile ˙Σext assuming mass is lost in the outermost 10 % where the gas disc is present at a given time (Paper I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Article number, page 2 of 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' : Towards a population synthesis of discs and planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The FRIED grid, in its current state, presents two shortcom- ings that we adapt here: (1) the lack of data for ambient fluxes below 10 G0 and (2) a floor evaporation rate of 10−10 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Both items lead to a significant external photoevaporation rate under any circumstances, which makes it very difficult to disen- tangle the effects of external photoevaporation from the rest (in- cluding internal photoevaporation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To remedy these problems, we make one addition and one change to the FRIED prescrip- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The change is to take the lower boundary of the external photoevaporation rate down to 1 × 10−15 M⊙ yr−1, which repre- sents a negligible mass-loss rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For this, we remove the floor value of 10−10 M⊙ yr−1 from the value returned from the interpo- lation in the grid and ensure that the resulting value is at least 1 × 10−15 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The change we make is to extend the domain down to 1 G0 to be able to study low-ambient-field cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In the region below 10 G0, we perform a linear interpolation between the value returned from the grid at that boundary and a fixed value of 1 × 10−15 M⊙ yr−1 at 1 G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Solids disc model The solid component of the disc is modelled using the two- population model of Birnstiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This approximates the full size distribution using only its two extremes: the smaller a0 well coupled to the gas (which can be seen as dust) and the larger, rapidly drifting a1 (which can be seen as pebbles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The smaller size a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1 µm is fixed while the larger size is con- strained by various limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The fragmentation limit is given by a1 = fF 2 3π ΣG ρsα v2 frag c2s , (2) where fF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='37 is a factor fitted to hydrodynamical simula- tions of Birnstiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2010) for the typical size of pebbles, ρs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='675 g cm−3 is the bulk density, and vfrag the fragmentation velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In the drift limit, the large size is given by a1 = fD 2Σdustv2 K πρsc2sζ ���∂ ln P ∂ ln r ���−1, (3) where vK is Keplerian velocity, Σ0 the surface density of dust only, and ζ is an efficiency parameter of the drift (to account for the fact that drift is more limited in discs with features such gaps created by planets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Zormpas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2022, while we only study smooth discs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The surface density of solids ΣD is divided into the two com- ponents with Σ0 = ΣD (1 − fm) and Σ1 = ΣD fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The factor fm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='75 when growth is fragmentation-limited and fm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='97 when drift-limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Gas drag onto both dust and pebbles is as- sumed to be in the Epstein regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The radial velocity of solids is made of two components, coupling to the radial gas flow and headwind (Nakagawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Paper I), u0/1 = uG,red 1 + St2 0/1 − 2udr St0/1 + (St0/1ϱ2)−1 , (4) where uG,red is the reduced radial gas velocity according to Gárate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2020) and Paper I, udr = − r 2vKρG ζ ∂P ∂r , St is the Stokes number, ϱ = ρG/(ρG + ρD), and ρD is the midplane dust density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Here, we introduce a drift efficiency ζ to parametrise mecha- nisms that reduce the headwind-induced drift velocity of dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In particular, it is possible to use this approach to represent the effect that radial substructures have on the drift of solids without modelling them in full detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The mass-averaged radial velocity is then given by ¯u = (1 − fm) u0 + fmu1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' As in the gas disc, time evolution is provided by an advection–diffusion equation, ∂ΣD ∂t = 1 r ∂ ∂r � r � ΣD¯u − DGΣG ∂ ∂r �ΣD ΣG ��� − ˙Σphoto − ˙Σrad − ˙Σpts, (5) where DG is the gas diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The terms ˙Σphoto and ˙Σrad are sink terms due to dust being entrained by photoevaporative winds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Facchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Franz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020) and ejected due to radiation pressure, respec- tively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' they are both described in Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In contrast to Paper I, we allow planetesimals to form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This is parametrised using the term ˙Σpts = ε d ˙MD 2πr = ε d|¯udr|ΣD, (6) which follows the prescription of Lenz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2019) as imple- mented and described in detail in Voelkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Here ε is a parameter that specifies the conversion efficiency into plan- etesimals over a length scale of d = 5H (Dittrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2013), ¯udr is the drift component of the mass-averaged radial velocity, and ˙MD the relative mass flux of dust and pebbles through the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Conversion into observed disc masses For consistency with disc mass observations, millimetre (mm) emission from dust and pebbles is computed from the disc sur- face density and temperature profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The method is similar to that of Birnstiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2018) and will be discussed in more de- tail in Paper III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The calculation is performed for a wavelength of λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='89 mm to reproduce ALMA observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The flux is converted back into a mass using a simple prescription assuming T = 20 K and the corresponding opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For comparison, we also provide the unbiased disc masses of gas and solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To be presented alongside disc gas masses, solid masses are multiplied by a factor 100, which is typically used as a gas-to-dust ratio in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Model parameters and initial conditions The evolution model requires several initial conditions and pa- rameters for evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' These are: the mass of the central star M⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' the mass of the gas disc MG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' the initial dust-to-gas ratio fD/G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' the power-law index for the initial profile β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' the inner edge of the disc rin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' the characteristic radius of the disc r1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' the tur- bulent viscosity parameter α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' the planetesimals formation effi- ciency ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' the fragmentation velocity vfrag;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' the efficiency of drift ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' the stellar X-ray luminosity LX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' and the ambient UV field strength F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The initial surface density profile of the gas disc is set as (Veras & Armitage 2004) ΣG(t = 0) = Σini � r r0 �−β exp �������− � r r1 �2−β������� � 1 − � rin r � , (7) where Σini is the surface density at r0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2 au, the reference dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The conversion between that and the total mass is obtained with MG = 2πΣini 2 − β (r0)β (r1)2−β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (8) The initial solid profile of the disc ΣD(t = 0) is set by multiplying the initial gas profile ΣG(t = 0) by the dust-to-gas ratio fD/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Article number, page 3 of 18 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' disc Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Parameter range for the main simulation grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Variable Sampling Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M⋆/M⊙ linear 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='4 MG/M⋆ logarithmic 10−3 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='5 β linear 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2 Pin/d logarithmic 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='15 3 × 101 r1/au logarithmic 3 3 × 102 α logarithmic 10−5 10−2 fD/G logarithmic 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='5 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3 vfrag/cm s−1 logarithmic 2 × 101 2 × 103 ε logarithmic 10−3 10−1 ζ logarithmic 10−2 1 LX/1030 erg s−1 logarithmic 10−2 102 F/G0 logarithmic 1 104 In the remainder of this work, we do not provide all param- eters as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For instance, the inner edge is parametrised by its period Pin, which we convert into distance by means of Kepler’s third law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Also, we generally set the initial disc mass by its solid content MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The ratio between the initial solids and gas masses is readily given by the dust-to-gas ratio, such that fD/G = MD/MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Simulation list To generate the list of the simulations to be performed, we se- lected the Latin hypercube sampling (LHS) method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' McKay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' By dividing each dimension into n intervals and then selecting one random sample from each interval, LHS en- sures that the entire range of possible values for each parameter is sampled with a uniform probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Additional criteria are re- quired to avoid correlation between selected values of different parameters (to disentangle their effects) and to ensure that the entire space is well sampled (to avoid locations with no results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To build the grid, we use the pyDOE2 Python package with the minmax setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Each generated grid contains values in the [0,1] range with uniform probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' These values have to be mapped into the range to be stud- ied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For our main grids, we outline these in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The selec- tion was made to encompass the needs of this and future works, as well as the limitations of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For instance, the stellar mass M⋆ is taken in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1 M⊙ to lie on the stellar evolu- tion tracks of Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2015), and the limits of the ambient UV field strength F match those of the FRIED grid (Haworth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2018) with the extrapolation for low field values from Pa- per I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The gas mass of the disc is given in terms of the stellar mass to roughly follow the scaling of Burn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The dust-to-gas ratio was selected to span the possible stellar metal- licities, with a reference stellar metallicity fD/G,⊙ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='0149 (Lod- ders 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The range power-law index was selected to cover the possible values of Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The lower boundary of the period at the inner edge Pin corresponds to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='7 d, which is nearly the maximum value of the stellar radii in the models of Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The fragmentation velocity was cho- sen to encompass the previously assumed value of ∼10 m s−1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Dr˛a˙zkowska & Alibert 2017, and references therein), and more current values of ∼1 m s−1, as ice was not found to be more sticky than silicates in recent experiments (Gundlach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Musiolik & Wurm 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Steinpilz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The range of planetesimal-formation efficiencies was selected to be able to study low efficiencies where only a small fraction of the mass of solids is converted into planetesimals and to cover the case ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='05, which forms a sufficient amount of planetesimals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Machine learning Surrogate models of disc evolution are obtained by means of a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' These neural networks are trained, validated, and tested using the scikit-learn Python package (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' scikit-learn uses cross-validation to train and validate the neural network with five passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This means that the combined training and validation set is divided into five equal- sized batches, and five successive training and validation steps are performed, each using four of the five batches for training and one batch for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The neural networks are fitted us- ing either the L-BFGS-B algorithm (Byrd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1995), which is part of the SciPy package (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020) or the ADAM method (Kingma & Ba 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Observational constraints To compute disc populations that are comparable to observa- tions, we must first describe the constraints on their initial prop- erties and their outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' These are then used to set the initial conditions and the comparison point for the outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Stellar mass The stellar initial mass function (IMF) has been determined (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Chabrier 2003), and so it could in principle be used to re- produce the stellar population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, the stellar mass func- tions for different star-forming regions deviate from the IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In the case of Taurus, Luhman (2000) found a peak around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='6 to 1 M⊙, while for the Orion Nebula Cluster (ONC), Da Rio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2012) found that the best log-normal fit has a mean at log10(M⋆/M⊙) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='45 (corresponding to M⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='35 M⊙), using the stellar evolution model of Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Cor- roborating this, the sample of Flaischlen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021), which is based on that of Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2012), has a stellar mass distri- bution peaking around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='4 M⊙: a simple log-normal fit to that data gives log10(µ/M⊙) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='481 and a narrower standard devi- ation of log10(σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' As the main aim of this work is to compare our model with disc lifetimes, their mass, and the stellar accretion rate of nearby star-forming regions, we chose to follow the stellar mass function of Da Rio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2012), with a mean of log10(µ/M⊙) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='45 and standard deviation log10(σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This should offer a distribution that is representative of both nearby clusters in general and of stars for which observations of disc masses and stellar accretion rates are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Our model uses the stellar evolution tracks of Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2015) to obtain the luminosity for disc irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' These are only defined for mass increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1 M⊙ from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='4 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To properly track stellar luminosities, we restricted ourselves to stellar masses that match these values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Initial dust mass Initial dust masses can be obtained from works targeting the youngest stars known to date, such as Tychoniec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2018), Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2019), or Tobin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021b) fitted the masses of the Class 0 discs of Tychoniec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2018), which gave log10(µ/M⊙) = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='49 and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='35 dex, taking out the conversion from dust mass to gas mass using the standard factor of 100 that was used there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' These disc masses Article number, page 4 of 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' : Towards a population synthesis of discs and planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Best-fit parameters for stellar X-ray luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Work a b Sca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Preibisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2005) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='10 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='65 Güdel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2007) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='12 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='54 Parameters a and b are those of the fit log � LX/erg s−1� = a × log (M⋆/M⊙) + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The ‘Sca.’ column provides the scatter of the residuals from the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' were used for a population of stars with masses of 1 M⊙ while the populations around lower-mass stars of Burn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021) scaled the disc masses proportionally to the stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, a complication arises from the fact that the mass of the central body is not known for the objects observed by Tychoniec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2018) and Tobin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To properly con- vert the absolute masses into disc-to-star mass ratios, as we do in this work, we must assume a reference stellar mass Mref ⋆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The method of Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021b) and Burn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021) was equivalent to setting Mref ⋆ = 1 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We use this as our default conversion factor, although consistency with the stellar mass dis- tribution discussed in the previous section would call for a lower value of Mref ⋆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We explore different values of this factor later in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Sizes Protoplanetary disc sizes have been found to be correlated with their mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2010) found that discs in the Ophi- uchus star-forming region have MD ∝ r(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3) 1 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' More recent studies, such as those of Tripathi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2017) and Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2018b), found that MD ∝ r2 1, while Hendler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2020) obtained different scalings across various star-forming re- gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For young and non-multiple discs, Tobin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2020) ob- tained r1 ∝ M(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='03) D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Adding a normalisation from the same work, we get r1 70 au = � MD 100 M⊕ �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='25 , (9) plus a residual scatter of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Dust-to-gas ratio The ratio between the initial masses of the gas and dust discs is given by fD/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We select this parameter as in Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021b), that is, we assume it is the same as the stellar metallicity (Gáspár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Thus, we can use the relation fD/G fD/G,⊙ = 10[Fe/H] (Murray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2001), where fD/G,⊙ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='0149 is the primordial solar value (Lodders 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The distribution of metallicity is chosen to be that of the CORALIE RV search sample (Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2005), which was modelled as a Gaus- sian with a mean of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='02 and a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To avoid extreme values, we restrict the parameter to within −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='6 < [Fe/H] < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Stellar X-ray luminosity A couple of surveys have been performed to determine the X- ray luminosities of young stars, the relevant results of which are provided in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' One is the Chandra Orion Ultradeep Project (COUP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Getman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Preibisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2005), which covers stellar masses M⋆ between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='9 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The survey found a stellar-mass dependency of LX ∝ M(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='44±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='10) ⋆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Another survey, the XMM-Newton Extended Survey of Taurus (XEST;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Güdel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2007), found that LX varies with stellar mass as LX ∝ M(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='54±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='12) ⋆ , which we used to correct for the stellar-mass effect and recompute the inherent scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The two surveys have similar stellar mass dependence, meaning that using one or the other to set the stellar X-ray luminosities should not affect the outcomes in any significant manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For this work, we compute LX using a log-normal distribution with the parameters selected following XEST (Güdel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2007), as the stellar mass depen- dence is consistent with the prescription used to compute the X-ray photoevaporation profiles in Picogna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Ambient FUV field strength The external photoevaporation prescription of Haworth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2018) requires the stellar mass, disc mass, outer radius, and am- bient FUV field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The first three can be readily obtained consistently from the simulation, but the latter, F, needs to be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Most stars are formed in stellar clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Lada & Lada 2003), which result in high stellar densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To retrieve the am- bient FUV relevant during the lifetime of protoplanetary discs, we use the simulation of Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The authors deter- mined that F is well described by a log-normal distribution with a median close to 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='25G0, where G0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='6 × 10−3 erg s−1 cm−2 is nearly the interstellar FUV field (Habing 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Inner edge of the gas disc The location of the inner edge of the gas disc is most relevant for the location of the close-in planets (such as hot Jupiters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' As we are mostly interested in warm giants further away than the inner edge, this parameter is of less importance in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We chose this parameter in the same way as in Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021b), that is, by assuming that the disc is truncated at the corotation ra- dius of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For the distribution of stellar rotation periods, we follow the results of Venuti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This gives a log-normal distribution with a median period of log10(µ/d) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='67617866 and deviation σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='305 673 3 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For comparison, the distribution of initial rotation periods used by Johnstone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021) has a median of log10(µ/d) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='5181 and a standard deviation of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3236 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The median rotation period here is smaller here (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3 d) than the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='7 d value of Venuti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2017) but not by a large amount, while the devia- tions are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The exact choice should therefore not affect the results significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Full model To generate the training, validation, and testing data for the sur- rogate models, we generated two sets of simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The first set contains 100 000 models that are used for the combined training and validation steps, while the second set contains 20 000 mod- els and is used for the testing step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The values of the first set are provided in Table 3 while the values of the second set are pro- vided in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Both tables are available at the CDS and have the same format;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' they contain the following columns: columns 1 to 12 are the initial conditions in the same order and units that are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Column 13 gives the disc lifetime accord- ing to when the mass becomes lower than 10−6M⋆ or when the surface density is lower than 1 × 10−3 g cm−2 inside 100 au (or Article number, page 5 of 18 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' disc 30 au for M⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1 M⊙ or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2 M⊙) and 1 × 10−2 g cm−2 outside that (this second criterion on the surface density is to avoid ex- cessively long-lived discs when photoevaporation rates, particu- larly external ones, are low).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Column 14 gives the lifetime us- ing the minimum value of the criterion of column 13 and the observability criterion in the near-infrared (NIR) from Kimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Columns 15-19 give the following outcomes at 1 × 105 yr: stellar accretion rate log10 � ˙M⋆/M⊙ yr−1� , the true gas mass log10 (MG/M⊙), the true solids mass log10 (MD/M⊙), the observed mass (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3) log10 (Mobs/M⊙) and the radius en- compassing 68 % of the flux log10 (r68/au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Columns 20-24 re- peat the same information, but at 2 × 106 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Two epochs (1 × 105 yr and 2 × 106 yr) were selected to be compatible with the observations we are comparing to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The first epoch is for comparison with early discs, such as their initial masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Its selection is a trade-off between two items: on the one hand, we would like to have the data as early as possible, while on the other hand, we need to wait until the initial dust growth has taken place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' From the analysis of individual discs, we found that 1 × 105 yr represents a good compromise in that sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The second epoch is for comparison with the star-forming regions of Lupus and Cha I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' As the stars in these regions are between 1 and 3 Myr old, we take the results at 2 Myr, as in Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Our results indicate that the two criteria for disc dispersal produce nearly identical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In only about 10 % of the cases, the NIR criterion predicts a lower disc lifetime than the crite- rion based on the mass, and the difference remains small when this occurs (we do not check for the reverse, as calculations stop when the mass criterion is reached).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' These results are consis- tent with the findings of Kunitomo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' As a conse- quence, hereafter we only report the disc lifetimes based on the NIR criterion of Kimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Also, we stop the calcu- lation at 100 Myr in any case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This affects some long-lived discs with minimal photoevaporation and accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In such cases, the lifetime based on the mass criterion is not reported while that based on the NIR emission is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Performance of the surrogate model We asses the performance of the surrogate models in terms of the best regression (obtained using ordinary least squares), the Pearson correlation coefficient R2, and the RMS of the differ- ences between the predicted and target lifetimes (the square root of the mean square error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' These were computed on the testing set (Table 4) that the surrogate model has not seen before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The hyper-parameters and results for all surrogate models that are part of this work are presented in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For three of them, we also show correlation plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In all cases, the fitting pro- cedure was performed on the logarithm (base 10) of the values, and so all the reported performances are given in these units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Concerning the different surrogate models, the ones for the disc lifetimes and for the stellar accretion rates provide the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The ones that are based on the dust disc, namely masses and radii, show a lower performance, especially at 2 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We note that these values are for each single prediction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' they rep- resent the level of additional uncertainty for the parameter stud- ies (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3 and Appendix A) while for the population studies (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 5) these errors can average out and result in an even better global accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The neural networks predicting the disc masses, their radii, and stellar accretion rates were fitted only on the discs that had not vanished at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This means that they are supported by a lower number of points than the ones predicting the life- times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This also implies that these surrogate models are only constrained in the region of the parameter space where lifetimes are larger than the time of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Thus, in the remainder of this work we only provide disc masses and stellar accretion rates for discs that have not yet dispersed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Effects of photoevaporation We began our investigations using the surrogate model, perform- ing a parameter study of the effects of the photoevaporation pre- scriptions on disc lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For this purpose, we generated two maps, one for internal photoevaporation and one for external photoevaporation, which vary the stellar mass and the control- ling parameter of each photoevaporation prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In each case, the value of the parameter controlling the other photoevap- oration prescription was set at the minimum of the studied range in order to avoid cross effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We assumed typical values of the remaining parameters: disc-to-star gas mass ratio MG/M⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1, dust-to-gas ratio fD/G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='0149, power-law index β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='9, period at the inner edge Pin = 10 d, characteristic radius r1 computed according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (9), viscosity parameter α = 1 × 10−3, fragmen- tation velocity vfrag = 2 m s−1, planetesimal formation efficiency ε = 1 × 10−3, and drift efficiency ζ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Internal photoevaporation The resulting map for internal photoevaporation is provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Here we observe that the surrogate model predicts sev- eral sharp transitions of disc lifetime with stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The most evident are those between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3 M⊙ and between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='8 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='9 M⊙ where disc lifetime increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' There are other transitions between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='5 M⊙ and between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='6 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='7 M⊙ where disc lifetime decreases, but only for large X-ray luminosities (LX > 1030 erg s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' These transitions match the switch from one photoevaporative profile to another, which are marked by the dashed white lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This indicates that the profile of surface- density loss has a strong effect on disc lifetime and not only the total mass-loss rate, which gradually changes between each stel- lar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Also, the further out the location of the peak of internal photoevaporation (which is for the profiles of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3 M⊙ and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='0 M⊙ stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' see top panel of Figure 7 of Picogna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021), the longer the disc lifetimes are in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We find that this effect is due to a larger inner region where material is not evaporated at all and can only be dispersed by viscous accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In this case, the ob- served disc lifetime is set by the dispersal timescale of the inner disc, which is given by the viscous timescale at the outer radius of the inner disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' As discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='5, the X-ray luminosity is correlated with stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To highlight this, we show in addition the me- dian stellar X-ray luminosity as a function of stellar mass from Güdel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2007) with the green dashed curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To determine the expected relationship between disc lifetime and stellar mass, one needs to follow this curve rather than a horizontal line on the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We see that internal photoevaporation leads to a lim- ited change in disc lifetime with stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This is because more massive stars lead to stronger mass-loss rates (owing to a corresponding increase in stellar X-ray luminosity), which com- pensates for the increase in disc mass (as we assume disc mass to be proportional to stellar mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This is shown by the black line that traces disc lifetimes of 3 Myr (a typical value in obser- vations), which is consistently lower than the median LX by a factor of a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Article number, page 6 of 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' : Towards a population synthesis of discs and planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' ��� ��� ��������������������� ��� ��� ��� ������������������������ � � ���� � � ���� �� � ����� ����������� �������������� �� �� �� �� ������ ���� ���� �� ���� �� �� �� �� �� � ��������� ���� ���� �� ���� � � ���� � ���� �� � ����� ����������� ����������������������� �� �� �� � ������ ���� �� � �� �� �� �� �� � ��������� ���� �� � � � ���� � ���� �� � ����� ����������� ����������������������� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Performance of three surrogate models based on the comparison of the predicted and actual values of the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The insert values show the best regression (ordinary least squares), the Pearson correlation coefficient R2, and the RMS of the differences between each predicted and actual value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Hyper parameters and performance of the surrogate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Age Model Solver Activ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' HLS alpha Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' R2 Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' R2 Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' MSE Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' MSE Lifetime L-BFGS logistic 25, 50, 45 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='592 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='99465 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='99436 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='00368 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='00389 100 kyr Accretion L-BFGS tahn 15, 30, 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='098 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='99909 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='99829 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='00163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='00300 Mass ADAM tahn 55, 40, 65 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='659 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='96330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='95040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='05690 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='07677 Radius ADAM tahn 60, 55, 45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='131 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='89222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='88045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='04460 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='04853 2 Myr Accretion L-BFGS tahn 60, 45, 70 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='407 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='99865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='99506 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='00349 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='01297 Mass ADAM tahn 65, 25, 55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='915 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='91565 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='89798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='25667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='32089 Radius ADAM tahn 70, 65, 35 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='901 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='85665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='80786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='11881 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='15925 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' External photoevaporation The resulting map for external photoevaporation is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Unlike internal photoevaporation, the prescription for ex- ternal photoevaporation provides for gradual changes of lifetime with stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, these changes lead to a larger depen- dency of disc lifetime on stellar mass than what is expected from internal photoevaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This is illustrated by the black line, which tracks a 3 Myr lifetime, as in the map for internal photo- evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Its position in terms of ambient UV field strength varies across the entire parameter range studied here, from less that 2 G0 for M⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1 M⊙ to about 4 × 103 G0 for M⋆ = 1 M⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' it becomes independent of stellar mass for M⋆ > 1 M⊙ and fluxes above ∼103 G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' While the trend of reduced disc lifetimes in regions with strong ambient UV fields (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Michel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021) is reproduced, the general behaviour of correlated disc lifetimes with stellar mass for a given ambient UV field strength is problematic for several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' First, this general behaviour is inconsistent with observations that suggest disc lifetimes are independent of, or slightly decreasing with, increasing stellar mass (Carpenter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Kennedy & Kenyon 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Bayo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Ribas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' There are several possibilities to remedy this, although they are unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To obtain a behaviour similar to observations, the mass loss would need to be correlated with stellar mass (Ko- maki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021), which in turn would require that the ambi- ent field be correlated with stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, the ambient field is usually dominated by the few most massive stars (Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2006), which means that it depends more on the cluster as a whole than on the mass of the star in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Another av- enue is that disc masses scale to a lesser extent with stellar mass than assumed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, this would not yield the expected behaviour of stellar accretion rates with stellar masses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Hart- mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Alcalá et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Flaischlen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We find that the FRIED grid prescription that we use in this work produces incompatible results that show at most a dependence of the disc lifetime on stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The second concern is that further lifetime analyses will be strongly affected by the selec- tion of the stellar masses, in contrast to internal photoevaporation where this dependency is weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Parameter sensitivity The sensitivity of disc lifetimes, disc masses, and stellar accre- tion rates at 2 Myr is studied in detail in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' These results can be summarised as follows: all the outcomes are in- sensitive to the power-law index β and the inner edge of the gas disc rin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The characteristic radius r1 and viscosity α control the viscous timescale of the disc, and therefore the stellar accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Disc lifetimes and observed dust masses are more strongly affected by the viscosity α than by the characteristic radius r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The twopop model parameters only affect the observed dust masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Observed dust masses are less affected by the dust-to- gas ratio fD/G than by the initial mass of the gas disc MG, except for discs close to dispersal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The fragmentation velocity vfrag and the drift efficiency strongly affect the observed dust masses, but only for vfrag ≳ 200 cm s−1, while the planetesimal formation efficiency ε only has a limited effect for values close to the max- imum we study, namely of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' These results narrow down the parameter space that we ex- plore in the remainder of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' First, we keep the values of the power-law index β and the inner edge of the gas disc rin as described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3, because they are of negligible importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We then only use the initial mass of the gas disc MG to con- trol disc masses, not the dust-to-gas ratio fD/G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' as the latter is in Article number, page 7 of 18 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' disc ��� ��� ��� ��� ��� ��� ��� ��������������� � �� � �� � ��� ��� ��� ������������������� ����� ������ ��������� ���� ����� ���� ����� ���� ����� ���� ����� ���� ������� ��� ��� ��� ��� ��� ��� ������������������������� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Map of disc lifetimes as a function of stellar mass and X-ray luminosity, which is the main driver of internal photoevaporation, ac- cording to the surrogate model described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' External photoe- vaporation was set to its minimum value (F = 1 G0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Other parameters were selected as MG/M⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1, fD/G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='0149, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='9, Pin = 10 d, r1 according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (9), α = 1 × 10−3, vfrag = 2 m s−1, ε = 1 × 10−3, and ζ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The green dashed line represents the dependency of LX on M⋆ from Güdel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2007) and the solid black line shows the location of a 3 Myr lifetime (a typical value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The results are discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' most cases of lower importance and well constrained by obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Also, we keep the planetesimal formation efficiency to the minimum value of ε = 1 × 10−3, the fragmentation velocity to vfrag = 200 cm s−1, and the drift efficiency to ζ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Disc populations We now compare synthetic disc populations with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For this, we proceed as follows: we draw 10 000 random discs whose initial conditions follow given distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The out- comes of each disc are obtained by means of the different sur- rogate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For the analysis, we first compare the cumulative distribution of disc lifetimes so that it can be compared to the fraction of stars that have a protoplanetary disc for stellar clus- ters with different ages, as in Haisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The second analysis is to compare observed disc masses and stellar accre- tion rates with the data of Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Here, we use the data at 2 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Further, we only use discs whose lifetime, as determined by the surrogate model from the previous analysis, is larger than the time of analysis in order to avoid being in the region where the surrogate model is not supported by any under- lying data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Canonical To determine if all the processes that are predicted from theory are able to reproduce disc observations, we compute a population of discs whose properties are as close as possible to observations ��� ��� ��� ��� ��� ��� ��� ��������������� � ��� ��� ��� ��� ��� ����������������������� ��� ��� ��� ��� ��� ��� ������������������������� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Map of disc lifetime as functions of stellar mass and ambient UV field strength, this latter being the main driver of external photoevapo- ration according to the surrogate model described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Internal photoevaporation is set to its minimum value (LX = 1 × 1028 erg s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Other parameters were selected as MG/M⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1, fD/G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='0149, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='9, Pin = 10 d, r1 according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (9), α = 1 × 10−3, vfrag = 2 m s−1, ε = 1 × 10−3, and ζ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The solid black line shows the location of a 3 Myr lifetime (a typical value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The results are discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Random distributions for the canonical population Variable Distribution log10(M⋆/M⊙) N(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='45, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='442) MG MD/fD/G β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='9 log10(Pin/d) N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='67617866, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='30567332) r1/au 70(MD/100 M⊕)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='25 × 10N(0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='12) log10(α) U(−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='5, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='0) log10( fD/G) N(−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='85, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='222) log10(MD/M⋆) N(−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='49, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='352) vfrag/cm s−1 200 ε 10−3 ζ 1 LX/1030 erg s−1 10N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='31,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='542) × (M⋆/1 M⊙)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='52 log10(F/G0) N(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='932) from early discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The only parameter that has some freedom is α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Here, we selected to draw log10 (α) with a uniform probabil- ity of between −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='5 and −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This was decided as a compromise between disc lifetime and stellar accretion rate, as we discuss be- low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For the other parameters, their distributions were selected as described in the discussion of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' for convenience, these are summarised in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The resulting distribution of disc lifetimes is shown with the blue curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' It becomes immediately apparent that the synthetic lifetimes are too short overall in comparison with observed discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The median lifetime of the synthetic discs is Article number, page 8 of 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' : Towards a population synthesis of discs and planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' �� � ��� ��� ��������� ��� ��� ��� ��� ��� ��� ���������������������������� ������� ��������� ��������� ���������� ����������������� �������������������� ����������� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Cumulative distribution of disc lifetimes for a population with canonical parameter distribution (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Two exponential decays fol- lowing Fedele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2010) with a characteristic time of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3 Myr (accre- tion) and 3 Myr (infrared excess) and the results of Ribas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2014) are shown as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='42 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Disc lifetime depends on the assumed distribution of α, which we chose such that it results in the largest stellar accretion rates at 2 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' A histogram of stellar accretion rate versus observed dust mass for the observed discs in the Lupus and Chamaeleon I star-forming regions is shown in the top-left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Only the synthetic discs that live beyond 2 Myr contribute to this histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The few remaining discs have low masses, as the discs are close to being dissipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Using larger values of α, for instance between roughly 10−3 and 10−2 as was proposed by Mulders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2017), would have resulted in even shorter disc lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This means that there would have been no discs that would live long enough to produce stellar accretion at 2 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Conversely, selecting a distribution of α with even lower values would allow disc lifetimes to be matched by observations, but this would result in even lower stellar accretion rates, which would be in tension with the results of Dullemond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2018), who concluded that α ≥ 1 × 10−4 from the sizes of disc substruc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The discrepancy between our modelled lifetimes and obser- vations arises from the strong mass-loss rates predicted for in- ternal and external photoevaporation, as we discuss in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' As disc lifetime depends on stellar mass (especially for exter- nal photoevaporation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3), this analysis is affected by the assumed stellar mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Had we selected larger stel- lar masses, the lifetimes would better match observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' How- ever, selecting stellar masses of around 1 M⊙, which would lead to a fairly good match including both photoevaporation prescrip- tions, is not representative of the star-forming regions that we are comparing to (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Effect of photoevaporation on disc populations The mismatch in disc lifetimes and disc masses that we obtained is in contrast with other studies, such as that of Mulders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2017), who did not include photoevaporation, Kunitomo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2020), who included only internal photoevaporation, and Weder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' ), who used an EUV-only internal photoevapora- tion prescription with much lower mass-loss rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Further, as already discussed, photoevaporation leads to considerable short- ening of disc lifetimes (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Therefore, here we investi- gate how the photoevaporation prescriptions affect the evolution of disc populations and determine whether they are responsible for the mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To this end, we created two additional popula- tions, each time neglecting one of the photoevaporation process by taken the corresponding controlling parameter to the mini- mum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For the population with internal photoevaporation only, the ambient flux has been set to F = 1G0, which corre- sponds to ˙M = 1 × 10−15 M⊙ yr−1 (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The results are shown as ‘Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' only’ in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For the population with external photoevaporation only, we set LX = 1028 erg s−1, the results of which are shown as ‘Ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' only’ in the same figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In each population, the disc lifetimes strongly increase com- pared to the canonical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, the distributions are differ- ent: internal photoevaporation leads to a relatively narrow distri- bution around 2 to 3 Myr, while with external photoevaporation disc lifetimes are more spread out, including very short-lived discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This leads to more discs being shown in the accretion ver- sus mass diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The population with only external photoevaporation has low disc masses combined with a narrow range of stellar accretion between 10−10 and 10−9 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This is because the mass loss occurs in the outer disc, which reduces its size, limiting the area from which the dust emits (as dust is also lost where there is no longer gas present).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' At the same time, the mass-accretion rate is weakly affected by the mass loss in the outer disc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' again because external photoevaporation affects only the outer disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Conversely, using internal photoevaporation leads to a larger spread in stellar accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' As internal photoevaporation re- moves material relatively close to the star, it competes with stel- lar accretion to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This, coupled with the spread of X- ray luminosities, leads to a spread in accretion rate (Owen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Thus, internal photoevaporation is needed to reproduce the observed spread in stellar accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We further note that neither population is able to reproduce the discs with an accre- tion rate larger than 10−8 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In addition, internal photoevaporation leaves discs with an inner cavity that have low accretion rates but where the outer disc is out of reach of internal photoevaporation and therefore takes a long time to dissipate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' these are what Owen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2011) refer to as relic discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' It is possible to avoid this situation to a large extent by having more compact discs initially, which leave nearly all of their mass within reach of internal photoevaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We conclude that previous studies managed to reproduce disc lifetimes because they used only one photoevaporation mechanism as the main loss mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, when both are accounted for, the combined mass-loss rate is so large that discs are very short lived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We discuss the implications of this and possible remedies in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Towards a best match Despite all the differences between models and observations highlighted so far, we now try to find a set of initial conditions that is able to better match disc evolution characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To this Article number, page 9 of 18 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' disc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='����������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='������������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='��������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='���������������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Histogram for stellar accretion rate vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' disc mass at 2 Myr and the same synthetic disc populations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The observed data from the Lupus (in red) and Chamaeleon I (orange) star forming regions from Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2019) are shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Two disc dispersal timescales τ = ˙MG/MG of 1 Myr and 10 Myr (assuming that the gas disc mass MG is 100 times the observed dust mass) and the best fit to the data from Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2016b) are shown as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' end, we investigate how the initial conditions can be modified from our canonical values provided in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The initial disc-to-star mass ratios were taken from Ty- choniec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2018) and Tobin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2020) assuming they were measured on stars of Mref ⋆ = 1 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, this assumption is inconsistent with the stellar mass distribution we selected, which has a median value of M⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='35 M⊙ (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Were we to se- lect a lower reference stellar mass, such as Mref ⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='33 M⊙, this would increase the disc masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' At the same time, selecting a reference stellar mass that is similar to the median value from our initial conditions results in an agreement between the disc masses in observations and our initial conditions, as shown with the dashed black and red lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In addition, to account for observational biases, we also want to check the observed dust masses after a short evolution time of 1 × 105 yr, which we provide with the solid lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The results show that the discs in the nominal populations have low masses com- pared to the non-multiple discs measured by Tobin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2020), Article number, page 10 of 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' : Towards a population synthesis of discs and planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' �� � �� � �� � �� � �� � ���� �� � ��� ��� ��� ��� ��� ��� ����������������� ���������������� ������������������� ��������������� �������� � � ����� ���� ������ � � ����� ����������� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Kernel density estimate for two populations from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4 and 5, both for their initial conditions (dashed lines) and retrieved disc masses at 100 kyr (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The initial mass distribution of the best-match population with Mref ⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='33 M⊙ (in red) is consistent with the fit by Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021b) to the data of Tychoniec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2018), which is shown as the dashed black line, while the retrieved masses at 100 kyr are compatible with the non-multiple discs of Tobin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' while a population with more massive initial discs has slightly larger masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Thus, the larger initial disc masses lead to a rea- sonable match with observations as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We point out that the disc-to-star mass ratio in the population with Mref ⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='33 M⊙ is about twice (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1 times) that of Emsen- huber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021b) and Burn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021), rather than a factor three as one might assume from the change of Mref ⋆ from 1 M⊙ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='33 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This is due to an inconsistency in the selection of the disc masses in the previous work: there, the gas masses were taken as a Monte Carlo variable that were converted from dust observations using a dust-to-gas ratio of 1 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, the ini- tial mass of solids in the model was recomputed from the gas mass using the same distribution as in this work (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='4), which has a median value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='42 % (fD/G,⊙ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='49 % with [Fe/H] = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In contrast, we use the solid disc mass as a Monte Carlo variable here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Another possibility is that early protoplanetary discs are not as extended as what is suggested by the findings of Tobin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' More compact discs are less susceptible to external pho- toevaporation, as there is less surface exposed to ambient radia- tion and they are more tightly bound to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' At the same time, a more compact disc means that more material is concentrated in the region where internal photoevaporation is most efficient, which allows the stellar accretion rate to remain larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Mag- netohydrodynamics models of protoplanetary disc formation by Hennebelle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2016), Lebreuilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021), and Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2021) could favour such a possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Finally, nearby star-forming regions have low masses, which results in a low ambient UV field strength F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' for instance, the value in Lupus is F ≈ 4G0 (Cleeves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Our nomi- nal distribution of ambient UV fields overestimates mass losses due to external photoevaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Therefore, we set F = 1G0, which results in negligible mass losses due to external photoe- vaporation ( ˙M = 1 × 10−15 M⊙ yr−1) and only internal photoe- vaporation remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Using internal photoevaporation confers the further advantage that a wider range of stellar accretion rates is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Below, we investigate whether more massive and compact discs are able to improve the match with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The ef- fects of the parameters mentioned above are discussed in Ap- pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' From these, we find that the following modifications to the initial conditions given in Table 6 are best able to repro- duce disc lifetimes and the accretion rate–mass relationship: – A decrease in the reference stellar mass to Mref ⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='33 M⊙, which corresponds to an increase in the disc mass by a factor of three compared to the nominal population;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' – r1 = 2/3 × 70 au (MD/100 M⊕)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='25, a factor 2/3 compared to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' and – only using internal photoevaporation (F = 1G0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The population using these distributions is shown as ‘Best match’ in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' While the overall stellar mass distribution we chose is repre- sentative of the observed stars, our visual optimisation approach to reproducing the observed disc masses and accretion rates is independent of the exact stellar mass dependency of the observ- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We will improve on this with a Bayesian framework in Paper III to optimise the initial parameters when reproducing a set of observations in four-dimensional space made up of stellar mass, disc mass, disc radius, and accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Compared to the population with only internal photoevap- oration (the other population that is closest in terms of initial conditions), we can see several differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' First, the larger disc masses result in an increased median lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' About 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2 % of the discs have a lifetime of greater than 10 Myr, representing a 100-fold increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Then, the combination of larger initial disc mass and smaller extent results in a certain number of cases with a stellar accretion rate of higher than 10−8 M⊙ yr−1, which was not previously seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The smaller extent of the disc also causes less discs to be relics (towards the bottom of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Here, the highest concentration of discs is found near the best-fit value of Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2016b, the pink dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 5) with a similar number on either side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We are still failing to reproduce the discs with the large accretion rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, part of these discs with large accretion rates could be due to binaries (Zagaria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2022), which we do not model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To corroborate this, the largest stellar accretion rates are biased to larger-mass stars (the mean stellar mass for systems with a stellar accretion rate higher than 3 × 10−9 M⊙ yr−1 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='74 M⊙ versus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='43 M⊙ for the gen- eral population), which at the same time are more likely to be in binary systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Duchêne & Kraus 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The mismatch should therefore not be the source of significant concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We find that this combination of parameters is able to reproduce the disc mass–accretion rate relationship and provide a reasonable match to most observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Discussion It is difficult to reconcile the results of our synthetic disc popu- lations with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We find that it is particularly hard to Article number, page 11 of 18 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' disc ���� ���� ���� ���� ���� ���� ���������� ��� ��� ��� ��� ��� ��� ������������� ������������ ������������ ������������ ������������ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Evolution of the relative gas mass: in the disc (blue), accreted onto the star (green), and lost by internal (orange) or external (red) pho- toevaporation until disc dispersal at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='38 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This represents a typi- cal disc, with M⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='5 M⊙, MG/M⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='9, Rin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1 au, r1 = 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='8 au, α = 1 × 10−3, LX = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='02 × 1029 erg s−1, and F = 10 G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The parameters for the solid disc are irrelevant, as this shows only the gas component, except that we used fD/G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='0149 to compute the solid mass needed to set the characteristic radius r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' obtain discs with characteristic lifetimes of 2 to 3 Myr accord- ing to Mamajek (2009) or Fedele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2010), and even less lifetimes of 5 to 10 Myr following Pfalzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Here, we assumed that the initial mass is constrained by the dust-mass measurements of Tychoniec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2018, MD/M⋆ ∼ 10−3) , and dust-to-gas ratios similar to the solar initial abundance, namely fD/G = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='49 % (Lodders 2003), combined with the predictions of internal and external photoevaporation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' To illustrate this issue, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 7 we provide the evolution of the gas mass still present in the disc and removed by the processes modelled here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Here we choose typical values for the initial conditions, with a star of mass M⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='5 M⊙, a gas disc with a mass of MG/M⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1, a power-law index of β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='9, an inner ra- dius of rin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1 au, and a characteristic radius of r1 = 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='8 au, which was computed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (9) assuming a dust-to-gas ratio of fD/G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='0149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The ambient UV field strength F = 10 G0 was set at the lower boundary of the computed grid while the X-ray luminosity LX = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='02 × 1029 erg s−1 follows the best fit of the XEST survey (Güdel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2007, Table 2) for the given stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Figure 7 shows that photoevaporation (both internal and ex- ternal) is responsible for the loss of nearly all the gas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' only 6 % of the gas is accreted onto the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The final disc lifetime of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='38 Myr is short compared to the characteristic lifetimes from observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We therefore have a problem with the mass bud- get, which could be resolved by (i) larger initial disc masses, (ii) mass replenishment after disc formation, or (iii) lower mass-loss rates by photoevaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Stellar accretion already plays a minor role in the mass bud- get and cannot be reduced further, in order to remain consis- tent with observed stellar accretion rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, radial mass transport could be the result of magnetically driven disc winds (Suzuki & Inutsuka 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Suzuki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2010) rather than pure viscous dissipation as we assumed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This would add another mass-loss channel, which would further exacerbate the problem (though it could shield stellar radiation, as discussed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We experiment with larger initial disc masses in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, even with gas masses on the order of 10 % of the stel- lar mass, disc lifetimes are not sufficiently long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Therefore, in- creasing the masses even more would be required, but this in- crease would bring another series of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For one, such large discs are likely gravitationally unstable and produce spi- ral density waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Second, such large discs would lead to strong gas-driven migration, which hinders planet formation (Nayak- shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Also, discs around 10 Myr-old stars HD163296 and TW Hya have at least 10 % of the stellar mass (Powell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2019) and it is unclear what their initial mass would have been for them to remain so large at their age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We therefore do not find that massive initial discs would be able to solve the conundrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Discs do not form instantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Rather, they grow from gas falling from the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This allows for disc masses to in- crease during the early stages of disc evolution, which is at odds with the models described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Modelling the infall stage would allow us to have longer-lived discs without them being very massive early on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Accretion can persist for several million years (Throop & Bally 2008), providing replenishment even at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The complex morphology of the gas disc around RU Lup (a ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='5 Myr old star) could be an outcome of this process (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The long-lived discs could also be second- generation discs, formed following accretion from the molecu- lar cloud (Kuffmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020) or the disruption of a planet (Nayakshin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' While none of these individual pro- cesses are sufficient to explain the presence of massive discs, they should still be explored if they can explain certain charac- teristics of the overall disc population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Finally, the photoevaporation rates used here could be over- estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Lower mass-loss rates would increase the disc life- times and masses at later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' An argument in favour of this hypothesis is that young discs can be shielded from the radiation of both their host and nearby stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The launching of magneti- cally driven disc winds occurs inside the location where internal photoevaporation is effective and would thus prevent EUV and X-ray photoevaporation during the early stages of disc evolution (Pascucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Similarly, early discs are likely embed- ded, preventing the radiation of nearby stars from reaching the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Both effects would reduce the photoevaporation rates dur- ing the early stage of disc evolution compared to what we model here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Summary and conclusion In this work, we investigate whether protoplanetary disc obser- vations can be reconciled with theoretical predictions of pro- cesses such as viscous accretion and photoevaporation (both in- ternal and external).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We first compute two sets of simulations that we use to fit neural networks (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' With these neural net- works, we can perform parameter studies and compute the out- comes of synthetic disc populations with limited computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We first compare how internal and external photoevaporation affect disc lifetime as a function of stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We find that be- cause of a direct link between stellar mass and X-ray luminos- ity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Preibisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Güdel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2007), which means mass-loss rate due to internal photoevaporation, discs around more massive stars are not significantly longer-lived than those Article number, page 12 of 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' : Towards a population synthesis of discs and planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' around low-mass stars (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Conversely, external photoevap- oration leads to a strong positive correlation between disc life- time and stellar mass (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 3), because gas is more bound for higher-mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This positive correlation is at odds with obser- vations that find that disc lifetimes are either independent of or anticorrelated with stellar mass (Carpenter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Kennedy & Kenyon 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Bayo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Ribas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Pfalzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2022) and should be investigated in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Turning to protoplanetary disc populations, we find that ac- counting for both internal and external photoevaporation accord- ing to theoretical predictions leads to disc lifetimes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4) that are much too short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Discs whose initial mass is 10 % of the stellar mass are dispersed in roughly 1 Myr under the combined effects of internal and external photoevaporation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Despite the dissimilarities, a reasonably good match to the disc properties of the Lupus and Chaemeleon I low-mass star- forming regions is obtained starting with more massive discs of smaller sizes, and with only internal photoevaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This is valid for clusters with low ambient field strengths, such as Lu- pus (4 G0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Cleeves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016), where losses due to external pho- toevaporation are low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The larger masses and smaller sizes are needed to improve the match in stellar accretion rates and ob- served masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The corresponding initial conditions and model parameters can be used to study planetary formation in similar environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' A more robust comparison with observations is performed in Paper III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, initial disc masses cannot be arbitrarily increased or discs would become gravitationally unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Instead, we sug- gest that future studies should include the modelling of the initial stages of disc formation, including the presence of an envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This envelope would allow discs to be replenished after their initial formation and provide shielding from UV radiation from nearby stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Also, magnetically driven disc winds would shield UV and X-ray radiation from the central star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This would provide a reduction of losses by both internal and external photoevapora- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Both effects allow for longer lifetimes and larger masses at later times without the need for extremely large masses at earlier times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We decided here to use observed dust masses as the main comparison point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, this is not the only possible avenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' For instance, disc radii could be less susceptible to the degen- eracy caused by regions that are optically thick (Pascucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' One likely difficulty would be in accounting for the large disc radii and sustained stellar accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Already with the comparatively small discs that we find to best match the disc mass–stellar accretion rate relationship, we are not able to re- produce the largest observed accretion rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Having larger discs would lead to a reduction of the stellar accretion rates for a given disc mass (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' which would lead to a mismatch with ob- servations of stellar accretion rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In this work, we assume that the gas discs evolve viscously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, simulations of disc evolution that account for non- ideal magnetohydrodynamical (MHD) effects find that the mag- netorotational instability (MRI), which is the likely mechanism generating the turbulence, is largely suppressed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Bai & Stone 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Lesur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Instead, it has also been pro- posed that the evolution is driven by magnetically driven disc winds (Suzuki & Inutsuka 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Magnetically driven disc wind prescriptions, such as those of Suzuki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2016) or Tabone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2022), include several model possibilities, which can be narrowed down by performing a similar comparison to that pre- sented here (Weder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Once such a model is properly coupled with internal photoevaporation to account for shielding, a similar study to that presented here is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The authors thank Christian Rab, Ilaria Pascucci, Susanne Pfalzner, and Aashish Gupta for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We also thank the anony- mous reviewer, whose comments and suggestions greatly helped improve the manuscript’s quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This work was funded by the Deutsche Forschungsge- meinschaft (DFG, German Research Foundation) - 362051796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This research was supported by the Excellence Cluster ORIGINS which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excelence Strategy – EXC-2094-390783311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The plots shown in this work were generated using matplotlib (Hunter 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' References Adams, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Proszkow, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Fatuzzo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Myers, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2006, ApJ, 641, 504 Alcalá, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Manara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Natta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2017, A&A, 600, A20 Alcalá, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Natta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Manara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2014, A&A, 561, A2 Alexander, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Clarke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Pringle, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2004, MNRAS, 354, 71 Alibert, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Mordasini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Benz, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2004, A&A, 417, L25 Alibert, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Mordasini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Benz, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Winisdoerffer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2005, A&A, 434, 343 Andrews, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Pérez, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2018a, ApJ, 869, L41 Andrews, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Terrell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Tripathi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2018b, ApJ, 865, 157 Andrews, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Wilner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Hughes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Qi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Dullemond, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2009, ApJ, 700, 1502 Andrews, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Wilner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Hughes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Qi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Dullemond, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2010, ApJ, 723, 1241 Ansdell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Williams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Manara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2017, AJ, 153, 240 Ansdell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Williams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', van der Marel, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016, ApJ, 828, 46 Bai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Stone, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2013, ApJ, 769, 76 Baraffe, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Chabrier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Allard, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Hauschildt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1998, A&A, 337, 403 Baraffe, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Homeier, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Allard, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Chabrier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2015, A&A, 577, A42 Bayo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Barrado, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Huélamo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2012, A&A, 547, A80 Beckwith, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Sargent, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1996, Nature, 383, 139 Birnstiel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Dullemond, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Brauer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2010, A&A, 513, A79 Birnstiel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Dullemond, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Zhu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2018, ApJ, 869, L45 Birnstiel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Klahr, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Ercolano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2012, A&A, 539, A148 Burn, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Emsenhuber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Melon Fuksman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', A&A Burn, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Emsenhuber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Weder, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2022, A&A, 666, A73 Burn, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Schlecker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Mordasini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021, A&A, 656, A72 Byrd, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Lu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Nocedal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Zhu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1995, SIAM Journal on Scientific Computing, 16, 1190 Cambioni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Asphaug, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Emsenhuber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2019a, ApJ, 875, 40 Cambioni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Delbo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Ryan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Furfaro, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Asphaug, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2019b, Icarus, 325, 16 Cambioni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Jacobson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Emsenhuber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021, PSJ, 2, 93 Carpenter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Mamajek, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Hillenbrand, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Meyer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2006, ApJ, 651, L49 Chabrier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2003, PASP, 115, 763 Clarke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Gendrin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Sotomayor, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2001, MNRAS, 328, 485 Cleeves, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Öberg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Wilner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016, ApJ, 832, 110 Da Rio, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Robberto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Hillenbrand, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Henning, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Stassun, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2012, ApJ, 748, 14 Dittrich, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Klahr, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Johansen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2013, ApJ, 763, 117 Dr˛a˙zkowska, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Alibert, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2017, A&A, 608, A92 Duchêne, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Kraus, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2013, ARA&A, 51, 269 Dullemond, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Birnstiel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2018, ApJ, 869, L46 Emsenhuber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Cambioni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Asphaug, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020, ApJ, 891, 6 Emsenhuber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Mordasini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Burn, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021a, A&A, 656, A69 Emsenhuber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Mordasini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Burn, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021b, A&A, 656, A70 Ercolano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Clarke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Drake, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2009, ApJ, 699, 1639 Ercolano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Clarke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Hall, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2011, MNRAS, 410, 671 Ercolano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Drake, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Raymond, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Clarke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2008, ApJ, 688, 398 Ercolano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Koepferl, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Owen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Robitaille, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2015, MNRAS, 452, 3689 Ercolano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Picogna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Monsch, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Drake, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Preibisch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021, MN- RAS, 508, 1675 Facchini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Clarke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Bisbas, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016, MNRAS, 457, 3593 Favata, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Micela, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2003, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', 108, 577 Fedele, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', van den Ancker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Henning, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Jayawardhana, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Oliveira, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2010, A&A, 510, A72 Feigelson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Montmerle, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1999, ARA&A, 37, 363 Flaischlen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Preibisch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Manara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Ercolano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021, A&A, 648, A121 Fortier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Alibert, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Carron, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Benz, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Dittkrist, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2013, A&A, 549, A44 Franz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Picogna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Ercolano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Birnstiel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020, A&A, 635, A53 Gárate, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Birnstiel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Dr˛a˙zkowska, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Stammler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020, A&A, 635, A149 Gáspár, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Rieke, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Ballering, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016, ApJ, 826, 171 Getman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Flaccomio, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Broos, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2005, ApJS, 160, 319 Gómez de Castro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2009, Ap&SS, 320, 97 Article number, page 13 of 18 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' disc Gorti, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Dullemond, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Hollenbach, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2009, ApJ, 705, 1237 Güdel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Briggs, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Arzner, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2007, A&A, 468, 353 Gundlach, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Schmidt, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Kreuzig, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2018, MNRAS, 479, 1273 Habing, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1968, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Netherlands, 19, 421 Haisch, Karl E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Lada, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Lada, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2001, ApJ, 553, L153 Hartmann, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Herczeg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Calvet, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016, ARA&A, 54, 135 Haworth, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Clarke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Rahman, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Winter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Facchini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2018, MNRAS, 481, 452 Hendler, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Pascucci, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Pinilla, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020, ApJ, 895, 126 Hennebelle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Commerçon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Chabrier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Marchand, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016, ApJ, 830, L8 Hollenbach, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Johnstone, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Lizano, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Shu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1994, ApJ, 428, 654 Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Andrews, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Öberg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020, ApJ, 898, 140 Hueso, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Guillot, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2005, A&A, 442, 703 Hunter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2007, Computing in Science and Engineering, 9, 90 Johnstone, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Bartel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Güdel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021, A&A, 649, A96 Kennedy, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Kenyon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2009, ApJ, 695, 1210 Kimura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Kunitomo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Takahashi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016, MNRAS, 461, 2257 Kingma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Ba, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2014, arXiv e-prints, arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='6980 Koepferl, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Ercolano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Dale, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2013, MNRAS, 428, 3327 Komaki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Nakatani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Yoshida, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021, ApJ, 910, 51 Kraus, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Ireland, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Hillenbrand, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Martinache, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2012, ApJ, 745, 19 Kuffmeier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Goicovic, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Dullemond, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020, A&A, 633, A3 Kunitomo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Ida, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Takeuchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021, ApJ, 909, 109 Kunitomo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Suzuki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Inutsuka, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020, MNRAS, 492, 3849 Lada, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Lada, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2003, ARA&A, 41, 57 Lebreuilly, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Hennebelle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Colman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021, ApJ, 917, L10 Lee, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Marchand, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Hennebelle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021, ApJ, 922, 36 Lenz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Klahr, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Birnstiel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2019, ApJ, 874, 36 Lesur, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Kunz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Fromang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2014, A&A, 566, A56 Lodders, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2003, ApJ, 591, 1220 Luhman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2000, ApJ, 544, 1044 Lüst, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1952, Zeitschrift Naturforschung Teil A, 7, 87 Lynden-Bell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Pringle, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1974, MNRAS, 168, 603 Mamajek, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2009, in American Institute of Physics Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1158, American Institute of Physics Conference Series, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Usuda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Tamura, & M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Ishii, 3–10 Manara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Fedele, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Herczeg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Teixeira, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016a, A&A, 585, A136 Manara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Mordasini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Testi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2019, A&A, 631, L2 Manara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Robberto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Da Rio, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2012, ApJ, 755, 154 Manara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Rosotti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Testi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016b, A&A, 591, L3 Manara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Testi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Herczeg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2017, A&A, 604, A127 Matsuyama, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Johnstone, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Hartmann, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2003, ApJ, 582, 893 McKay, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Beckman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Conover, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1979, Technometrics, 21, 239 Michel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', van der Marel, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Matthews, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021, ApJ, 921, 72 Mordasini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Alibert, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Benz, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2009, A&A, 501, 1139 Mulders, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Pascucci, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Manara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2017, ApJ, 847, 31 Murray, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Chaboyer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Arras, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Hansen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Noyes, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2001, ApJ, 555, 801 Musiolik, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Wurm, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2019, ApJ, 873, 58 Nakagawa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Sekiya, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Hayashi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1986, Icarus, 67, 375 Nakamoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Nakagawa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1994, ApJ, 421, 640 Nayakshin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Elbakyan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Rosotti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2022, MNRAS, 512, 6038 Nayakshin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Tsukagoshi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Hall, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020, MNRAS, 495, 285 Owen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Clarke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Ercolano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2012, MNRAS, 422, 1880 Owen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Ercolano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Clarke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2011, MNRAS, 412, 13 Pascucci, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Cabrit, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Edwards, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='10068 Pascucci, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Testi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Herczeg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016, ApJ, 831, 125 Pedregosa, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Varoquaux, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Gramfort, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2011, Journal of Machine Learning Research, 12, 2825 Pfalzner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Dehghani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Michel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2022, ApJ, 939, L10 Picogna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Ercolano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Espaillat, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021, MNRAS, 508, 3611 Picogna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Ercolano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Owen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Weber, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2019, MNRAS, 487, 691 Powell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Murray-Clay, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Pérez, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Schlichting, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Rosenthal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2019, ApJ, 878, 116 Preibisch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Favata, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2005, ApJS, 160, 401 Preibisch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Zinnecker, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Herbig, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1996, A&A, 310, 456 Qiao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Haworth, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Sellek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Ali, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2022, MNRAS, 512, 3788 Ribas, Á.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Bouy, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Merín, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2015, A&A, 576, A52 Ribas, Á.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Merín, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Bouy, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Maud, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2014, A&A, 561, A54 Santos, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Israelian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Mayor, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2005, A&A, 437, 1127 Sellek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Booth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Clarke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020, MNRAS, 498, 2845 Shakura, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Sunyaev, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 1973, A&A, 500, 33 Steinpilz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Teiser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Wurm, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2019, ApJ, 874, 60 Suzuki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Inutsuka, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2009, ApJ, 691, L49 Suzuki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Muto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Inutsuka, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2010, ApJ, 718, 1289 Suzuki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Ogihara, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Morbidelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Crida, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Guillot, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2016, A&A, 596, A74 Tabone, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Rosotti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Cridland, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Armitage, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Lodato, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2022, MNRAS, 512, 2290 Throop, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Bally, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2008, AJ, 135, 2380 Tobin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Sheehan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Megeath, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020, ApJ, 890, 130 Toci, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Rosotti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Lodato, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Testi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Trapman, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2021, MNRAS, 507, 818 Tripathi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Andrews, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Birnstiel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Wilner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2017, ApJ, 845, 44 Tychoniec, Ł.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Manara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Rosotti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020, A&A, 640, A19 Tychoniec, Ł.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Tobin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Karska, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2018, ApJS, 238, 19 van Terwisga, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', van Dishoeck, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Mann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020, A&A, 640, A27 Venuti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Bouvier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Cody, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2017, A&A, 599, A23 Veras, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Armitage, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2004, MNRAS, 347, 613 Virtanen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Gommers, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Oliphant, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020, Nature Methods, 17, 261 Voelkel, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Klahr, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Mordasini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Emsenhuber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Lenz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2020, A&A, 642, A75 Weder, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Mordasini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Emsenhuber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', A&A Williams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Cieza, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Hales, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2019, ApJ, 875, L9 Winter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' & Haworth, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2022, European Physical Journal Plus, 137, 1132 Zagaria, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Clarke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Rosotti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Manara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2022, MNRAS, 512, 3538 Zormpas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Birnstiel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', Rosotti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=', & Andrews, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 2022, A&A, 661, A66 Article number, page 14 of 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' : Towards a population synthesis of discs and planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Appendix A: Parameter study The surrogate models allow us to study the effects of the differ- ent parameters, as shown for stellar mass and photoevaporation in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Here, we expand the study to other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The parameters that are not varied are selected as in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='3, ex- cept for the photoevaporation-related parameters, which are set as LX = 1 × 1029 erg s−1 and F = 10 G0 to provide lifetimes that are globally in line with the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We study in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1 the effects of the parameters of the gas disc model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The top row shows the outcomes as functions of the power-law index of the initial profile β and the inner edge rin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' These two parameters have very little effect on the final life- times, as all values lie within about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1 dex, corresponding to a maximum relative difference of 26 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The same applies to disc masses and stellar accretion rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Thus, the choice of these pa- rameters has negligible effects on the final properties and we do not discuss these parameters further in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The bottom row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1 shows the effect of the viscosity parameter α and the disc’s characteristic radius r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The charac- teristic radius has limited effect on the disc lifetimes while α, which affects the whole viscous evolution, has an important ef- fect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, both parameters affect the stellar accretion rate, making it possible to combine these two parameters to set the behaviour of stellar accretion versus lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2 shows the same analysis but for the parameters of the solid disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' As such, only the observed dust masses are shown for each parameter combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The left panel features the dust-to-gas ratio fD/G and the disc’s gas mass MG/M⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The results show that the dust-to-gas ratio is only important to con- trol the observed disc masses for discs that are close to disper- sal (towards the left of the panel) while it has a lower effect on more massive discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The centre panel shows two of the twopop model parameters, planetesimal formation efficiency and dust- to-gas ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The panel shows that the planetesimal formation efficiency parameter ε is of lower importance than the initial dust mass (which is controlled by the dust-to-gas ratio fD/G) for the observed disc masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The right panels show two other twopop model parameters, the drift efficiency ζ and the fragmen- tation velocity vfrag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Here we see that fragmentation velocities vfrag ≳ 300 cm s−1 lead to lower disc masses because drift be- comes efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' This effect can be counterbalanced by reducing the drift efficiency ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, for a value of vfrag = 200 cm s−1, the drift efficiency has a small effect on the observed disc masses, which indicate that drift is already inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' One might expect observed dust masses to be affected equally by the two parameters shown in the left panel (as the initial solid mass is the product of the dust-to-gas ratio and the gas mass MG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, our results indicate that the initial gas mass has a greater effect than the dust-to-gas ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Article number, page 15 of 18 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' disc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='��� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='���� ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Gas disc lifetime (left), observed dust masses at 2 Myr (centre), and stellar accretion rates at 2 Myr (right) as functions of the power-law index of the initial profile β and the inner edge rin (top) or viscosity parameter α and characteristic radius r1 (bottom) In the bottom row, the white region is where disc lifetimes are less than 2 Myr and the values cannot be constrained by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' �� � �� � �� � �������������� �� � ���������������������� ��� ��� ��� ��� ���������� �� �� �� � ���������������������� �� � �� � �� � ����������������� ��� ��� ��� ��� ���������� �� �� ��� ��� ����������������������� �� � �� � ��� ����������� ��� ��� ��� ���������� �� �� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Observed mass at 2 Myr as function of the parameters of the twopop model: gas-disc mass and dust-to-gas ratio (left), planetesimal formation efficiency and fragmentation velocity (centre), and drift efficiency and fragmentation velocity (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Article number, page 16 of 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' : Towards a population synthesis of discs and planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Appendix B: Best parameters for population To show how the different model parameters affect the mass– accretion rate relationship and how we came to select the best model parameters, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1 we provide several 2D histograms similar to those shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 5, but varying one parameter at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' These were generated with the parameters of the ‘best match’ population, except for the parameter being varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The first row of the figure shows the effect of varying the disc mass (both gas and solids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We see that an increase in the initial mass is recovered in the observed mass after 2 Myr of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In addition, the stellar accretion rate is correlated with the disc mass similarly to the best fit of Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The initial disc mass can therefore be used to set observed masses and an in- crease from the canonical value is required to obtain disc masses consistent with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, this parameter cannot be used to control the behaviour of the stellar accretion rates for given disc masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The second row of the same figure shows the effect of reduc- ing the disc’s characteristic radius by a certain factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Smaller discs will result in increased stellar accretion rates for a given disc mass while also reducing the occurrence of non-accreting discs (the ones at the bottom of the plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In addition, smaller discs will also tend to have greater disc lifetimes because they are less affected by photoevaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Discs that follow the re- lationship of Tobin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2020) are found to have overly low accretion rates for their masses, while when reduced by a factor two, stellar accretion rates are now too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The best-fit param- eter is therefore between these two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Then, the third row shows the effect of the α viscosity pa- rameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The synthetic populations shown here show less spread due to the use of a fixed value of α, whereas the ones above have the individual values selected from a distribution that goes from 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='5 to 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The results are in line with the discus- sion in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content='1: low values of α result in low accretion rates with a large amount of remaining discs, while large values re- sult in a small number of discs with excessive stellar accretion rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Therefore, while large values of α could be used to have large accretion rates, they would also require a corresponding in- crease in the initial disc masses to maintain the expected lifetime distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' We find that values around α = 1 × 10−3 provide a reasonable match to disc lifetimes and stellar accretion rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' In addition, we also note that there is an increase in the observed dust masses along with α, which is related to the extent of the solid disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' With large α, the grains do not grow as much as with lower α, and as a consequence do not drift as fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Hence, the dust emits from a larger area, which is then reflected in the emit- ted flux and the observed mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' However, this effect only lasts until the dispersal of the gas disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Article number, page 17 of 18 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' disc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} 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accretion rate versus disc mass at 2 Myr, showing the effects of the different parameters, as given above each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The top row shows the effect of increasing the initial disc mass MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The middle row shows the effect of reducing the disc characteristic radius r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The bottom row shows the effect of the disc viscosity parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' The observed data from the Lupus (in red) and Chamaeleon I (orange) star-forming regions from Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' (2019) are shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} +page_content=' Article number, page 18 of 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf'} diff --git a/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf b/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c67cffbe43a762f677ea300aebc827f1c3bc2fb9 --- /dev/null +++ b/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:914f886ee0a2c70d77202ab44913996ef5770141a6ae4bafa7afbd53808cf018 +size 12367653 diff --git a/C9AzT4oBgHgl3EQfwf5z/content/tmp_files/2301.01723v1.pdf.txt b/C9AzT4oBgHgl3EQfwf5z/content/tmp_files/2301.01723v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b25f2c5134def3a1b74bd386e9553aed2a7d2a94 --- /dev/null +++ b/C9AzT4oBgHgl3EQfwf5z/content/tmp_files/2301.01723v1.pdf.txt @@ -0,0 +1,846 @@ +Instability of the cosmological DBI-Galileon in the +non-relativistic limit +C. Leloup1,2, L. Heitz3 and J. Neveu3,4 +1 Universit´e Paris-Cit´e, CNRS, Astroparticule et Cosmologie, 75013 Paris, France +2 Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, +WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan +3 Universit´e Paris-Saclay, CNRS, IJCLab, 91405, Orsay, France +4 Sorbonne Universit´e, CNRS, Universit´e de Paris, LPNHE, 75252 Paris Cedex 05, +France +Abstract. +The DBI-Galileon model is a tensor-scalar theory of gravity which finds its +foundation as the most general theory of the dynamics of a 4D brane embedded in +a 5D bulk. +It is of particular interest as it provides a few free parameters with a +physical meaning, such as the cosmological constant which is there related to the +brane tension. Most studies of this model have been performed assuming a maximally +symmetric geometry for the 5D bulk, in which it has been shown that the theory +reduces to various types of Galileon. In contrast, the general case for the geometry of +the bulk provides a different covariantization of the Galileon model than the covariant +Galileon: the DBI-Galileon. From the tight constraints on the gravitational waves +speed, we are naturally led to consider the non-relativistic limit of the model where +the kinetic energy of the brane is small compared to its tension, that we study in +the context of late-time cosmology. +The DBI-Galileon in the non-relativistic limit +is simply an expansion around General Relativity (GR) which can be expressed as +a shift-symmetric Horndeski theory. We developed the description of this theory at +the background and perturbation level. However, by studying the scalar and tensor +perturbations around a flat FLRW background, we found that they contain a ghost +degree of freedom leading to fatal instability of the vacuum for every combination of the +free parameters. As a lesson, we emphasized which of the Horndeski terms competes +to avoid this instability in more general cases. +1. Introduction +Dark energy has been modelled by a large variety of theories since decades. Among +these, many rely on the introduction of additional scalar fields whose dynamics, at the +origin of the late-time acceleration of the expansion of the Universe, is determined by +arbitrary parametric functions, potentials and/or coupling (see e.g. [1]). These are the +so-called scalar-tensor theories of modified gravity. In particular, the class of Horndeski +theories is of great interest as it contains all models of modified gravity with a single +arXiv:2301.01723v1 [hep-th] 4 Jan 2023 + +Instability of the cosmological DBI-Galileon in the non-relativistic limit +2 +additional scalar field leading to second-order equations of motion [2, 3]. Extensions of +Horndeski theories to scalar-tensor theories of one scalar field with equations of motion of +higher orders have also been explored [4, 5]. Particular Horndeski theories are described +by the specification of four arbitrary functions of the scalar field and its kinetic energy, +leading to a huge variety of models and phenomenological behaviours. +Among these wide classes of models, some can be built from first physical principles +or arguments of symmetry. +For instance, the Galileon model [6] and its covariant +extension [7] was built by imposing a galilean symmetry for the scalar field, leaving +only five free numerical parameters. We can also cite, among many others, the pure +kinetic gravity theory [8], massive gravity in the non-relativistic limit [9, 10] and the +DBI-Galileon [11] which is the main object of this paper. +The DBI-Galileon is a model that falls into the class of Brane-world scenarios of +extra-dimension theories, where the matter fields are confined on a 4D brane while +gravity can propagate into the additional spatial dimensions. Of most interest for the +DBI-Galileon is the case of a single extra-dimension as it has been shown that theories +with more co-dimensions exhibit ghosts either in the flat or self-accelerating de Sitter +solution [12]. The action include a volume term for the 4D brane in the 5D bulk which +leads to the well-known Dirac-Born-Infeld (DBI) action. +This action, and DBI-like +extensions, can lead to a self-accelerating solution and has been thoroughly studied as +a candidate model in the early Universe cosmic inflation paradigm [13, 14]. In addition, +the DBI-Galileon model exhibits the Galileon Lagrangians in the non-relativistic limit +[11] but giving a physical meaning to their free parameters: the Planck mass in the +brane, the Planck mass in the bulk, etc. In particular, the brane tension here plays the +role of the cosmological constant which brings a possible interpretation of its nature. +The original probe brane construction has been revisited in [15] where the matter +metric is disformally related to a standard gravitational metric, or in [16] in the +framework of spontaneous symmetry breaking for the 5D space-time symmetries broken +by the presence of the brane, bridging the gap with Brane-world scenarios developed +in the context of quantum field theory and an interpretation of the scalar field as a +Nambu-Goldstone boson [17, 18]. The DBI-Galileon model has been studied extensively +in special cases of the maximally symmetric bulk geometry [19, 20]. However, to our +knowledge, no study of the DBI-Galileon in the late-time cosmology setting as a potential +candidate for Dark Energy has been performed so far. +In this paper we develop the DBI-Galileon theory in the non relativistic limit +(Section 2) and study its dynamics in the Friedmann-Lemaˆıtre-Robertson-Walker flat +metric (Section 3). The perturbation stability is explored in Section 4 and then discussed +in Section 5. +2. DBI-Galileon in the non-relativistic limit +DBI-Galileon +We are interested in the description of a four dimensional brane universe embedded in + +Instability of the cosmological DBI-Galileon in the non-relativistic limit +3 +a five dimensional bulk from the cosmological perspective. In this context, it has been +shown in [11] that the most general action on the brane is given by the 4D Lovelock +terms [21] inside the brane and the boundary terms associated to the 5D Lovelock terms +in the bulk: +S = +� +dx4√−g +� +−Λ − M 3 +5K + M 2 +P +2 R − β M 3 +5 +m2 KGB + Lm (˜qµν, ψm) +� +, +(1) +where g is the induced metric on the brane, Λ the brane cosmological constant, K is the +extrinsic curvature of the brane, R is the Ricci scalar on the brane, KGB is the boundary +term on the brane of the Gauss-Bonnet scalar in the bulk and Lm is the Lagrangian +density of matter that lives confined in the brane. In the above action has been defined +the 4D Planck mass MP, its 5D counterpart M5 and their ratio m = M 3 +5/M 2 +P, and β is +an arbitrary constant parameter. +Because the action is defined on the 4-dimensional brane, the above quantities are +expressed in terms of the induced metric g, as opposed to the 5-dimensional bulk metric +G. Assuming we have a coordinate system (xµ, y) in the bulk, Greek letters being defined +to span from 0 to 3, such that the length element in this frame is ds2 = qµνdxµdxν+(dy)2. +The brane position in the bulk is defined by y = π (xµ) such that we can express the +induced metric from the bulk metric: +gµν = qµν + ∂µπ∂νπ +and +gµν = qµν − γ2∂µπ∂νπ +(2) +with the Lorentz factor γ = +� +1 + (∂π)2�−1/2. From this expression of the induced metric, +we can explicitly write the action (1) using the bulk metric q and the scalar field π. As +an illustration, see how the cosmological constant part of the action on the brane can +be expressed: +SΛ = −Λ +� +d4x√−g = −Λ +� +d4x√−q +� +1 + (∂π)2 +(3) +As was pointed out in the original paper by de Rham and Tolley [11], we recover a DBI +term in the action. This leads us to associate the cosmological constant Λ with a brane +tension f, following Λ = f 4 for unit convenience. +Non-relativistic limit +If the derivatives of the field vanish, ∂µπ = 0, then we recover standard GR. Because +we are interested in the cosmological setting, where predictions from the ΛCDM model +based on GR match precisely a large range of observations, we consider only small +corrections to GR. Therefore, we consider the DBI-Galileon in the so-called non- +relativistic limit where (∂π)2 ≪ 1. In particular, the DBI part of the action in the +non-relativistic limit becomes: +Sf = −f 4 +� +d4x√−q +� +1 + ∇µπ∇µπ +2 +− (∇µπ∇µπ)2 +8 ++ . . . +� +(4) + +Instability of the cosmological DBI-Galileon in the non-relativistic limit +4 +We see that, to have a canonically normalized field, we can proceed to the field +redefinition π → ϕ = f 2π. Up to degree 5 in ∂ϕ/f 2, we find the following Lagrangian +operators for the DBI-Galileon in the non-relativistic limit: +Lf = 1 + L2 +2f 4 − X2 +2 + . . . +(5) +LK = − L3 +2f 6 − 2X +f 6 [ψ] + . . . +(6) +LR = ¯R + 1 +f 4 +� +[Φ]2 − +� +Φ2� +− f 4X +2 +¯R − 2 ¯Rµν∇µϕ∇νϕ +� ++ L4 +4f 8 + . . . +(7) +LKGB = 2 +f 6 +�� +− ¯Rµν [Φ] + 2 ¯RµρΦρ +ν + ¯RµρνλΦρλ� +∇µϕ∇νϕ + 1 +3 [Φ]3 − [Φ] +� +Φ2� ++ 2 +3 +� +Φ3� +−1 +2 +¯R [ψ] +� ++ L5 +3f 10 + . . . +(8) +We defined on the above expressions the tensor Φµν = ∇µ∇νϕ, and the three scalars +[Φ] = Φµ +µ, [ψ] = ∂µϕ·Φµν·∂νϕ and X = − (∂ϕ)2 /2f 4. Furthermore, the L2...5 Lagrangian +operators are the covariant Galileon model Lagrangian operators as defined in [6, 7]. +Therefore, the theory described here is a generalization of the Galileon model. We can +note that the additional terms in LR and LKGB compared to the Galileon Lagrangian +operators vanish in the particular case of a flat geometry. We conclude that the DBI- +Galileon in the non-relativistic limit is a different coviariantization of the flat Galileon +than the covariant Galileon [7] or dRGT massive gravity [9, 10], with an additional +terms in Lf and another one in LK. That point noted, we will stay at leading order +in X in the following. Being a theory of a scalar field interacting with a metric, with +equations of motion of at most second order, it can be described as a Horndeski theory +[2, 3] with the following Horndeski functions: +G2 = A (ϕ) − f 4 (1 − X + . . .) +(9) +G3 = M 3 +5 +f 2 (X + . . .) +(10) +G4 = M 2 +P +2 +� +1 − X − X2/2 . . . +� +(11) +G5 = −2β M 3 +5 +m2f 2 +� +X + X2 + . . . +� +(12) +3. Cosmological background evolution +We expand the dynamics of the fields around a flat FLRW background on the 4D slice +of the bulk along xµ where the properties of the brane, gµν and ϕ, are defined. On this +slice, the length element is: +ds2 = qµνdxµdxν = −dt2 + a2 (t) δijdxidxj +(13) +It might appear more natural to expand around a flat FLRW background on the +brane with its induced metric gµν. However, the two metrics qµν and gµν are related + +Instability of the cosmological DBI-Galileon in the non-relativistic limit +5 +by a disformal transformation involving the scalar field which, at the background level, +depends only on the physical time: +¯gµν = ¯qµν + ( ˙¯ϕ (t)) +2 +f 4 +δµ0δν0 +(14) +where barred quantities are taken at the background level. Therefore, the geometry on +the brane is also of the FLRW type, but with a different definition for the physical time, +leading to a different scale factor and expansion history. Because in our case ˙¯ϕ ≪ f 2, +the expansion history on the 4D slice and on the brane will, then, be approximately the +same. In addition, equations (9)-(12) determine a self-consistent Horndeski theory of +gravity with the metric qµν and the scalar field ϕ without referring to the induced metric +gµν. Thus, in the following we apply the well-known techniques used in the context of +Horndeski theories. +In the non-relativistic limit, action (1) reads: +S = +� +dx4√−q (Lf + LK + LR + LKGB + Lm (qµν, ψm)) . +(15) +We define Ω0 +m and Ω0 +r the standard present energy density parameters for pressureless +matter and radiation respectively, and ¯H the normalized Hubble rate H/H0 with H0 +the present Hubble constant. Prime symbol denotes the derivative with respect to ln a. +We set: +˜x = ϕ′H0 +f 2 , +Ω0 +Λ = +Λ +3H2 +0M 2 +P += +f 4 +3H2 +0M 2 +P +, +η = +M 3 +5 +M 2 +PH0 +, +ξ = β +η , +κ = MPH0 +f 2 +. +(16) +Then, the two Friedmann equations derived from action (15) are: +¯H2 = −3 +2 +¯H4˜x2 − 15 +8 +¯H6˜x4 + η ¯H4˜x3 − 10 +3 ξ ¯H6˜x3 − 14 +3 ξ ¯H8˜x5 ++ Ω0 +m +a3 + Ω0 +r +a4 + Ω0 +Λ +� +1 + 1 +2 +¯H2˜x2 +� +(17) +¯H2 + 2 +3 +¯H ¯H′ = −2 +3ξ +� +2 ¯H6˜x3 + 5 ¯H5˜x3 ¯H′ + 3 ¯H6˜x2˜x′ + 2 ¯H8˜x5 + 5 ¯H8˜x4˜x′ + 7 ¯H7˜x5 ¯H′� +− 1 +2 +¯H4˜x2 − ¯H3˜x2 ¯H′ − 2 +3 +¯H4˜x˜x′ + 1 +3η ¯H3˜x2( ¯H˜x)′ − 3 +8 +¯H6˜x4 − 5 +4 +¯H5˜x4 ¯H′ − ¯H6˜x3˜x′ +− Ω0 +r +3a4 + Ω0 +Λ +� +1 − 1 +2 +¯H2˜x2 +� +(18) +We see that we recover the ΛCDM equations when setting ˜x to zero, but with a physical +interpretation of the cosmological constant as the brane tension. +The DBI model +proposed here is then an extension of the standard model of cosmology, and as such +follows a late accelerated expansion but with a physical interpretation of the origin of +Λ as a brane tension. Using the same methodology and similar notations as in [22], we +derive the field ϕ equation of motion from action (15). This leads to a system of two + +Instability of the cosmological DBI-Galileon in the non-relativistic limit +6 +coupled equations +˜x′ += −˜x + αλ − σγ +σβ − αω +¯H′ += ωγ − βλ +σβ − αω +(19) +with +β = 2 ¯H4 + 9 +2 +¯H6˜x2 − Ω0 +Λ ¯H2 − 2η ¯H4˜x + 4ξ ¯H6˜x + 40 +3 ξ ¯H8˜x3 +α = 6 ¯H3˜x − Ω0 +Λ ¯H˜x + 15 +2 +¯H5˜x3 − 3η ¯H3˜x2 + 10ξ ¯H5˜x2 + 70 +3 ξ ¯H7˜x4 +γ = 4 ¯H4˜x − 2Ω0 +Λ ¯H2˜x − η ¯H4˜x2 + 2ξ ¯H6˜x2 − 10 +3 ξ ¯H8˜x4 +ω = 4 +3 +¯H4˜x + ¯H6˜x3 − 1 +3η ¯H4x2 + 2ξ ¯H6˜x2 + 10 +3 ξ ¯H8˜x4 +σ = 2 +3 +¯H + 2 ¯H3˜x2 + 15 +12 +¯H5˜x4 − 1 +3η ¯H3˜x3 + 10 +3 ξ ¯H5˜x3 + 14 +3 ξ ¯H7˜x5 +λ = ¯H2 + Ω0 +r +3a4 − Ω0 +Λ + 1 +2Ω0 +Λ ¯H2˜x2 − 1 +3 +¯H4˜x2 − 5 +8 +¯H6˜x4 + 1 +3η ¯H4˜x3 − 2 +3ξ ¯H6˜x3 − 2ξ ¯H8˜x5 +Given values for the parameters Ω0 +m, η, ξ and κ, and initial conditions ˜x0 and H0, +this system can be integrated to compute background cosmology observables like the +distance moduli of type Ia supernovae. In Figure 1 we illustrate this with a Hubble +diagram prediction compared with recent type Ia supernova data [23]. As an initial +condition for ˜x, we chose to set ˜x0 = 6 × 10−8 today. +This is the maximum value +allowed by the constraint on gravitational wave speed (see Section 4) coming from the +quasi simultaneous observation of photons and gravitational waves after neutron star +merger event GW170817A [24]. Nevertheless, before discussing more the cosmological +scenarios proposed by the DBI-Galileon model, stability conditions much be computed +first to assess the viability of the models for any set of parameters at the perturbation +level. +4. Stability conditions +To be viable as a description of our Universe, the model has to fulfill stability +conditions. +These requirements apply to degrees of freedom propagating around +the fixed background, i.e. +to the cosmological perturbations, that are capable of +undermining the stability of the Universe. +In the determination of the stability +conditions for the DBI-Galileon model in the non-relativistic limit, we use the formalism +described in [25] for Horndeski theories. In our case, these stability conditions are defined + +Instability of the cosmological DBI-Galileon in the non-relativistic limit +7 +14 +16 +18 +20 +22 +24 +26 +Distance moduli [mag] +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +Redshift z +0.2 +0.0 +Residuals [mag] +Figure 1. +Hubble diagram prediction for the non-relativistic DBI-Galileon model for Ω0 +Λ = 0.7, +˜x0 = 6 × 10−8, η = ξ = 0 (blue) compared with binned Pantheon data (black points) [23]. We used +M = 23.81 for the offset magnitude of the diagram. Residuals to the fit are presented in the bottom +panel. +from the following quantities derived from the particular Horndeski functions (9) to (12): +ω1 = 1 + 1 +2 +¯H2˜x2 + 3 +8 +¯H4˜x4 + 2ξ ¯H4˜x3 + 2ξ ¯H6˜x5 +(20) +ω2 = 2 ¯H + 3 ¯H3˜x2 + 15 +4 +¯H5˜x4 − η ¯H3˜x3 + 10ξ ¯H5˜x3 + 14ξ ¯H7˜x5 +(21) +ω3 = 9 +� +− ¯H2 + 1 +2Ω0 +Λ ¯H2˜x2 − 3 ¯H4˜x2 + 2η ¯H4˜x3 − 10ξ ¯H6˜x3 − 45 +8 +¯H6˜x4 − 56 +3 ξ ¯H8˜x5 +� +(22) +ω4 = 1 − 1 +2 +¯H2˜x2 − 1 +8 +¯H4˜x4 + 2ξ ¯H3˜x2( ¯H˜x)′ + 2ξ ¯H5˜x4( ¯H˜x)′ +(23) +Tensorial stability conditions +In order to avoid ghosts and Laplacian instabilities, we impose the following constraints +on the sign of the kinetic term and on the sign of the gravitational waves speed squared: +Qt ≡ ω1 +4 = 1 +4 + 1 +8 +¯H2˜x2 + 3 +32 +¯H4˜x4 + 1 +2ξ ¯H4˜x3 + 1 +2ξ ¯H6˜x5 > 0 +(24) +c2 +t ≡ ω4 +ω1 += 1 − 1 +2 ¯H2˜x2 − 1 +8 ¯H4˜x4 + 2ξ ¯H3˜x2( ¯H˜x)′ + 2ξ ¯H5˜x4( ¯H˜x)′ +1 + 1 +2 ¯H2˜x2 + 3 +8 ¯H4˜x4 + 2ξ ¯H4˜x3 + 2ξ ¯H6˜x5 +≥ 0 +(25) +In particular, we see that the gravitational wave speed depends on ˜x, and tends to +1 when ˜x → 0 : +4Qt ≃ 1 + 2( ¯H˜x)2 +(26) +ct ≃ 1 − 1 +2( ¯H˜x)2 +(27) + +Instability of the cosmological DBI-Galileon in the non-relativistic limit +8 +Given the very tight constraint on the speed of gravitational waves, equal to the speed +of light up to a ∼ 10−15 difference [24, 26, 27], this justifies a posteriori the relevance +of the non-relativistic limit where ˜x ≪ 1. Moreover, we see that tensorial perturbations +are stable in this limit since Qt > 0. +Scalar stability conditions +Similar stability conditions apply to the scalar degrees of freedom, here including the +scalar perturbations of matter components: +Qs ≡ ω1 (4ω1ω3 + 9ω2 +2) +3ω2 +2 +> 0 +(28) +c2 +s ≡ 3 (2ω2 +1ω2H − ω2 +2ω4 + 4ω1ω2 ˙ω1 − 2ω2 +1 ˙ω2) − 6ω2 +1 +� (1 + wi) ρi +ω1 (4ω1ω3 + 9ω2 +2) +≥ 0 +(29) +where wi and ρi are respectively the equation of state parameter and the energy density +of the fluid i, and the sum runs over all the components of the Universe (here only +pressureless matter and radiation). At the lowest order in ˜x, we get: +Qs ≃ 3 +2(Ω0 +Λ − ¯H2)˜x2 +(30) +c2 +s ≃ 1 + 2 +� +η ¯H2˜x′ − ¯H ¯H′ − 2ξ ¯H4˜x′� +3 +� +Ω0 +Λ − ¯H2� +(31) +With ˜x ≪ 1, a fit of the DBI-Galileon model to data leads to cosmological +parameters close to the standard model ones: Ω0 +m ≈ 0.3 and Ω0 +Λ ≈ 0.7 [28]. Therefore, +from the first Friedmann equation, we get Ω0 +Λ < ¯H2 for all relevant models in agreement +with cosmological observations. As Qs ≤ 0, the DBI-Galileon model contains scalar +instabilities unless it reduces to GR. One way to avoid this would be to add a spatial +curvature to the metric, but with a strong energy density (at least ∼ 0.3) which is also +excluded by observations [28]. +5. Discussion +Physical interpretation +From the definition (28), we see that the dominant terms come G2 (giving the Ω0 +Λ term) +and G4 (giving the ¯H2 term). The competition between the two terms leads to the +ghost-like behaviour in a cosmological setting: Qs ≤ 0. In other words, it is the result +of the competition between the DBI and the Einstein-Hilbert terms. The DBI action +will have the effect of stretching the brane towards an extremal surface, whereas the +Einstein-Hilbert term on the brane will tend to make the brane contract on itself from +the effect of curvature. However, in the non-relativistic limit of the DBI-Galileon, the +Einstein-Hilbert term destabilizes the scalar field perturbations and the stretching effect +from the cosmological constant is not strong enough to counterbalance, leading to an +instantaneous decay of the vacuum state. + +Instability of the cosmological DBI-Galileon in the non-relativistic limit +9 +Because the DBI-Galileon action is the most general one can find of a 4D probe +brane in a 5D bulk, we expect this statement to be quite general for all such theories +studied in the current context. Indeed, this is true in a standard cosmological setting +which is realised with an FLRW slicing of the bulk space-time (equivalent to an FLRW +background on the brane). Therefore, the only way to evade this ghostly behaviour in +cosmology is to include the full relativistic dynamics of the theory (˜x ∼ 1). We have +seen that, in this case, we expect significant deviations of the speed of gravitational +waves ct from c. This is not a definitive impossibility though if the full DBI-Galileon +is viewed as an effective theory valid only at cosmological scales for which the speed +of gravitational waves has not been probed [29, 30]. +Indeed, the constraint on the +gravitational speed from the observation of GW170817 in coincidence with GRB170817A +[24] is only valid on small scales probed by LIGO and Virgo. A modification of the +dispersion relation of gravitational waves at small scales from operators present in the +UV complete theory could allow ct ̸= c on cosmological scales while being compatible +with current astrophysical observations. Waiting for the next generation of gravitational +wave interferometers, in particular LISA, which will be able to probe this relation at +larger scales [31], this possibility remains open. +Direct coupling to matter +In the context of cosmology, where standard model matter is present, there might be +direct coupling to the scalar field. In that case, the metric ˜q to which matter is sensitive +is different than the space-time metric q: +S = +� +dx4√−q (Lf + LK + LR + LKGB) + +� +dx4� +−˜qLm (˜qµν, ψm) . +(32) +It has been shown in [32] that the two metrics are related by a disformal transformation +of the following form: +qµν = A +� +ϕ, ˜X +� +˜qµν + B +� +ϕ, ˜X +� ∂µϕ∂νϕ +f 4 +(33) +where A and B are arbitrary functions of the scalar field and ˜X = −˜qµν∂µϕ∂νϕ/2f 4. +For simplicity and following the treatment of the covariant Galileon [22], we assume +that A and B are constant parameters. This can be further justified by the fact that, +a dependency on X would introduce, in general, higher order terms which would go +beyond the framework of Horndeski theories [33], and a dependency on ϕ would, in +general, break the shift symmetry followed by the scalar field ϕ in the probe brane +context. Note that, when A = −B, matter is coupled to the induced metric on the +brane. +Contrary to the covariant Galileon, the DBI-Galileon action is not invariant by +such a change of reference frame. However, new terms that can not be absorbed into +a redefinition of the parameters arise only at higher order in ˜X. Therefore, the non- +relativistic dynamics is not change by the introduction of a direct coupling between the + +Instability of the cosmological DBI-Galileon in the non-relativistic limit +10 +scalar field and matter of the form (33) with constant parameters. In particular, this +does not prevent the perturbations around the FLRW background from showing ghost +instabilities. +Generalization +The DBI-Galileon is a particular example of the more general class of shift-symmetric +Horndeski theories. These are subclass of Horndeski theories which are invariant under +a shift symmetry of the scalar field ϕ → ϕ + c [34, 35]. In these theories, the arbitrary +Horndeski functions are restricted to be functions of X alone. In order to make the +non-relativistic limit apparent, we Taylor expand these arbitrary functions around GR: +G2 ≡ Λ + ++∞ +� +n=1 +g(n) +2 Xn +(34) +G3 ≡ ++∞ +� +n=1 +g(n) +3 Xn +(35) +G4 ≡ M 2 +P +2 ++ ++∞ +� +n=1 +g(n) +4 Xn +(36) +G5 ≡ ++∞ +� +n=1 +g(n) +5 Xn +(37) +The constant terms in G3 and G5 do not appear in the expansion as they lead +to total derivative terms. Because the Horndeski functions depend only on X, the ω +functions that determine the stability conditions reduce to: +ω1 ≡ 2G4 − 2X +� +2G4,X + ˙φHG5,X +� +(38) +ω2 ≡ 4HG4 − 2X +� +˙φG3,X + 8HG4,X + 5 ˙φH2G5,X +� +− 4X2H +� +4G4,XX + ˙φHG5,XX +� +(39) +ω3 ≡ − 18H2G4 + 3X +� +G2,X + 12 ˙φHG3,X + 42H2G4,X + 30 ˙φH3G5,X +� ++ 6X2 � +G2,XX + 3 ˙φHG3,XX + 48H2G4,XX + 13H3 ˙φG5,XX +� ++ 12X3H2 � +6G4,XXX + H ˙φG5,XXX +� +(40) +ω4 ≡ 2G4 − 2X ¨φG5,X +(41) +From these, we can compute the quantity Qs up to first order in X: +Qs = X +H2 +� +g(1) +2 ++ 6H2g(1) +4 +� ++ O +� +X +3 +2 +� +(42) +This leads to a very simple formulation of the no-ghost condition, independent of +X, in the context of Shift-Symmetric Horndeski theories in the non-relativistic limit: +g(1) +2 ++ 6H2g(1) +4 +> 0 +(43) + +Instability of the cosmological DBI-Galileon in the non-relativistic limit +11 +In the context of the brane galileon, where g(1) +4 += −M 2 +P/2 and g(1) +2 += Λ, this is +equivalent to the inequality which is never fulfilled in flat space: +Λ − 3M 2 +PH2 > 0 +⇔ +Ω0 +Λ > ¯H2 +(44) +Other stability conditions are given by: +c2 +s ≃ 1 + 2¨φg(1) +3 ++ 4 ˙Hg(1) +4 ++ 2¨φH2g(1) +5 +g(1) +2 ++ 6H2g(1) +4 +> 0 +(45) +Qt ≃ M 2 +P +4 +(46) +c2 +t ≃ 1 +(47) +where we expressed these quantities at the lowest order. The two tensorial conditions +are, thus, automatically satisfied in this context. +On the other hand, the stability +conditions for scalar perturbations at the lowest order give a simple inequality involving +the parameters of the Taylor expansion, that can be easily checked at the background +level. +6. Conclusion +We described the DBI-Galileon theory of a four-dimensional brane evolving in a 5D bulk +space-time in the non-relativistic limit where its local kinetic energy is small compared +to its tension. This model belongs to the class of shift-symmetric Horndeski theories, +themselves being a subclass of the more general family of Horndeski theories. From +the construction of the DBI-Galileon model, the free parameters of the model acquire a +physical meaning. In particular, the interpretation of the cosmological constant is linked +to the brane tension energy density. We derived the equations driving the evolution +of the late-time Universe around a spatially flat FLRW cosmological background and +studied the stability of scalar and tensorial perturbations. +This model reduces to +an expansion around standard GR, and therefore around standard ΛCDM in the +cosmological context. As such, it is naturally compatible with data in the non-relativistic +limit provided the effect of the scalar field is small enough, even considering the speed +of the gravitational waves. +However, it revealed fatal ghostly behaviour for scalar +perturbations around the FLRW background. From there, we derived the corresponding +stability conditions for shift-symmetric Horndeski theories in the non-relativistic limit +in the cosmological context and found very simple formulations for these conditions. +Acknowledgements +We would like to thank Marc Besan¸con, Arnaud de Mattia and Vanina Ruhlmann- +Kleider for their comments on the present paper. We also want to thank David Langlois +for useful and interesting comments and suggestions. + +Instability of the cosmological DBI-Galileon in the non-relativistic limit +12 +References +[1] Brax P 2017 Reports on Progress in Physics 81 016902 URL https://dx.doi.org/10.1088/ +1361-6633/aa8e64 +[2] Horndeski G W 1974 Int. J. Theor. Phys. 10 363–384 +[3] Deffayet C, Gao X, Steer D A and Zahariade G 2011 Phys. Rev. D 84 064039 (Preprint 1103.3260) +[4] Langlois D and Noui K 2016 JCAP 07 016 (Preprint 1512.06820) +[5] Langlois D 2019 Int. J. Mod. Phys. D 28 1942006 (Preprint 1811.06271) +[6] Nicolis A, Rattazzi R and Trincherini E 2009 Phys. Rev. D 79 064036 (Preprint 0811.2197) +[7] Deffayet C, Esposito-Farese G and Vikman A 2009 Phys. Rev. D 79 084003 (Preprint 0901.1314) +[8] Gubitosi G and Linder E V 2011 Physics Letters B 703 113–118 ISSN 0370-2693 URL https: +//www.sciencedirect.com/science/article/pii/S0370269311008707 +[9] de Rham C and Gabadadze G 2010 Phys. Rev. D 82 044020 (Preprint 1007.0443) +[10] de Rham C, Gabadadze G and Tolley A J 2011 Phys. Rev. Lett. 106 231101 (Preprint 1011.1232) +[11] de Rham C and Tolley A J 2010 JCAP 05 015 (Preprint 1003.5917) +[12] Hinterbichler K, Trodden M and Wesley D 2010 Phys. Rev. D 82 124018 (Preprint 1008.1305) +[13] Silverstein E and Tong D 2004 Phys. Rev. D 70 103505 (Preprint hep-th/0310221) +[14] Langlois D, Renaux-Petel S, Steer D A and Tanaka T 2008 Phys. Rev. D 78 063523 (Preprint +0806.0336) +[15] Zumalacarregui M, Koivisto T S and Mota D F 2013 Phys. Rev. D 87 083010 (Preprint 1210.8016) +[16] Goon G, Hinterbichler K, Joyce A and Trodden M 2012 JHEP 06 004 (Preprint 1203.3191) +[17] Dobado A and Maroto A L 2001 Nucl. Phys. B 592 203–218 (Preprint hep-ph/0007100) +[18] Cembranos J A R, Dobado A and Maroto A L 2001 Goldstone bosons and solitons on the brane +20th International Symposium on Lepton and Photon Interactions at High Energies (LP 01) +(Preprint hep-ph/0107155) +[19] Burrage C, de Rham C and Heisenberg L 2011 JCAP 05 025 (Preprint 1104.0155) +[20] Goon G, Hinterbichler K and Trodden M 2011 JCAP 07 017 (Preprint 1103.5745) +[21] Lovelock D 1971 J. Math. Phys. 12 498–501 +[22] Appleby S and Linder E V 2012 Journal of Cosmology and Astroparticle Physics 2012 043 URL +https://dx.doi.org/10.1088/1475-7516/2012/03/043 +[23] Scolnic D M et al. 2018 The Astrophysical Journal 859 101 ISSN 15384357 (Preprint 1710.00845) +URL http://arxiv.org/abs/1710.00845http://dx.doi.org/10.17909/T95Q4X +[24] Abbott B P et al. (LIGO Scientific, Virgo, Fermi-GBM, INTEGRAL) 2017 Astrophys. J. Lett. +848 L13 (Preprint 1710.05834) +[25] De Felice A and Tsujikawa S 2012 JCAP 02 007 (Preprint 1110.3878) +[26] Creminelli P and Vernizzi F 2017 Phys. Rev. Lett. 119 251302 (Preprint 1710.05877) +[27] Ezquiaga J M and Zumalac´arregui M 2017 Phys. Rev. Lett. 119 251304 (Preprint 1710.05901) +[28] Planck Collaboration 2020 A&A 641 A6 URL https://doi.org/10.1051/0004-6361/201833910 +[29] de Rham C and Melville S 2018 Phys. Rev. Lett. 121 221101 (Preprint 1806.09417) +[30] Ezquiaga J M and Zumalac´arregui M 2018 Front. Astron. Space Sci. 5 44 (Preprint 1807.09241) +[31] Barausse E et al. 2020 Gen. Rel. Grav. 52 81 (Preprint 2001.09793) +[32] Bekenstein J D 1993 Phys. Rev. D 48 3641–3647 (Preprint gr-qc/9211017) +[33] Bettoni D and Liberati S 2013 Phys. Rev. D 88 084020 (Preprint 1306.6724) +[34] Sotiriou T P and Zhou S Y 2014 Phys. Rev. Lett. 112 251102 (Preprint 1312.3622) +[35] Sotiriou T P and Zhou S Y 2014 Phys. Rev. D 90 124063 (Preprint 1408.1698) + diff --git a/C9AzT4oBgHgl3EQfwf5z/content/tmp_files/load_file.txt b/C9AzT4oBgHgl3EQfwf5z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cd997e878ff94a4db60da0b429394cdd1fc8cf30 --- /dev/null +++ b/C9AzT4oBgHgl3EQfwf5z/content/tmp_files/load_file.txt @@ -0,0 +1,457 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf,len=456 +page_content='Instability of the cosmological DBI-Galileon in the non-relativistic limit C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Leloup1,2, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Heitz3 and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Neveu3,4 1 Universit´e Paris-Cit´e, CNRS, Astroparticule et Cosmologie, 75013 Paris, France 2 Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan 3 Universit´e Paris-Saclay, CNRS, IJCLab, 91405, Orsay, France 4 Sorbonne Universit´e, CNRS, Universit´e de Paris, LPNHE, 75252 Paris Cedex 05, France Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' The DBI-Galileon model is a tensor-scalar theory of gravity which finds its foundation as the most general theory of the dynamics of a 4D brane embedded in a 5D bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' It is of particular interest as it provides a few free parameters with a physical meaning, such as the cosmological constant which is there related to the brane tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Most studies of this model have been performed assuming a maximally symmetric geometry for the 5D bulk, in which it has been shown that the theory reduces to various types of Galileon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In contrast, the general case for the geometry of the bulk provides a different covariantization of the Galileon model than the covariant Galileon: the DBI-Galileon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' From the tight constraints on the gravitational waves speed, we are naturally led to consider the non-relativistic limit of the model where the kinetic energy of the brane is small compared to its tension, that we study in the context of late-time cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' The DBI-Galileon in the non-relativistic limit is simply an expansion around General Relativity (GR) which can be expressed as a shift-symmetric Horndeski theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' We developed the description of this theory at the background and perturbation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' However, by studying the scalar and tensor perturbations around a flat FLRW background, we found that they contain a ghost degree of freedom leading to fatal instability of the vacuum for every combination of the free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' As a lesson, we emphasized which of the Horndeski terms competes to avoid this instability in more general cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Introduction Dark energy has been modelled by a large variety of theories since decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Among these, many rely on the introduction of additional scalar fields whose dynamics, at the origin of the late-time acceleration of the expansion of the Universe, is determined by arbitrary parametric functions, potentials and/or coupling (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' These are the so-called scalar-tensor theories of modified gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In particular, the class of Horndeski theories is of great interest as it contains all models of modified gravity with a single arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='01723v1 [hep-th] 4 Jan 2023 Instability of the cosmological DBI-Galileon in the non-relativistic limit 2 additional scalar field leading to second-order equations of motion [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Extensions of Horndeski theories to scalar-tensor theories of one scalar field with equations of motion of higher orders have also been explored [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Particular Horndeski theories are described by the specification of four arbitrary functions of the scalar field and its kinetic energy, leading to a huge variety of models and phenomenological behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Among these wide classes of models, some can be built from first physical principles or arguments of symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' For instance, the Galileon model [6] and its covariant extension [7] was built by imposing a galilean symmetry for the scalar field, leaving only five free numerical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' We can also cite, among many others, the pure kinetic gravity theory [8], massive gravity in the non-relativistic limit [9, 10] and the DBI-Galileon [11] which is the main object of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' The DBI-Galileon is a model that falls into the class of Brane-world scenarios of extra-dimension theories, where the matter fields are confined on a 4D brane while gravity can propagate into the additional spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Of most interest for the DBI-Galileon is the case of a single extra-dimension as it has been shown that theories with more co-dimensions exhibit ghosts either in the flat or self-accelerating de Sitter solution [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' The action include a volume term for the 4D brane in the 5D bulk which leads to the well-known Dirac-Born-Infeld (DBI) action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' This action, and DBI-like extensions, can lead to a self-accelerating solution and has been thoroughly studied as a candidate model in the early Universe cosmic inflation paradigm [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In addition, the DBI-Galileon model exhibits the Galileon Lagrangians in the non-relativistic limit [11] but giving a physical meaning to their free parameters: the Planck mass in the brane, the Planck mass in the bulk, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In particular, the brane tension here plays the role of the cosmological constant which brings a possible interpretation of its nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' The original probe brane construction has been revisited in [15] where the matter metric is disformally related to a standard gravitational metric, or in [16] in the framework of spontaneous symmetry breaking for the 5D space-time symmetries broken by the presence of the brane, bridging the gap with Brane-world scenarios developed in the context of quantum field theory and an interpretation of the scalar field as a Nambu-Goldstone boson [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' The DBI-Galileon model has been studied extensively in special cases of the maximally symmetric bulk geometry [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' However, to our knowledge, no study of the DBI-Galileon in the late-time cosmology setting as a potential candidate for Dark Energy has been performed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In this paper we develop the DBI-Galileon theory in the non relativistic limit (Section 2) and study its dynamics in the Friedmann-Lemaˆıtre-Robertson-Walker flat metric (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' The perturbation stability is explored in Section 4 and then discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' DBI-Galileon in the non-relativistic limit DBI-Galileon We are interested in the description of a four dimensional brane universe embedded in Instability of the cosmological DBI-Galileon in the non-relativistic limit 3 a five dimensional bulk from the cosmological perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In this context,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' it has been shown in [11] that the most general action on the brane is given by the 4D Lovelock terms [21] inside the brane and the boundary terms associated to the 5D Lovelock terms in the bulk: S = � dx4√−g � −Λ − M 3 5K + M 2 P 2 R − β M 3 5 m2 KGB + Lm (˜qµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' ψm) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' (1) where g is the induced metric on the brane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Λ the brane cosmological constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' K is the extrinsic curvature of the brane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' R is the Ricci scalar on the brane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' KGB is the boundary term on the brane of the Gauss-Bonnet scalar in the bulk and Lm is the Lagrangian density of matter that lives confined in the brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In the above action has been defined the 4D Planck mass MP, its 5D counterpart M5 and their ratio m = M 3 5/M 2 P, and β is an arbitrary constant parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Because the action is defined on the 4-dimensional brane, the above quantities are expressed in terms of the induced metric g, as opposed to the 5-dimensional bulk metric G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Assuming we have a coordinate system (xµ, y) in the bulk, Greek letters being defined to span from 0 to 3, such that the length element in this frame is ds2 = qµνdxµdxν+(dy)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' The brane position in the bulk is defined by y = π (xµ) such that we can express the induced metric from the bulk metric: gµν = qµν + ∂µπ∂νπ and gµν = qµν − γ2∂µπ∂νπ (2) with the Lorentz factor γ = � 1 + (∂π)2�−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' From this expression of the induced metric, we can explicitly write the action (1) using the bulk metric q and the scalar field π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' As an illustration, see how the cosmological constant part of the action on the brane can be expressed: SΛ = −Λ � d4x√−g = −Λ � d4x√−q � 1 + (∂π)2 (3) As was pointed out in the original paper by de Rham and Tolley [11], we recover a DBI term in the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' This leads us to associate the cosmological constant Λ with a brane tension f, following Λ = f 4 for unit convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Non-relativistic limit If the derivatives of the field vanish, ∂µπ = 0, then we recover standard GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Because we are interested in the cosmological setting, where predictions from the ΛCDM model based on GR match precisely a large range of observations, we consider only small corrections to GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Therefore, we consider the DBI-Galileon in the so-called non- relativistic limit where (∂π)2 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In particular, the DBI part of the action in the non-relativistic limit becomes: Sf = −f 4 � d4x√−q � 1 + ∇µπ∇µπ 2 − (∇µπ∇µπ)2 8 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' � (4) Instability of the cosmological DBI-Galileon in the non-relativistic limit 4 We see that, to have a canonically normalized field, we can proceed to the field redefinition π → ϕ = f 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Up to degree 5 in ∂ϕ/f 2, we find the following Lagrangian operators for the DBI-Galileon in the non-relativistic limit: Lf = 1 + L2 2f 4 − X2 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' (5) LK = − L3 2f 6 − 2X f 6 [ψ] + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' (6) LR = ¯R + 1 f 4 � [Φ]2 − � Φ2� − f 4X 2 ¯R − 2 ¯Rµν∇µϕ∇νϕ � + L4 4f 8 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' (7) LKGB = 2 f 6 �� − ¯Rµν [Φ] + 2 ¯RµρΦρ ν + ¯RµρνλΦρλ� ∇µϕ∇νϕ + 1 3 [Φ]3 − [Φ] � Φ2� + 2 3 � Φ3� −1 2 ¯R [ψ] � + L5 3f 10 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' (8) We defined on the above expressions the tensor Φµν = ∇µ∇νϕ, and the three scalars [Φ] = Φµ µ, [ψ] = ∂µϕ·Φµν·∂νϕ and X = − (∂ϕ)2 /2f 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Furthermore, the L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='5 Lagrangian operators are the covariant Galileon model Lagrangian operators as defined in [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Therefore, the theory described here is a generalization of the Galileon model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' We can note that the additional terms in LR and LKGB compared to the Galileon Lagrangian operators vanish in the particular case of a flat geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' We conclude that the DBI- Galileon in the non-relativistic limit is a different coviariantization of the flat Galileon than the covariant Galileon [7] or dRGT massive gravity [9, 10], with an additional terms in Lf and another one in LK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' That point noted, we will stay at leading order in X in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Being a theory of a scalar field interacting with a metric, with equations of motion of at most second order, it can be described as a Horndeski theory [2, 3] with the following Horndeski functions: G2 = A (ϕ) − f 4 (1 − X + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=') (9) G3 = M 3 5 f 2 (X + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=') (10) G4 = M 2 P 2 � 1 − X − X2/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' � (11) G5 = −2β M 3 5 m2f 2 � X + X2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' � (12) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Cosmological background evolution We expand the dynamics of the fields around a flat FLRW background on the 4D slice of the bulk along xµ where the properties of the brane, gµν and ϕ, are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' On this slice, the length element is: ds2 = qµνdxµdxν = −dt2 + a2 (t) δijdxidxj (13) It might appear more natural to expand around a flat FLRW background on the brane with its induced metric gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' However, the two metrics qµν and gµν are related Instability of the cosmological DBI-Galileon in the non-relativistic limit 5 by a disformal transformation involving the scalar field which, at the background level, depends only on the physical time: ¯gµν = ¯qµν + ( ˙¯ϕ (t)) 2 f 4 δµ0δν0 (14) where barred quantities are taken at the background level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Therefore, the geometry on the brane is also of the FLRW type, but with a different definition for the physical time, leading to a different scale factor and expansion history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Because in our case ˙¯ϕ ≪ f 2, the expansion history on the 4D slice and on the brane will, then, be approximately the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In addition, equations (9)-(12) determine a self-consistent Horndeski theory of gravity with the metric qµν and the scalar field ϕ without referring to the induced metric gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Thus, in the following we apply the well-known techniques used in the context of Horndeski theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In the non-relativistic limit, action (1) reads: S = � dx4√−q (Lf + LK + LR + LKGB + Lm (qµν, ψm)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' (15) We define Ω0 m and Ω0 r the standard present energy density parameters for pressureless matter and radiation respectively, and ¯H the normalized Hubble rate H/H0 with H0 the present Hubble constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Prime symbol denotes the derivative with respect to ln a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' We set: ˜x = ϕ′H0 f 2 , Ω0 Λ = Λ 3H2 0M 2 P = f 4 3H2 0M 2 P , η = M 3 5 M 2 PH0 , ξ = β η , κ = MPH0 f 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' (16) Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' the two Friedmann equations derived from action (15) are: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H2 = −3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H4˜x2 − 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H6˜x4 + η ¯H4˜x3 − 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3 ξ ¯H6˜x3 − 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3 ξ ¯H8˜x5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='+ Ω0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='a3 + Ω0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='a4 + Ω0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='Λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='1 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H2˜x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='(17) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H2 + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H ¯H′ = −2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3ξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='2 ¯H6˜x3 + 5 ¯H5˜x3 ¯H′ + 3 ¯H6˜x2˜x′ + 2 ¯H8˜x5 + 5 ¯H8˜x4˜x′ + 7 ¯H7˜x5 ¯H′� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H4˜x2 − ¯H3˜x2 ¯H′ − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H4˜x˜x′ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3η ¯H3˜x2( ¯H˜x)′ − 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H6˜x4 − 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H5˜x4 ¯H′ − ¯H6˜x3˜x′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='− Ω0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3a4 + Ω0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='Λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='1 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H2˜x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='(18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='We see that we recover the ΛCDM equations when setting ˜x to zero,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' but with a physical interpretation of the cosmological constant as the brane tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' The DBI model proposed here is then an extension of the standard model of cosmology, and as such follows a late accelerated expansion but with a physical interpretation of the origin of Λ as a brane tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Using the same methodology and similar notations as in [22], we derive the field ϕ equation of motion from action (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' This leads to a system of two ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='Instability of the cosmological DBI-Galileon in the non-relativistic limit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='coupled equations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='˜x′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='= −˜x + αλ − σγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='σβ − αω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='= ωγ − βλ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='σβ − αω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='(19) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='β = 2 ¯H4 + 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H6˜x2 − Ω0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='Λ ¯H2 − 2η ¯H4˜x + 4ξ ¯H6˜x + 40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3 ξ ¯H8˜x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='α = 6 ¯H3˜x − Ω0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='Λ ¯H˜x + 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H5˜x3 − 3η ¯H3˜x2 + 10ξ ¯H5˜x2 + 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3 ξ ¯H7˜x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='γ = 4 ¯H4˜x − 2Ω0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='Λ ¯H2˜x − η ¯H4˜x2 + 2ξ ¯H6˜x2 − 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3 ξ ¯H8˜x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='ω = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H4˜x + ¯H6˜x3 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3η ¯H4x2 + 2ξ ¯H6˜x2 + 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3 ξ ¯H8˜x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='σ = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H + 2 ¯H3˜x2 + 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H5˜x4 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3η ¯H3˜x3 + 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3 ξ ¯H5˜x3 + 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3 ξ ¯H7˜x5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='λ = ¯H2 + Ω0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3a4 − Ω0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='Λ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='2Ω0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='Λ ¯H2˜x2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H4˜x2 − 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H6˜x4 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3η ¯H4˜x3 − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3ξ ¯H6˜x3 − 2ξ ¯H8˜x5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='Given values for the parameters Ω0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' ξ and κ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' and initial conditions ˜x0 and H0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' this system can be integrated to compute background cosmology observables like the distance moduli of type Ia supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In Figure 1 we illustrate this with a Hubble diagram prediction compared with recent type Ia supernova data [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' As an initial condition for ˜x, we chose to set ˜x0 = 6 × 10−8 today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' This is the maximum value allowed by the constraint on gravitational wave speed (see Section 4) coming from the quasi simultaneous observation of photons and gravitational waves after neutron star merger event GW170817A [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Nevertheless, before discussing more the cosmological scenarios proposed by the DBI-Galileon model, stability conditions much be computed first to assess the viability of the models for any set of parameters at the perturbation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Stability conditions To be viable as a description of our Universe, the model has to fulfill stability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' These requirements apply to degrees of freedom propagating around the fixed background, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' to the cosmological perturbations, that are capable of undermining the stability of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In the determination of the stability conditions for the DBI-Galileon model in the non-relativistic limit, we use the formalism described in [25] for Horndeski theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In our case, these stability conditions are defined Instability of the cosmological DBI-Galileon in the non-relativistic limit 7 14 16 18 20 22 24 26 Distance moduli [mag] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='75 Redshift z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='0 Residuals [mag] Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Hubble diagram prediction for the non-relativistic DBI-Galileon model for Ω0 Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='7, ˜x0 = 6 × 10−8, η = ξ = 0 (blue) compared with binned Pantheon data (black points) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' We used M = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='81 for the offset magnitude of the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Residuals to the fit are presented in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='from the following quantities derived from the particular Horndeski functions (9) to (12): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='ω1 = 1 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H2˜x2 + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H4˜x4 + 2ξ ¯H4˜x3 + 2ξ ¯H6˜x5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='(20) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='ω2 = 2 ¯H + 3 ¯H3˜x2 + 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H5˜x4 − η ¯H3˜x3 + 10ξ ¯H5˜x3 + 14ξ ¯H7˜x5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='(21) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='ω3 = 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='− ¯H2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='2Ω0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='Λ ¯H2˜x2 − 3 ¯H4˜x2 + 2η ¯H4˜x3 − 10ξ ¯H6˜x3 − 45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H6˜x4 − 56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3 ξ ¯H8˜x5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='(22) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='ω4 = 1 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H2˜x2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='¯H4˜x4 + 2ξ ¯H3˜x2( ¯H˜x)′ + 2ξ ¯H5˜x4( ¯H˜x)′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='(23) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='Tensorial stability conditions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='In order to avoid ghosts and Laplacian instabilities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' we impose the following constraints on the sign of the kinetic term and on the sign of the gravitational waves speed squared: Qt ≡ ω1 4 = 1 4 + 1 8 ¯H2˜x2 + 3 32 ¯H4˜x4 + 1 2ξ ¯H4˜x3 + 1 2ξ ¯H6˜x5 > 0 (24) c2 t ≡ ω4 ω1 = 1 − 1 2 ¯H2˜x2 − 1 8 ¯H4˜x4 + 2ξ ¯H3˜x2( ¯H˜x)′ + 2ξ ¯H5˜x4( ¯H˜x)′ 1 + 1 2 ¯H2˜x2 + 3 8 ¯H4˜x4 + 2ξ ¯H4˜x3 + 2ξ ¯H6˜x5 ≥ 0 (25) In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' we see that the gravitational wave speed depends on ˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' and tends to 1 when ˜x → 0 : 4Qt ≃ 1 + 2( ¯H˜x)2 (26) ct ≃ 1 − 1 2( ¯H˜x)2 (27) Instability of the cosmological DBI-Galileon in the non-relativistic limit 8 Given the very tight constraint on the speed of gravitational waves,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' equal to the speed of light up to a ∼ 10−15 difference [24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 26,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' this justifies a posteriori the relevance of the non-relativistic limit where ˜x ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Moreover, we see that tensorial perturbations are stable in this limit since Qt > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Scalar stability conditions Similar stability conditions apply to the scalar degrees of freedom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' here including the scalar perturbations of matter components: Qs ≡ ω1 (4ω1ω3 + 9ω2 2) 3ω2 2 > 0 (28) c2 s ≡ 3 (2ω2 1ω2H − ω2 2ω4 + 4ω1ω2 ˙ω1 − 2ω2 1 ˙ω2) − 6ω2 1 � (1 + wi) ρi ω1 (4ω1ω3 + 9ω2 2) ≥ 0 (29) where wi and ρi are respectively the equation of state parameter and the energy density of the fluid i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' and the sum runs over all the components of the Universe (here only pressureless matter and radiation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' At the lowest order in ˜x, we get: Qs ≃ 3 2(Ω0 Λ − ¯H2)˜x2 (30) c2 s ≃ 1 + 2 � η ¯H2˜x′ − ¯H ¯H′ − 2ξ ¯H4˜x′� 3 � Ω0 Λ − ¯H2� (31) With ˜x ≪ 1, a fit of the DBI-Galileon model to data leads to cosmological parameters close to the standard model ones: Ω0 m ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3 and Ω0 Λ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='7 [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Therefore, from the first Friedmann equation, we get Ω0 Λ < ¯H2 for all relevant models in agreement with cosmological observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' As Qs ≤ 0, the DBI-Galileon model contains scalar instabilities unless it reduces to GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' One way to avoid this would be to add a spatial curvature to the metric, but with a strong energy density (at least ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3) which is also excluded by observations [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Discussion Physical interpretation From the definition (28), we see that the dominant terms come G2 (giving the Ω0 Λ term) and G4 (giving the ¯H2 term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' The competition between the two terms leads to the ghost-like behaviour in a cosmological setting: Qs ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In other words, it is the result of the competition between the DBI and the Einstein-Hilbert terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' The DBI action will have the effect of stretching the brane towards an extremal surface, whereas the Einstein-Hilbert term on the brane will tend to make the brane contract on itself from the effect of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' However, in the non-relativistic limit of the DBI-Galileon, the Einstein-Hilbert term destabilizes the scalar field perturbations and the stretching effect from the cosmological constant is not strong enough to counterbalance, leading to an instantaneous decay of the vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Instability of the cosmological DBI-Galileon in the non-relativistic limit 9 Because the DBI-Galileon action is the most general one can find of a 4D probe brane in a 5D bulk, we expect this statement to be quite general for all such theories studied in the current context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Indeed, this is true in a standard cosmological setting which is realised with an FLRW slicing of the bulk space-time (equivalent to an FLRW background on the brane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Therefore, the only way to evade this ghostly behaviour in cosmology is to include the full relativistic dynamics of the theory (˜x ∼ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' We have seen that, in this case, we expect significant deviations of the speed of gravitational waves ct from c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' This is not a definitive impossibility though if the full DBI-Galileon is viewed as an effective theory valid only at cosmological scales for which the speed of gravitational waves has not been probed [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Indeed, the constraint on the gravitational speed from the observation of GW170817 in coincidence with GRB170817A [24] is only valid on small scales probed by LIGO and Virgo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' A modification of the dispersion relation of gravitational waves at small scales from operators present in the UV complete theory could allow ct ̸= c on cosmological scales while being compatible with current astrophysical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Waiting for the next generation of gravitational wave interferometers, in particular LISA, which will be able to probe this relation at larger scales [31], this possibility remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Direct coupling to matter In the context of cosmology, where standard model matter is present, there might be direct coupling to the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In that case, the metric ˜q to which matter is sensitive is different than the space-time metric q: S = � dx4√−q (Lf + LK + LR + LKGB) + � dx4� −˜qLm (˜qµν, ψm) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' (32) It has been shown in [32] that the two metrics are related by a disformal transformation of the following form: qµν = A � ϕ, ˜X � ˜qµν + B � ϕ, ˜X � ∂µϕ∂νϕ f 4 (33) where A and B are arbitrary functions of the scalar field and ˜X = −˜qµν∂µϕ∂νϕ/2f 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' For simplicity and following the treatment of the covariant Galileon [22], we assume that A and B are constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' This can be further justified by the fact that, a dependency on X would introduce, in general, higher order terms which would go beyond the framework of Horndeski theories [33], and a dependency on ϕ would, in general, break the shift symmetry followed by the scalar field ϕ in the probe brane context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Note that, when A = −B, matter is coupled to the induced metric on the brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Contrary to the covariant Galileon, the DBI-Galileon action is not invariant by such a change of reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' However, new terms that can not be absorbed into a redefinition of the parameters arise only at higher order in ˜X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Therefore, the non- relativistic dynamics is not change by the introduction of a direct coupling between the Instability of the cosmological DBI-Galileon in the non-relativistic limit 10 scalar field and matter of the form (33) with constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In particular, this does not prevent the perturbations around the FLRW background from showing ghost instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Generalization The DBI-Galileon is a particular example of the more general class of shift-symmetric Horndeski theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' These are subclass of Horndeski theories which are invariant under a shift symmetry of the scalar field ϕ → ϕ + c [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In these theories, the arbitrary Horndeski functions are restricted to be functions of X alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In order to make the non-relativistic limit apparent, we Taylor expand these arbitrary functions around GR: G2 ≡ Λ + +∞ � n=1 g(n) 2 Xn (34) G3 ≡ +∞ � n=1 g(n) 3 Xn (35) G4 ≡ M 2 P 2 + +∞ � n=1 g(n) 4 Xn (36) G5 ≡ +∞ � n=1 g(n) 5 Xn (37) The constant terms in G3 and G5 do not appear in the expansion as they lead to total derivative terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Because the Horndeski functions depend only on X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' the ω functions that determine the stability conditions reduce to: ω1 ≡ 2G4 − 2X � 2G4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='X + ˙φHG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='X � (38) ω2 ≡ 4HG4 − 2X � ˙φG3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='X + 8HG4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='X + 5 ˙φH2G5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='X � − 4X2H � 4G4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='XX + ˙φHG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='XX � (39) ω3 ≡ − 18H2G4 + 3X � G2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='X + 12 ˙φHG3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='X + 42H2G4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='X + 30 ˙φH3G5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='X � + 6X2 � G2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='XX + 3 ˙φHG3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='XX + 48H2G4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='XX + 13H3 ˙φG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='XX � + 12X3H2 � 6G4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='XXX + H ˙φG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='XXX � (40) ω4 ≡ 2G4 − 2X ¨φG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='X (41) From these,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' we can compute the quantity Qs up to first order in X: Qs = X H2 � g(1) 2 + 6H2g(1) 4 � + O � X 3 2 � (42) This leads to a very simple formulation of the no-ghost condition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' independent of X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' in the context of Shift-Symmetric Horndeski theories in the non-relativistic limit: g(1) 2 + 6H2g(1) 4 > 0 (43) Instability of the cosmological DBI-Galileon in the non-relativistic limit 11 In the context of the brane galileon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' where g(1) 4 = −M 2 P/2 and g(1) 2 = Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' this is equivalent to the inequality which is never fulfilled in flat space: Λ − 3M 2 PH2 > 0 ⇔ Ω0 Λ > ¯H2 (44) Other stability conditions are given by: c2 s ≃ 1 + 2¨φg(1) 3 + 4 ˙Hg(1) 4 + 2¨φH2g(1) 5 g(1) 2 + 6H2g(1) 4 > 0 (45) Qt ≃ M 2 P 4 (46) c2 t ≃ 1 (47) where we expressed these quantities at the lowest order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' The two tensorial conditions are, thus, automatically satisfied in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' On the other hand, the stability conditions for scalar perturbations at the lowest order give a simple inequality involving the parameters of the Taylor expansion, that can be easily checked at the background level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Conclusion We described the DBI-Galileon theory of a four-dimensional brane evolving in a 5D bulk space-time in the non-relativistic limit where its local kinetic energy is small compared to its tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' This model belongs to the class of shift-symmetric Horndeski theories, themselves being a subclass of the more general family of Horndeski theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' From the construction of the DBI-Galileon model, the free parameters of the model acquire a physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' In particular, the interpretation of the cosmological constant is linked to the brane tension energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' We derived the equations driving the evolution of the late-time Universe around a spatially flat FLRW cosmological background and studied the stability of scalar and tensorial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' This model reduces to an expansion around standard GR, and therefore around standard ΛCDM in the cosmological context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' As such, it is naturally compatible with data in the non-relativistic limit provided the effect of the scalar field is small enough, even considering the speed of the gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' However, it revealed fatal ghostly behaviour for scalar perturbations around the FLRW background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' From there, we derived the corresponding stability conditions for shift-symmetric Horndeski theories in the non-relativistic limit in the cosmological context and found very simple formulations for these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Acknowledgements We would like to thank Marc Besan¸con, Arnaud de Mattia and Vanina Ruhlmann- Kleider for their comments on the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' We also want to thank David Langlois for useful and interesting comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Instability of the cosmological DBI-Galileon in the non-relativistic limit 12 References [1] Brax P 2017 Reports on Progress in Physics 81 016902 URL https://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='1088/ 1361-6633/aa8e64 [2] Horndeski G W 1974 Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 10 363–384 [3] Deffayet C, Gao X, Steer D A and Zahariade G 2011 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' D 84 064039 (Preprint 1103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3260) [4] Langlois D and Noui K 2016 JCAP 07 016 (Preprint 1512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='06820) [5] Langlois D 2019 Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' D 28 1942006 (Preprint 1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='06271) [6] Nicolis A, Rattazzi R and Trincherini E 2009 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' D 79 064036 (Preprint 0811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='2197) [7] Deffayet C, Esposito-Farese G and Vikman A 2009 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' D 79 084003 (Preprint 0901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='1314) [8] Gubitosi G and Linder E V 2011 Physics Letters B 703 113–118 ISSN 0370-2693 URL https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='com/science/article/pii/S0370269311008707 [9] de Rham C and Gabadadze G 2010 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' D 82 044020 (Preprint 1007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='0443) [10] de Rham C, Gabadadze G and Tolley A J 2011 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 106 231101 (Preprint 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='1232) [11] de Rham C and Tolley A J 2010 JCAP 05 015 (Preprint 1003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='5917) [12] Hinterbichler K, Trodden M and Wesley D 2010 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' D 82 124018 (Preprint 1008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='1305) [13] Silverstein E and Tong D 2004 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' D 70 103505 (Preprint hep-th/0310221) [14] Langlois D, Renaux-Petel S, Steer D A and Tanaka T 2008 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' D 78 063523 (Preprint 0806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='0336) [15] Zumalacarregui M, Koivisto T S and Mota D F 2013 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' D 87 083010 (Preprint 1210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='8016) [16] Goon G, Hinterbichler K, Joyce A and Trodden M 2012 JHEP 06 004 (Preprint 1203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3191) [17] Dobado A and Maroto A L 2001 Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' B 592 203–218 (Preprint hep-ph/0007100) [18] Cembranos J A R, Dobado A and Maroto A L 2001 Goldstone bosons and solitons on the brane 20th International Symposium on Lepton and Photon Interactions at High Energies (LP 01) (Preprint hep-ph/0107155) [19] Burrage C, de Rham C and Heisenberg L 2011 JCAP 05 025 (Preprint 1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='0155) [20] Goon G, Hinterbichler K and Trodden M 2011 JCAP 07 017 (Preprint 1103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='5745) [21] Lovelock D 1971 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 12 498–501 [22] Appleby S and Linder E V 2012 Journal of Cosmology and Astroparticle Physics 2012 043 URL https://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='1088/1475-7516/2012/03/043 [23] Scolnic D M et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 2018 The Astrophysical Journal 859 101 ISSN 15384357 (Preprint 1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='00845) URL http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='org/abs/1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='00845http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='17909/T95Q4X [24] Abbott B P et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' (LIGO Scientific, Virgo, Fermi-GBM, INTEGRAL) 2017 Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 848 L13 (Preprint 1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='05834) [25] De Felice A and Tsujikawa S 2012 JCAP 02 007 (Preprint 1110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3878) [26] Creminelli P and Vernizzi F 2017 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 119 251302 (Preprint 1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='05877) [27] Ezquiaga J M and Zumalac´arregui M 2017 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 119 251304 (Preprint 1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='05901) [28] Planck Collaboration 2020 A&A 641 A6 URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='1051/0004-6361/201833910 [29] de Rham C and Melville S 2018 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 121 221101 (Preprint 1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='09417) [30] Ezquiaga J M and Zumalac´arregui M 2018 Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 5 44 (Preprint 1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='09241) [31] Barausse E et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 2020 Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 52 81 (Preprint 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='09793) [32] Bekenstein J D 1993 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' D 48 3641–3647 (Preprint gr-qc/9211017) [33] Bettoni D and Liberati S 2013 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' D 88 084020 (Preprint 1306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='6724) [34] Sotiriou T P and Zhou S Y 2014 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' 112 251102 (Preprint 1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='3622) [35] Sotiriou T P and Zhou S Y 2014 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content=' D 90 124063 (Preprint 1408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} +page_content='1698)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'} diff --git a/CdE2T4oBgHgl3EQf9AmF/vector_store/index.pkl b/CdE2T4oBgHgl3EQf9AmF/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..b08b15ac9234d895ce9f7750a09710cfab7116f6 --- /dev/null +++ b/CdE2T4oBgHgl3EQf9AmF/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b1615a59309dd5ac5f8ba4473c2a7efd6e34590dcaf0d4319b61f1166c84d59b +size 195964 diff --git a/DNFQT4oBgHgl3EQfPTY7/content/tmp_files/2301.13278v1.pdf.txt b/DNFQT4oBgHgl3EQfPTY7/content/tmp_files/2301.13278v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3fb47293e9691c012c3229495748b6b5f880f736 --- /dev/null +++ b/DNFQT4oBgHgl3EQfPTY7/content/tmp_files/2301.13278v1.pdf.txt @@ -0,0 +1,1485 @@ +The younger flagellum coordinates the beating in +C. reinhardtii +Da Wei1,3,Greta Quaranta2, Marie-Eve Aubin-Tam1†, Daniel S.W. Tam2∗ +1Department of Bionanoscience, Delft University of Technology, +2628CJ Delft, Netherlands. +2Laboratory for Aero and Hydrodynamics, Delft University of Technology, +2628CD Delft, Netherlands. +3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, +Chinese Academy of Sciences; Beijing 100190, China. +†Corresponding author. Email: m.e.aubin-tam@tudelft.nl; +∗Corresponding author. Email: d.s.w.tam@tudelft.nl. +1 +arXiv:2301.13278v1 [physics.bio-ph] 30 Jan 2023 + +Abstract +Eukaryotes swim with coordinated flagellar (ciliary) beating and steer by fine-tuning +the coordination. The model organism for studying flagellate motility, C. reinhardtii (CR), +employs synchronous, breast-stroke-like flagellar beating to swim, and it modulates the +beating amplitudes differentially to steer. This strategy hinges on both inherent flagellar +asymmetries (e.g. different response to chemical messengers) and such asymmetries be- +ing effectively coordinated in the synchronous beating. In CR, the synchrony of beating is +known to be supported by a mechanical connection between flagella, however, how flagellar +asymmetries persist in the synchrony remains elusive. For example, it has been speculated +for decades that one flagellum leads the beating, as its dynamic properties (i.e. frequency, +waveform, etc.) appear to be copied by the other one. In this study, we combine experi- +ments, computations, and modeling efforts to elucidate the roles played by each flagellum +in synchronous beating. With a non-invasive technique to selectively load each flagellum, +we show that the coordinated beating essentially responds to only load exerted on the cis +flagellum; and that such asymmetry in response derives from a unilateral coupling between +the two flagella. Our results highlight a distinct role for each flagellum in coordination and +have implication for biflagellates’ tactic behaviors. +One-Sentence Summary: The younger flagellum of C. reinhardtii coordinates the synchronous +beating and couples to external forces. +2 + +Introduction +The ability to swim towards desirable environments and away from hazardous ones is funda- +mental to the survival of many microorganisms. These so-called tactic behaviors are exhib- +ited by many motile microorganisms ranging from bacteria (1, 2) to larger flagellates and cili- +ates (3–5). Different microorganisms have developed specific strategies for steering, depending +on the tactic behavior and on their specific sensory and motility repertoire. For example, bacte- +ria modulate the tumbling rate (1) while flagellates and ciliates modulate the waveform (6–9), +amplitude (10, 11) and frequency of their flagellar/ciliary (4, 12) beating. The goal of these +active modulations of the motility is to achieve a spatially asymmetric generation of propulsive +force to steer the organism. +C. reinhardtii (CR), the model organism for studies of flagellar motility, achieves tactic nav- +igation by a fine-tuned differential modulation on its two flagella. Studying this organism offers +great opportunities to look into how flagella coordinate with each other and how such coordi- +nation helps facilitate targeted steering. CR has a symmetric cell body and two near-identical +flagella inherited from the common ancestors of land plants and animals (13). It swims by +beating its two flagella synchronously and is capable of photo- and chemotaxis (10, 14). For +this biflagellated organism, effective steering hinges on both flagellar asymmetry and flagellar +coordination. On the one hand, the two flagella must be asymmetric to respond differentially +to stimuli (10,15); on the other hand, the differential responses must be coordinated by the cell +such that the beating would remain synchronized to guarantee effective swimming. Understand- +ing this remarkable feat requires knowledge about both flagellar asymmetry and coordination. +The two flagella are known to be asymmetric in several, possibly associated, aspects. First +of all, they differ in developmental age (16, 17). The flagellum closer to the eyespot, the cis(- +eyespot) flagellum, is always younger than the other one, the trans(-eyespot) flagellum. This +is because the cis is organized by a basal body (BB) that develops from a pre-matured one in +the mother cell; and this younger BB also organizes the flagellar root (D4 rootlet) that dictates +the eyespot formation (18). Second, the two flagella have asymmetric protein composition (19– +21). For example, the trans flagellum is richer in CAH6, a protein possibly involved in CO2 +sensing (14,20). Finally, the flagella have different dynamic properties (22–24). Their beating is +modulated differentially by second messengers such as calcium (22,23) and cAMP (25). When +beating alone, the trans beats at a frequency 30%-40% higher than the cis (23,26–28); the trans +also displays an attenuated waveform (29) and a much stronger noise (29,30). +3 + +Remarkably, despite these inherent asymmetries, CR cells establish robust synchronization +between the flagella. Such coordination enables efficient swimming and steering of the cells +and takes basis on the fibrous connections between flagellar bases (31,32). Intriguingly, in the +coordinated beating, both flagella display dynamic properties, i.e., flagellar waveform, beating +frequency (∼50 Hz), and frequency fluctuation, that are more similar to those of the cis flag- +ellum (26, 28–30, 33). This has led to a long-standing hypothesis that ”the cis somehow tunes +the trans flagellum” (26). This implies that the symmetric flagellar beating (”breast-stroke”) +observed is the result of interactions between two flagella playing differential roles in coordi- +nation. How does the basal coupling make this possible? Recent theoretical efforts show that +the basal coupling can give rise to different synchronization modes (34–36); and that flagellar +dynamics, such as beating frequency, may simply emerge from the interplay between mechan- +ics of basal coupling and bio-activity (36). Yet, most theoretical efforts examining flagellar +synchronization have assumed two identical flagella, limiting the results’ implication for the +realistic case. Moreover, little experiments directly probe the flagella’s differential roles during +synchronous beating (37). Therefore, flagellar coordination in this model organism remains un- +clear. To clarify the picture experimentally, one needs to selectively force each flagellum, and +characterize the dynamics of the flagellar response. +In this study, we address this challenge and devise a non-invasive approach to apply external +forces selectively on the cis- or the trans-flagella. Oscillatory background flows are imposed +along an angle with respect to the cell’s symmetry axis. Such flows result in controlled hydro- +dynamic forces, which are markedly different on the two flagella. With experiments, hydrody- +namic computations, and modeling, we show definitively that the two flagella are unilaterally +coupled, such that the younger flagellum (cis) coordinates the beating, whereas the elder one +simply copies the dynamic properties of the younger. This also means that only external forces +on the cis may mechanically fine-tune the coordination. We also study the effect of calcium +in the cis’ leading role as calcium is deeply involved in flagellar asymmetry and hence photo- +tactic steering. In addition, a well-known mutant that lacks flagellar dominance (ptx1) (23,38) +is examined. Results show that the coordinating role of cis does not need environmental free +calcium, whereas it does require the genes lost in ptx1. Our results discern the differential roles +of CR’s flagella, highlight an advanced function of the inter-flagellar mechanical coupling, and +have implications for biflagellates’ tactic motility. +4 + +Experimental scheme for selective loading +We set out to establish a non-invasive experimental technique that exerts differential loads on the +flagella of CR. Following the study by Quaranta et al. (31), we induce oscillatory background +flows to exert hydrodynamic forcing to flagella of captured cells. With programmed oscillations +of the piezoelectric stage, the amplitude, frequency, and direction of the background flows are +all controlled, enabling selective loading. +To quantitatively estimate the selectivity of the flows along different angles (θ), we compute +the flagellar loads under the flows along θ = −45◦, 0◦, and 45◦, see Fig. 1A. Computations +based on boundary element methods (BEM) and slender-body theory (SBT) give the real-time +drag force F on each flagellum and the power P exerted by the viscous forces on each flagellum. +For given realistic flagellar shapes, we compare computed loads with and without external flows. +From these we isolate the loads from the induced flows FFlow and PFlow (Methods). +Loads on each flagellum under flows of θ = 0◦, −45◦, 45◦ are presented in Fig. 2. Upper +panels display the magnitude of the drag force FFlow = |FFlow|; while lower panels show viscous +power PFlow. Force magnitudes are scaled by F0 = 6πµRU0 = 9.9 pN; while the powers by +P0 = F0U0 = 1.1 fW. F0 is the Stokes drag on a typical free-swimming cell (radius R = 5 µm, +speed U0 = 110 µm/s, water viscosity µ = 0.95 mPa·s). +Evidently, along θ = 0◦, flows load the flagella equally (Fig. 2A). However, at θ = −45◦, +flows load the cis flagellum ∼ 2 times larger than the trans (Fig. 2B, F c +Flow ≈ 2F t +Flow); whereas +flows at θ = 45◦ do the opposite (Fig. 2C). The selectivity also manifests in (the absolute values +of) PFlow. We do notice that flows along θ = +45◦ are able to synchronize the flagella with +PFlow < 0, meaning that the flagella are working against the flows, and this shall be discussed +in later sections. +Hereon forward, we refer to θc-flows, flows for which θ = −45◦ and the cis-flagellum is +selectively loaded. Likewise, θt-flows denote flows on θ = +45◦ that selectively load the trans. +θa-flows denote the axial flow along θ = 0◦. We next introduce how we quantify the flows’ +effective forcing strength (ε) on the cell. +Phase dynamics of flagellar beating is extracted from videography (31,39,40). Recordings +are masked and thresholded to highlight the flagella (Fig. 1B-C). Then the mean pixel values +over time within two sampling windows (Fig. 1D) are converted to observable-invariant flagellar +phases (41), Fig. 1E. Throughout this study, as cis and trans always beat synchronously (Fig. 1E +inset), their phases ϕc,t are used interchangeably as the flagellar phase ϕ. The flagellar phase +5 + +dynamics under external periodic forcing is described by Adler equation (42–44): +d∆ϕ +dt += −2πν − 2πε sin(∆ϕ) + ζ(t). +(1) +∆ϕ = ϕ − 2πfft is the phase difference between the beating and the forcing, with ff the +forcing frequency, and ε the forcing strength. The detuning ν = ff − f0 is the frequency +mismatch between the beating (f0) and forcing. ζ(t) represents a white noise that satisfies +⟨ζ(τ + t)ζ(τ)⟩ = 2Teffδ(t), with Teff an effective temperature and δ(t) the Dirac delta function. +When the forcing strength outweighs the detuning (ε > |ν|), synchronization with the flow +(d∆ϕ/dt = 0) emerges, see the plateaus marked black in Fig. 1F. We characterize synchro- +nization with τ = tsync/ttot, where tsync is the total time of flow synchronization and ttot the +flow duration. Fig. 1F presents the phase dynamics which are representative and range from: +no synchronization (τ=0, i), unstable synchronization (0 < τ < 1, ii-iii), and stable synchro- +nization (τ=1, iv). In this study, the frequency range in ν for which τ ≥ 0.5 is used to measure +ε (see Fig. 1F inset). This method is equivalent to previous fitting-based methods (28, 31), see +SM. Sec.S1. +Asymmetric susceptibility to flow synchronization +Now we examine cell responses to flows of various amplitudes and along different directions. +First we explore flow synchronization over a broad range of amplitudes and frequencies. θa- +flows with frequencies ff ∈ [40, 75] Hz and amplitudes U ∈ [390, 2340] µm/s are imposed. The +scanned range covers reported intrinsic frequencies of both the cis and trans flagellum (22,24, +26, 27); while the amplitude reaches the maximum instantaneous speed of a beating flagellum +(∼ 2000 µm/s). Fig. 3A displays the resultant flow-synchronized time fractions τ. Up until the +strongest flow amplitude, the large forces cannot disrupt the synchronized flagellar beating. In +addition, synchronization is never established around frequencies other than f0. This shows that +the inter-flagella coupling is much stronger than the maximum amplitude of forcing. +Next we examine the synchronization with the θc-flows and θt-flows. Flows of a fixed +amplitude (∼ 7U0) but varying frequencies around f0 are applied to each captured cell (see +Methods). With these, the flow-synchronized time fraction τ as a function of the detuning (ν) +and flow direction (θc,a,t) is recorded and helps quantify the flows’ effective forcing ε(θ). +Comparing τ(ν; θc) to τ(ν; θt), with τ(ν; θa) as reference, we find that θc-flows are the most +effective in synchronizing the beating (Fig. 3B). We illustrate this point with the profiles of an +6 + +exemplary cell (Fig. 3B inset). First, although both the θc-flow (red) and the θt-flow (blue) +can synchronize the cell at small detunings (|ν| <0.5Hz), the θc-flow maintains the synchro- +nization for the whole time ( τ(θc) =1), while the θt-flow for a slightly smaller time fraction +( τ(θc) ≈0.85). This is due to phase-slips (step-like changes in ∆ϕ(t) in Fig. 1F) between flag- +ella and the flow, and means that the θt-flow synchronization is less stable. Additionally, for +intermediate detuning (0.5 Hz< |ν| <4 Hz), τ(θc) is always larger than τ(θt) . In some cases, +the θc-flow synchronizes the cell fully whereas the θt-flow fails completely (e.g., at ν = −2 +Hz). Together, these results imply that a flow of given amplitude synchronizes flagellar beating +more effectively if it selectively loads the cis. +We repeat the experiments with cells from multiple cultures, captured on different pipettes, +and with different eyespot orientations (∼50% heading rightward in the imaging plane) to rule +out possible influence from the setup. τ(ν; θ) of N=11 wt cells tested in the TRIS-minimal +medium (pH=7.0) are displayed in Fig. 3B (labeled as ”TRIS”). On average, ε(θc) = 2.9 Hz +and is 70% larger than ε(θt) = 1.7 Hz. It bears emphasis that ε(θc) > ε(θt) holds true for every +single cell tested (11/11). In Fig. 3C, we show this by representing each cell as a point in the +ε(θc) - ε(θt) plane. A point being below the first bisector line ( ε(θc) = ε(θt) ) indicates that +ε(θc) > ε(θt) for this cell. All cells cluster clearly below the line. This asymmetry manifest +equivalently through τ. In Fig. 3D, each point represents the time fractions of the same cell +synchronized by the θc-flow and the θt-flow at the same frequency. Most points (>90%) are +below the first bisector line, meaning that τ(θc) > τ(θt) . Altogether, all results show that +selectively loading the cis flagellum establishes synchronization with the flow more effectively, +pointing to cis and trans playing differential roles in the coordinated beating. +We next study whether this newly observed cis-trans asymmetry is affected by calcium +depletion. Calcium is a critical second messenger for modulating flagellates motility and is +deeply involved in phototaxis (45). The depletion of the free environmental calcium is known +to degrade flagellar synchronization and exacerbate flagellar asymmetry (22). Here we focus +on whether calcium depletion affects the asymmetry ε(θc) > ε(θt) . We deplete environmental +calcium by EGTA-chelation, following the protocol in Ref. (46). Similar to previous reports (22, +47), the number of freely swimming cells drops significantly in EGTA-containing medium. +However, the remaining cells beat synchronously for hours after capture. For these beating cells, +calcium depletion is first confirmed by characterizing their deflagellation behavior. Indeed, +calcium depletion is reported to inhibit deflagellation (28, 48). In experiments with standard +calcium concentration, all cells deflagellated under pipette suction (20/20). For experiments +7 + +conducted in calcium depleting EGTA-containing medium, we observe deflagellation to occur +in none but one cell (1/19). +After confirming the calcium depletion, we perform the same sets of flow synchronization +experiments. The dashed lines in Fig. 3B show the median synchronization profiles τ(ν; θ) +(N=6 cells, labeled as ”EGTA”). The flagellar asymmetry is unaffected, see also Fig. 3E. Note +that ε(θc) > ε(θt) again applies for every single cell tested. The mean values of ε drop slightly. +However, the different effectiveness between θc-flows and θt-flows, ε(θc) − ε(θt) , is not af- +fected, see Fig. 3E inset. +Finally, we determine how the forcing strength of the flow depends on the hydrodynamic +forces exerted by the flow on the flagella. We compute the hydrodynamic beat-averaged loads, +F Flow = +� 2π +0 +FFlowdϕ/2π, P Flow = +� 2π +0 +PFlowdϕ/2π, induced by the flow on the trans and on +the cis flagella, see the horizontal lines in Fig. 2. These loads are computed for the θc-flow, +θt-flow, θa-flow and we also include experiments and computations performed with flows along +θ = 90◦ (circles), see SM. Sec.S2. Fig. 3F and G represent ε as a function of the loads on the +cis and trans flagellum respectively, with each symbol representing one of the four different +flow directions, see the drawings. We find that the effective forcing strength scales with the +time-averaged drag on the cis, ε ∼ F +c +Flow, while we find no such correlation between ε and +F +t +Flow. The linear relation between ε and F +c +Flow has an intercept near zero (ε|F c +Flow=0 ≈ 0). +Given the total forces on both flagella (F +c +Flow + F +t +Flow) for these flows remains almost constant +(0.74-0.79F0), the zero-intercept implies that for a hypothetical flow that exerts no load on the +cis but solely forces the trans, it will not be able to synchronize the cell at all. This suggests a +negligible contribution of the forcing on the trans in establishing synchronization with flows. +The asymmetry is lost in ptx1 mutants +Furthermore, we examine the flagellar dominance mutant ptx1. In this mutant, both flagella re- +spond similarly to changes of calcium concentrations (38) and have similar beating frequencies +when demembranated and reactivated (23). +Ptx1 mutants have two modes of coordinated beating, namely, the in-phase (IP) synchro- +nization and the anti-phase (AP) synchronization (29, 49). First, we apply θa-flow in the same +frequency and amplitude ranges as for wt cells. We find that the IP mode around f0 ≈ 50 Hz +is the only mode that can be synchronized by external flows. We focus on this mode and report +τ as τ = tsync/tIP for this mutant, where tIP is the total time of IP-beating under the applied +8 + +flows, see Fig. 4A. Synchronization profiles τ(ν; θ) of ptx1 are shown in Fig. 4B. The median +profiles are of similar width and height, indistinguishable from each other, and hence indicate +a loss of asymmetric susceptibility to flow synchronization. The loss is further confirmed by +the extracted ε(θ) (31) and τ(θ) (Fig. 4C-D). Cells and synchronization attempts are distributed +evenly across the first bisector lines (7/14 cells are below ε(θc) = ε(θt) in Fig. 4C, and ∼50% +points are below τ(θc) = τ(θt) in Fig. 4D). Altogether, all results show consistently that the +asymmetry is lost in ptx1. +Modeling +Framework +To investigate the implications of our experimental results on the coupling between flagella +and their dynamics, we develop a model for the system (SM. Sec.S3), representing flagella and +external flows as oscillators with directional couplings: +� +� +� +� +� +˙ϕf = 2πff +˙ϕc = 2π[fc − λt sin(ϕc-ϕt) − εc sin(ϕc-ϕf)] + ζc(t) +˙ϕt = 2π[ft − λc sin(ϕt-ϕc) − εt sin(ϕt-ϕf)] + ζt(t). +(2) +ϕf,c,t(t) respectively represent the phase of the flow, the cis, and the trans flagellum. ff,c,t +represents the inherent frequency of the forcing (flow), the cis, and the trans respectively. The +phase dynamics of each flagellum is modulated by its interactions with the other flagellum as +well as the background flow. Take the cis ( ˙ϕc) for example, the effect of the trans and the forcing +on the cis are respectively accounted for by the λt-term and the εc-term, see Eq. (2). In other +words, λt and εc measure the sensitivity of the actual cis-frequency to the phase differences +between oscillators (ϕc − ϕt,f), see the arrows in Fig. 5A. Lastly, ζc,t represent the white noise +of the cis and trans flagellum respectively. In the following parts, without loss of generality, the +noise are assumed equally strong and uncorrelated (⟨ζ2 +c ⟩ = ⟨ζ2 +t ⟩, or T c +eff = T t +eff). Nuanced phase +dynamics under differential noise levels can be found in SM. Sec.S4. +Eq. (2) can be readily reduced to Eq. (1), which allows us to write the experimentally mea- +sured values (f0, ε(θ), Teff) analytically with εc,t, λc,t, and ζc,t. The asymptotic behavior of the +9 + +model under the condition ϕc ≈ ϕt ≈ ϕf are (SM. Sec.S3): +� +� +� +� +� +f0 += αfc + (1 − α)ft, +Teff += α2T c +eff + (1 − α)2T t +eff, +ε(θ) += αεc(θ) + (1 − α)εt(θ), +(3) +with α = λc/(λc + λt) representing the dominance of cis. It is then clear that when α ≈ 1, the +coordinated beating will display dynamic properties of the cis flagellum. +Fig. 5A illustrates an exemplary modeling scheme describing flagellar beating subjected +to θc-flows. The direction and thickness of arrows represent coupling direction and strength +respectively. The selective loading on the cis is represented by εc > εt; while λc > λt reflects +that the cis has a more dominant role in the coordinated beating. We run Monte-Carlo simulation +with Eq. (2) using customized MATLAB scripts. +Coordinated beating under symmetric forcing +We first model the flow synchronization induced by θa-flow (symmetric flagellar loads). In this +case, ε(θ) = αεc(θ) + (1 − α)εt(θ) (Eq. (3)) reduces to ε = εc,t and is independent of α. We +set εc,t as 2.4 Hz to match the measured ε(θa) =2.4 Hz (Fig. 3B). +At similar detunings as in the experimental results in Fig. 1F, our Monte-Carlo simulations +reproduces the phase dynamics with: (i) no flow synchronization, (ii-iii) unstable synchroniza- +tion, and (iv) stable synchronization (Fig. 5B). Repeating the simulations for varying forcing +strength ε (= εc,t) and frequency ff yields Arnold tongue diagrams in agreement with those +reported from our experiments. The Arnold Tongue for wt in Fig. 3A and ptx1 in Fig. 4A are +reproduced with simulations shown in Fig. 5C and D respectively. The only parameter value +changed between Fig. 5C and D is the level of noise (T c,t +eff ), which is increased by an order of +magnitude. The differences in phase dynamics between wt and ptx1, when subjected to sym- +metric external loading, are therefore accounted by solely varying the noise. +Coordinated beating under selective loading +We next model flow synchronization by the θc-flows and the θt-flows. The selective forcing +(εc ̸= εt) allows the effect of flagellar dominance (λc ̸= λt) to manifest in the effective forcing +strength ε(θ) and hence in the synchronization profiles τ(ν; θ), Fig. 5E. Similar to our exper- +imental observations, θc-flow synchronizes the coordinated beating over the broadest range of +ν (i.e. largest ε). This is directly attributed to the dominance λc > λt: by setting λc = λt, +10 + +the differences between τ(θc) and τ(θt) disappear even under selective loading (Fig. 5E inset). +Fig. 5F details how the asymmetry of inter-flagellar coupling (λc/λt) affects the asymmetry +between τ(θc) and τ(θt) . The open symbols represent ε(θ) measured from modeled τ(ν; θ) +and the lines represent Eq. (3). The difference between ε(θc) and ε(θt) increases with λc/λt, +and they each saturates to reflect only the forcing on the cis (εc, the grey dashed lines). With +fc = 45 Hz, ft = 65 Hz (23, 26), and f0 ≈ 50 Hz, we deduce from Eq. (3) that λc = 4λt for +wt cells. For wt cells under calcium depletion, experimental results are reproduced with a lower +total forcing strength (Fig. 5G). εc + εt is set to 4.08 Hz (15% lower) to reflect the 7% − 20% +decrease in ε(θ) induced by calcium depletion. +The ptx1 results are reproduced with a stronger noise (T c,t +eff = 9.42 rad2/s) and a symmetric +inter-flagellar coupling λc/λt = 1, see Fig. 5H and Table. 1. Both changes are necessary for +reproducing the synchronization profiles of ptx1 in Fig. 5H: while the stronger noise lowers +the maximal values of τ(θ, ν), setting λc/λt = 4 would still result in τ(θc) > τ(θt) in the +central range (|ν| ≲ 2.4 Hz). Finally, it is noteworthy that the noise in ptx1 increases not only +because a higher noise value for individual flagella, but also because the cis-trans coupling has +become symmetric. As shown by Eq. (3), the unilateral coupling promotes not only the cis- +frequency in the synchrony but also the cis-noise. Given T c +eff ≪ T t +eff and λc = 4λt, we confirm +with simulations that the cis stabilizes the beating frequency of the trans and decreases its +beating noise. The simulations are in good agreement with experimental noise measurements, +see SM. Sec.S4 for details. +Discussion +The two flagella of C. reinhardtii have long been known to have inherently different dynamic +properties such as frequency, waveform, level of active noise, and responses to second messen- +gers (23,25,26,29,30). Intriguingly, when connected by basal fibers and beating synchronously, +they both adopt the kinematics of the cis-(eyespot) flagellum, which led to the assumption that +the flagella may have differential roles in coordination. In this work, we test this hypothesis by +employing oscillatory flows applied from an angle with respect to the cells’ symmetry axis and +thus exert biased loads on one flagellum. +Without an exception, in wt cells, θc-flows, the ones that selectively load the cis flagellum, +are always more effective in synchronizing the flagellar beating than the θt-flows. This is shown +by the larger effective forcing strengths ( ε(θc) > ε(θt) , Fig. 3B-C) and larger synchronized time +11 + +fractions ( τ(θc) > τ(θt) , Fig. 3D). Mapping the measured forcing strength ε(θ) as a function +of the loads, we find empirically that ε ∝ F +c +Flow (Fig. 3F) and that trans-loads appear to mat- +ter negligibly. These observations all indicate that the cis-loads determine whether an external +forcing can synchronize the cell. Moreover, this point is further highlighted by an unexpected +finding: when θt-flows are applied, the trans flagellum always beats against the external flow +(P t +Flow < 0) and the only stabilizing factor for flow synchronization is the cis flagellum working +along with the flow during the recovery stroke (Fig. 2C lower panel). These observations defini- +tively prove that the two flagella have differential roles in the coordination and interestingly +imply that flagella are coupled to external flow only through the cis. +To have a mechanistic understanding of this finding, we model the system with Eq. (2). In +the model, selective hydrodynamic loading and flagellar dominance in the coordinated beating +are respectively represented by εc ̸= εt and λc ̸= λt. Setting out from the model, we obtain +closed-form expressions for observables such as f0 and ε (Eq. (3)), which illustrate how flag- +ellar dominance and selective loading affect the coordinated flagellar beating. Moreover, with +Monte-Carlo simulation, we clarified the interplay between flows and flagella (SM. Sec.S3), +and reproduces all experimental observations. +With the model, we show that a ”dominance” of the cis (λc > λt) is sufficient to explain +why the coordinated flagellar beating bears the frequency and the noise level of the cis flag- +ellum. In the model, such dominance means that the cis-phase is much less sensitive to the +trans-phase than the other way around. We then reproduce the phase dynamics of flow synchro- +nization at varying detunings (Fig. 5B), amplitudes (Fig. 5C), and noise (Fig. 5D). Exploiting +the observation that the coordination between flagella cannot be broken by external flows up +to the strongest ones tested (εmax ∼ 10 Hz, Fig. 3A), we quantify the lower limit of the total +basal coupling, λc + λt, to be approximately 40 Hz (deduced in SM. Sec.S3), which is an order +magnitude larger than the hydrodynamic inter-flagellar coupling (31,50–52). +The modulation of flagellar dominance mediates tactic behaviors (22, 23, 38, 47). Calcium +is hypothesized to be underlying the modulation of dominance, as it causes the connecting +fiber between flagella to contract (53), modulates the cis- and trans activity (e.g. beating am- +plitude) differentially (22), and calcium influx comprises the initial step of CR’s photo- (54) +and mechanoresponses (45). We therefore investigate flagellar coupling in the context of tactic +steering by depleting the environmental free calcium and hence inhibiting signals of calcium +influxes. Cells are first acclimated to calcium depletion, and then tested with the directional +flows. Our results show that the cis dominance does not require the involvement of free envi- +12 + +ronmental calcium. Calcium depletion merely induces an overall drop in the forcing strength +perceived by the cell ε(θ) (7% − 20%), which is captured by reducing εc + εt for 15% (mean +drop) in the model (Fig. 5G). Together, our results indicate that the leading role of cis, is an +inherent property, that does not require active influx of external calcium, and possibly reflects +an intrinsic mechanical asymmetry of the cellular mesh that anchors the two flagella into the +cell body. +In ptx1 cells, a lack of flagellar dominance (λc = λt) and a stronger noise level help repro- +duce our experimental observations. Previous studies suggested that both flagella of ptx1 are +similar to the wildtype trans (23), and that the noise levels of this mutant’s synchronous beating +are much greater than those of wt (29) (see also SM. Sec.S4). If both flagella and their anchoring +roots indeed have the composition of the wildtype trans, such symmetry would predict λc = λt. +This symmetric coupling renders the noise of ptx1 Teff = T t +eff (Eq. (3)), which is about an order +of magnitude larger than the noise of wt Teff ≈ T c +eff. +The comparison between ptx1 and wt highlights an intriguing advantage of the observed +unilateral coupling (λc ≫ λt); that is, it strongly suppresses the high noise of the trans. Consid- +ering that the trans is richer in CAH6 protein and this protein’s possible role in inorganic carbon +sensing (14,20), the potential sensing role of the trans is worth noticing. Assuming the strong +noise present in the trans originates from the biochemical processes related to sensing, then +the unilateral coupling effectively prevents such noise from perturbing the cell’s synchronous +beating and effective swimming. In this way, the asymmetric coupling may combine the benefit +of having a stable cis as the driver while equipping a noisy trans as a sensor. +Material and methods +Cell culture +CR wildtype (wt) strain cc125 (mt+) and flagellar dominance mutant ptx1 cc2894 (mt+) are +cultured in TRIS-minimal medium (pH=7.0) with sterile air bubbling, in a 14h/10h day-night +cycle. Experiments are performed on the 4th day after inoculating the liquid culture, when the +culture is still in the exponential growth phase and has a concentration of ∼ 2 × 105 cells/ml. +Before experiments, cells are collected and resuspended in fresh TRIS-minimal (pH=7.0). +13 + +Calcium depletion +In calcium depletion assays, cells are cultured in the same fashion as mentioned above but +washed and resuspended in fresh TRIS-minimal medium + 0.5 mM EGTA (pH=7.0). Free +calcium concentration is estimated to drop from 0.33 mM in the TRIS-minimal medium, to +0.01 µM in the altered medium (46). Experiments start at least one hour after the resuspension +in order to acclimate the cells. +Experimental setup +Single cells of CR are studied following a protocol similar to the one described in (31). Cell +suspensions are filled into a customized flow chamber with an opening on one side. The air- +water interface on that side is pinned on all edges and is sealed with silicone oil. A micropipette +held by micromanipulator (SYS-HS6, WPI) enters the chamber and captures single cells by as- +piration. The manipulator and the captured cell remain stationary in the lab frame of reference, +while the flow chamber and the fluid therein are oscillated by a piezoelectric stage (Nano-Drive, +Mad City Labs), such that external flows are applied to the cell. Frequencies and amplitudes of +the oscillations are individually calibrated by tracking micro-beads in the chamber. Bright field +microscopy is performed on an inverted microscope (Nikon Eclipse Ti-U, 60× water immersion +objective). Videos are recorded with a sCMOS camera (LaVision PCO.edge) at 600-1000 Hz. +Measurement scheme +The flagellar beating of each tested cell is recorded before, during, and after the application +of the flows. We measure the cell’s average beating frequency f0 over 2 s (∼100 beats). For +ptx1 cells, f0 is reported for the in-phase (IP) synchronous beating. Unless otherwise stated, +directional flows (θ = 0, ±45◦) are of the same amplitude (780±50 µm/s, mean±std), similar +to those used in Ref. (31). Flow frequencies ff are scanned over [f0 − 7, f0 + 7] Hz for each +group of directional flows. +Computation of the flagellar loads +To quantify the hydrodynamic forces on the flagella, we first track realistic flagellar deforma- +tion from videos wherein background flows are applied. Then we employ a hybrid method +combining boundary element method (BEM) and slender-body theory (40, 55) to compute the +14 + +drag forces exerted on each flagellum and the forces’ rates of work. In this approach, each flag- +ellum is represented as a slender-body (55) with 26 discrete points along its centerline and the +time-dependent velocity of each of the 26 points is calculated by its displacement across frames. +The cell body and the pipette used to capture the cell are represented as one entity with a com- +pleted double layer boundary integral equation (56). Stresslet are distributed on cell-pipette’s +surface; while stokeslet and rotlet of the completion flow are distributed along cell-pipette’s +centerline (57). The no-slip boundary condition on the cell-pipette surface is satisfied at col- +location points. Lastly, stokeslets are distributed along the centerlines of the flagella, so that +no-slip boundary conditions are met on their surfaces. Integrating the distribution of stokeslets +f(s) over a flagellar shape, one obtains the total drag force F = +� +f(s)ds is obtained. Similarly, +the force’s rate of work is computed as P = +� +f(s) · U(s)ds, where U(s) is the velocity of the +flagellum at the position s along the centerline. +The computations shown in this study are based on videos of a representative cell which +originally beats at ∼50 Hz. The cell is fully synchronized by flows along different directions +(θ = 0◦, ±45◦ and 90◦) at 49.2 Hz. In the computations, the applied flows are set to have an +amplitude of 780 µm/s to reflect the experiments. Computations begin with the onset of the +background flows (notified experimentally by a flashlight event), and last for ∼30 beats (500 +frames sampled at 801 fps). Additionally, we confirm the results of θt-flow-synchronization, +that both flagella spend large fractions of time beating against the flows, with other cells and +with θt-flows at other frequencies. +Isolate loads of external flows +The total loads (F and P) computed consist of two parts, one from the flow created by the two +flagella themselves and the other from the applied flow. In the low Reynolds number regime, +the loads of the two parts add up directly (linearity): F = FSelf + FFlow, and P = PSelf + PFlow. +To isolate FFlow and PFlow, we compute F′ = FSelf and P ′ = PSelf by running the computation +again but without the external flows, and obtain FFlow = F − F′ and PFlow = P − P ′. +Modeling parameters +We assume the flagellar intrinsic frequencies fc and ft to be 45 Hz and 65 Hz respectively +(23, 26, 28). On this basis, λc : λt is assumed to be 4:1 to account for the observed f0 (∼ 50 +Hz). εc : εt is set as 2:1, 1:1, and 1:2 for the θc-flows, the θa-flows, and the θt-flows respectively, +15 + +see Fig. 2A-C. Additionally, εc + εt is assumed to be constant to reflect the fact that F +c +Flow + +F +t +Flow approximately does not vary with flow directions. We take a typical value of T c,t +eff = +1.57 rad2/s (31). The sum of inter-flagellar coupling λtot = λc + λt is set to be large enough, +i.e., λtot = 3νct with νct = |ft − fc|, to account for the fact that: 1) the coordinated beating +is approximated in-phase, and 2) up until the strongest flow applied, the coordinated beating +cannot be broken (quantitative evaluation is detailed in SM. Sec.S3). To model wt cells under +calcium depletion, we decrease εc + εt by 15% - which is the mean decrease in the observed +ε(θc) , ε(θa) , and ε(θt) (Fig. 3E). For ptx1 cells, we assume a symmetric inter-flagellar coupling +(λc = λt) and a stronger noise level (SM. Sec.S4). The parameters are summarized in Table. 1. +Table 1: Modeling parameters +variable +symbol (unit) +TRIS +EGTA +ptx1 +Intrinsic freq. (23,26) +fc, ft (Hz) +45,65 +45,65 +45,65 +Basal coupling∗ +λc + λt (Hz) +60 +60 +60 +cis dominance (23,38) +λc : λt (-) +4:1 +4:1 +1:1 +Flow detuning +ν (Hz) +[-10,10] +[-10,10] +[-10,10] +Total forcing (51) +εc + εt (Hz) +4.8 +4.08 +4.8 +Noise∗ (31) +T c,t +eff (rad2/s) +1.57 +1.57 +9.42 +∗ detailed in SM. Sec.S3 +16 + +References +1. Berg, H. C. & Brown, D. A. Chemotaxis in escherichia coli analysed by three-dimensional +tracking. Nature 239, 500–504 (1972). +2. Smriga, S., Fernandez, V. I., Mitchell, J. G. & Stocker, R. Chemotaxis toward phytoplank- +ton drives organic matter partitioning among marine bacteria. Proceedings of the National +Academy of Sciences 113, 1576–1581 (2016). +3. Hegemann, P. & Berthold, P. Chapter 13 - sensory photoreceptors and light control of +flagellar activity. In Harris, E. H., Stern, D. B. & Witman, G. B. (eds.) The Chlamydomonas +Sourcebook (Second Edition), vol. 3, 395–429 (Academic Press, London, 2009). +4. Ueki, N., Matsunaga, S., Inouye, I. & Hallmann, A. How 5000 independent rowers coor- +dinate their strokes in order to row into the sunlight: Phototaxis in the multicellular green +alga volvox. BMC Biology 8, 103 (2010). +5. Stehnach, M. R., Waisbord, N., Walkama, D. M. & Guasto, J. S. Viscophobic turning +dictates microalgae transport in viscosity gradients. Nature Physics 17, 926–930 (2021). +6. Brokaw, C., Josslin, R. & Bobrow, L. Calcium ion regulation of flagellar beat symmetry in +reactivated sea urchin spermatozoa. Biochemical and Biophysical Research Communica- +tions 58, 795–800 (1974). +7. Gong, A. et al. The steering gaits of sperm. Philosophical Transactions of the Royal Society +B: Biological Sciences 375, 20190149 (2020). +8. Gadˆelha, H., Hern´andez-Herrera, P., Montoya, F., Darszon, A. & Corkidi, G. Human sperm +uses asymmetric and anisotropic flagellar controls to regulate swimming symmetry and cell +steering. Science Advances 6, eaba5168 (2020). +9. Bennett, R. R. & Golestanian, R. A steering mechanism for phototaxis in Chlamydomonas. +Journal of The Royal Society Interface 12, 20141164 (2015). +10. R¨uffer, U. & Nultsch, W. Flagellar photoresponses of Chlamydomonas cells held on mi- +cropipettes: II. Change in Flagellar Beat Pattern. Cell Motility and the Cytoskeleton 18, +269–278 (1991). +17 + +11. Ueki, N. & Wakabayashi, K.-i. Dynein-mediated photobehavioral responses in Chlamy- +domonas. In Dyneins, 368–385 (Elsevier Ltd., 2017), second edition edn. +12. Naitoh, Y. & Kaneko, H. Reactivated triton-extracted models of paramecium: Modification +of ciliary movement by calcium ions. Science 176, 523–524 (1972). +13. Merchant, S. S. et al. The Chlamydomonas genome reveals the evolution of key animal and +plant functions. Science 318, 245–250 (2007). +14. Choi, H. I., Kim, J. Y. H., Kwak, H. S., Sung, Y. J. & Sim, S. J. Quantitative analysis of the +chemotaxis of a green alga, Chlamydomonas reinhardtii, to bicarbonate using diffusion- +based microfluidic device. Biomicrofluidics 10, 014121–014121 (2016). +15. R¨uffer, U. & Nultsch, W. Flagellar photoresponses of Chlamydomonas cells held on mi- +cropipettes: I. change in flagellar beat frequency. Cell Motility and the Cytoskeleton 15, +162–167 (1990). +16. Holmes, J. & Dutcher, S. Cellular asymmetry in C. reinhardtii. Journal of Cell Science 94, +273–285 (1989). +17. Dutcher, S. K. & O’Toole, E. T. The basal bodies of C. reinhardtii. Cilia 5, 1–7 (2016). +18. Mittelmeier, T. M., Boyd, J. S., Lamb, M. R. & Dieckmann, C. L. Asymmetric properties of +the Chlamydomonas reinhardtii cytoskeleton direct rhodopsin photoreceptor localization. +Journal of Cell Biology 193, 741–753 (2011). +19. Sakakibara, H., Mitchell, D. R. & Kamiya, R. A Chlamydomonas outer arm dynein mutant +missing the alpha heavy chain. The Journal of cell biology 113, 615–622 (1991). +20. Mackinder, L. C. et al. A spatial interactome reveals the protein organization of the algal +co2-concentrating mechanism. Cell 171, 133–147.e14 (2017). +21. Yu, K., Liu, P., Venkatachalam, D., Hopkinson, B. M. & Lechtreck, K. F. The bbsome +restricts entry of tagged carbonic anhydrase 6 into the cis-flagellum of chlamydomonas +reinhardtii. PLOS ONE 15, 1–21 (2020). +22. Kamiya, R. & Witman, G. B. Submicromolar levels of calcium control the balance of +beating between the two flagella in demembranated models of Chlamydomonas. Journal +of Cell Biology 98, 97–107 (1984). +18 + +23. Okita, N., Isogai, N., Hirono, M., Kamiya, R. & Yoshimura, K. Phototactic activity in +Chlamydomonas ’non-phototactic’ mutants deficient in Ca2+-dependent control of flagellar +dominance or in inner-arm dynein. Journal of Cell Science 118, 529–537 (2005). +24. Takada, S. & Kamiya, R. Beat frequency difference between the two flagella of Chlamy- +domonas depends on the attachment site of outer dynein arms on the outer-doublet micro- +tubules. Cell Motility and the Cytoskeleton 36, 68–75 (1997). +25. Saegusa, Y. & Yoshimura, K. cAMP controls the balance of the propulsive forces generated +by the two flagella of Chlamydomonas. Cytoskeleton 72, 412–421 (2015). +26. Kamiya, R. & Hasegawa, E. Intrinsic difference in beat frequency between the two flagella +of Chlamydomonas reinhardtii. Experimental Cell Research 173, 299–304 (1987). +27. R¨uffer, U. & Nultsch, W. Comparison of the beating of cis- and trans-flagella of Chlamy- +domonas cells held on micropipettes. Cell Motility and the Cytoskeleton 7, 87–93 (1987). +28. Wan, K. Y., Leptos, K. C. & Goldstein, R. E. Lag, lock, sync, slip: the many phases of +coupled flagella. Journal of The Royal Society Interface 11, 20131160 (2014). +29. Leptos, K. C. et al. Antiphase synchronization in a flagellar-dominance mutant of Chlamy- +domonas. Physical Review Letters 111, 1–5 (2013). +30. Wan, K. Y. Coordination of eukaryotic cilia and flagella. Essays In Biochemistry 62, 829– +838 (2018). +31. Quaranta, G., Aubin-Tam, M.-E. & Tam, D. Hydrodynamics versus intracellular coupling +in the synchronization of eukaryotic flagella. Physical Review Letters 115, 238101 (2015). +32. Wan, K. Y. & Goldstein, R. E. Coordinated beating of algal flagella is mediated by basal +coupling. Proceedings of the National Academy of Sciences 113, E2784–E2793 (2016). +33. R¨uffer, U. & Nultsch, W. +High-speed cinematographic analysis of the movement of +Chlamydomonas. Cell Motility 5, 251–263 (1985). +34. Klindt, G. S., Ruloff, C., Wagner, C. & Friedrich, B. M. In-phase and anti-phase flagellar +synchronization by waveform compliance and basal coupling. New Journal of Physics 19, +113052 (2017). +19 + +35. Liu, Y. et al. Transitions in synchronization states of model cilia through basal connection +coupling. Journal of The Royal Society Interface 15 (2018). +36. Guo, H., Man, Y., Wan, K. Y. & Kanso, E. Intracellular coupling modulates biflagellar +synchrony. Journal of The Royal Society Interface 18, 20200660 (2021). +37. Wan, K. Y. & Goldstein, R. E. Rhythmicity, recurrence, and recovery of flagellar beating. +Phys. Rev. Lett. 113, 238103 (2014). +38. Horst, J. & Witman, G. B. Ptx1, a nonphototactic mutant of Chlamydomonas, lacks control +of flagellar dominance. The Journal of Cell Biology 120, 733–741 (1993). +39. Wei, D., Dehnavi, P. G., Aubin-Tam, M.-E. & Tam, D. Is the zero reynolds number approx- +imation valid for ciliary flows? Physical Review Letters 122, 124502 (2019). +40. Wei, D., Dehnavi, P. G., Aubin-Tam, M.-E. & Tam, D. Measurements of the unsteady flow +field around beating cilia. Journal of Fluid Mechanics 915, A70 (2021). +41. Kralemann, B., Cimponeriu, L., Rosenblum, M., Pikovsky, A. & Mrowka, R. Phase dy- +namics of coupled oscillators reconstructed from data. Phys. Rev. E 77, 066205 (2008). +42. Pikovsky, A., Rosenblum, M. & Kurths, J. Synchronization: A Universal Concept in Non- +linear Sciences. Cambridge Nonlinear Science Series (Cambridge University Press, 2001). +43. Polin, M., Tuval, I., Drescher, K., Gollub, J. P. & Goldstein, R. E. Chlamydomonas swims +with two “gears” in a eukaryotic version of run-and-tumble locomotion. Science 325, 487– +490 (2009). +44. Friedrich, B. Hydrodynamic synchronization of flagellar oscillators. The European Physi- +cal Journal Special Topics 225, 2353–2368 (2016). +45. Yoshimura, K. Stimulus Perception and Membrane Excitation in Unicellular Alga Chlamy- +domonas, 79–91 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2011). +46. Wakabayashi, K.-i., Ide, T. & Kamiya, R. Calcium-dependent flagellar motility activation +in C. reinhardtii in response to mechanical agitation. Cell Motility and the Cytoskeleton +66, 736–742 (2009). +20 + +47. Pazour, G. J., Agrin, N., Leszyk, J. & Witman, G. B. Proteomic analysis of a eukaryotic +cilium. The Journal of Cell Biology 170, 103–113 (2005). +48. Quarmby, L. & Hartzell, H. Two distinct, calcium-mediated, signal transduction pathways +can trigger deflagellation in C. reinhardtii. +The Journal of Cell Biology 124, 807–815 +(1994). +49. R¨uffer, U. & Nultsch, W. +Flagellar coordination in Chlamydomonas cells held on mi- +cropipettes. Cell Motility and the Cytoskeleton 41, 297–307 (1998). +50. Brumley, D. R., Wan, K. Y., Polin, M. & Goldstein, R. E. Flagellar synchronization through +direct hydrodynamic interactions. eLife 3, e02750 (2014). +51. Klindt, G. S., Ruloff, C., Wagner, C. & Friedrich, B. M. Load response of the flagellar beat. +Physical Review Letters 258101, 1–5 (2016). +52. Pellicciotta, N. et al. Entrainment of mammalian motile cilia in the brain with hydrody- +namic forces. Proceedings of the National Academy of Sciences 117, 8315–8325 (2020). +53. Hayashi, M., Yagi, T., Yoshimura, K. & Kamiya, R. Real-time observation of Ca2+-induced +basal body reorientation in Chlamydomonas. Cell Motility and the Cytoskeleton 41, 49–56 +(1998). +54. Harz, H. & Hegemann, P. Rhodopsin-regulated calcium currents in Chlamydomonas. Na- +ture 351, 489–491 (1991). +55. Keller, J. B. & Rubinow, S. I. Slender-body theory for slow viscous flow. Journal of Fluid +Mechanics 75, 705–714 (1976). +56. Power, H. & Miranda, G. Second kind integral equation formulation of stokes’ flows past +a particle of arbitrary shape. SIAM Journal on Applied Mathematics 47, 689–698 (1987). +57. Keaveny, E. E. & Shelley, M. J. Applying a second-kind boundary integral equation for +surface tractions in stokes flow. Journal of Computational Physics 230, 2141 – 2159 (2011). +58. Kamiya, R. Analysis of cell vibration for assessing axonemal motility in Chlamydomonas. +Methods 22, 383–387 (2000). +21 + +Acknowledgments +The authors thank Roland Kieffer for technical support. D.W. thanks Ritsu Kamiya for helpful +discussions. The authors acknowledge support by the European Research Council (ERC starting +grants no. 716712 and no. 101042612). +Author Contributions +D.W. performed experiments, computations, designed the model, and drafted the manuscript. +G.Q. performed early experiments and obtained preliminary results. M.A. and D.T. conceived +the study, supervised the project and critically revised the manuscript. +Competing interests +Authors declare that they have no competing interests. +Supplementary materials +Supplementary Text +Figs. S1 to S5 +References (23,26,28,29,31,38,43,58) +22 + +Figure 1: Experimental workflow. (A) Captured CR cells are subjected to sinusoidal flows of +frequency ff along given angles (θ) in the xy-plane. Flows along θ = −45◦, 0◦, 45◦ of same +amplitude (780±50 µm/s, mean±std.) are used and termed as shown. (B-E) Extracting flagellar +phase ϕc and ϕt by image processing. Raw images (B) are thresholded and contrast-adjusted to +highlight the flagella (C). Mean pixel values within the user-defined interrogation windows (red +and blue circles) capture the raw phases of beating (D), which are then converted to observable- +independent phases (E). Inset: phase difference ϕc − ϕt. (F) Flagella-flow phase dynamics at +decreasing detuning ν = ff − f0 with f0 the cell’s beating frequency without external flow. +Traces i to iv are taken at detunings marked in the inset. Plateaus marked black represent +flow synchronization, whose time fractions τ = tsync/ttot are noted. ttot is the total time of +recording. Inset: the flow synchronization profile, τ(ν), reports the effective forcing strength +2ε by its width. +23 + +A +Oa- flow,0 +B +eyespot +Ot- flow, 45 +c -flow,-45° +A +(s/ur) +-780μm/s +D +D +t () +E +c +pt +(2元) +0.5 +0.5S +0 +4444442 +0.5 +0 +4 +8 +12 +16 +t (beat) +F 12 +IV +ii +() - ) +i, T=0 +T +2 +8 +ii +ii, T = 0.18 +1 +0 +-6 +0 +ii, T = 0.80 +v (Hz) +4 +iv, T = 1.00 +0 +0 +2 +4 +6 +8 +10 +t (s)Figure 2: External flagellar loads when beating is synchronized. Force magnitude (upper pan- +els) and power (lower panels) exerted by external flows of θ = 0◦ (A, θa-flow), −45◦ (B, +θc-flow), and +45◦ (C, θt-flow). The medians (solid lines) and interquartile ranges (shadings) +are computed over ∼20 synchronized beats. Dashed horizontal lines: loads averaged over a +synchronized beat. Force magnitudes and powers are scaled by F0=9.9 pN and P0=1.1 fW +respectively. Flagellar phase corresponds to the displayed shapes in the middle x-axis. +24 + +I cis loads + trans loads += Median +Interquartile +:.: Beat-averaged +A +B +FFlow/Fo +0.5 +0 +H +/Po +10 +0 +-10 +0元/2元3元/22元 +0元/2元3元/2 2元 +0元/2元3元/2 2元 +Flagellar phase (rad) +Flagellar phase (rad) +Flagellar phase (rad)Figure 3: Flow synchronization of wt cells. (A) Arnold tongue of a representative cell tested +with θa-flow. The contour is interpolated from N=132 measurements (6 equidistant amplitudes +× 22 equidistant frequencies), and color-coded by the entrained time fraction τ. (B) The syn- +chronization profiles τ(ν; θ) of a representative wt cell (inset), the median profile of the TRIS +group wt cells (N=11, solid lines) and the EGTA group (N=6, dashed lines), with either θc- +flows (red), θa-flows (yellow) or θt-flows (blue). Shaded areas are the interquartile ranges for +the TRIS group. (C) Tested wt cells represented on the ε(θc) − ε(θt) plane (TRIS group). Solid +line: the first bisector line (y = x). (D) Comparing τ(ν; θc) and τ(ν; θt) for each cell at each ap- +plied frequency. N=132 pairs of experiments are represented on the τ(θc) − τ(θt) plane. More +than 90% of them are below the first bisector line. (E) The coupling strengths ε(θ) of the TRIS +group (black) and the EGTA group (gray). Bars and error bars: mean and 1 std., respectively. +Inset: δε = ε(θc) − ε(θt) . NS: not significant, p>0.05, Kruskal-Wallis test, One-Way ANOVA. +Relations between the forcing strength ε and the loads on the cis (F) and the trans flagellum +(G). Markers represent different flow angles, see the drawings. +25 + +20 +C +4 +0 +(zH) +(10)3 +2 +8 +0 +0 +-10-5 +0 +510 +15 +2025 +0 +2 +4 +6 +v (Hz) +ε(0c) (Hz) +B +D +A single celi +4 +0 +4 +Median over population +T=1 +TRIS +0 +EGTA : +2 = 6.0 Hz +T +0 +t(Gc) (-) +1 +E +6 +(zH) +3NS +5.3 Hz +E +3 +3.8 Hz +m +0 +-8 +-4 +0 +4 +8 +v (Hz) +TRIS EGTA + cis loads + trans loads +F +4 +G 4 +2 +2 +口 +m +3 +0 +0 +0 +0.5 +0 +5 +0 +0.5 +0 +5 +FFlow/Fo +Plow/Po +FFlow/Fo +Pflow/PoFigure 4: The asymmetric susceptibility to flow synchronization is lost in the flagellar domi- +nance mutant ptx1. (A) Arnold tongue of a representative ptx1 cell tested with θa-flow. The +contour is interpolated from N=132 measurements (6 equidistant amplitudes × 22 equidistant +frequencies). Color bar: the entrained time fraction τ = tsync/tIP. (B) Flow synchronization +profiles τ(ν; θ) of N=14 ptx1 cells, tested with θc-flows (red), θa-flows (yellow) and θt-flows +(blue). (C) ε(θc) and ε(θt) of the tested cells. The first bisector line (solid): y = x. (D) +τ(ν; θc,t) for each cell at each applied frequency. N=154 points are present. +26 + +A +20 +6 +() n/n +口 +10 +(zH) +4 +口 +口 +口 +8 +口 +@2 +口 +口 +-10 +-5 +0 +5 +10 +15 +20 25 +口 +3 +v (Hz) +0 +B +Median +Interquartile +0 +2 +4 +6 +(0c) (Hz) +0 +(-) (0)1 +T +0 +1 +0 +0 +-8 +-4 +0 +4 +8 +0 +1 +v (Hz) +T(c) (-)Figure 5: Modeling the asymmetric flow synchronization. (A) Modeling scheme describing a +cell beating under directional flow (θc-flow as an example). Arrows represent the directional +coupling coefficients with line thickness representing the relative strength. For example, λc +points from cis to trans, representing how the latter (ϕc) is sensitive to the former (ϕt); mean- +while, the arrow of λc being thicker than λt means that ϕt is much more sensitive to ϕc than +the other way around. (B) Modeled phase dynamics of flow synchronization under θa-flows, +analogous to Fig. 1F. Reproducing the Arnold tongue diagrams at the noise level of wt (C) +and ptx1 (D), analogous to Fig. 3A and Fig. 4A respectively. (E) Flow synchronization profiles +τ(ν; θ) obtained experimentally (upper panel) and by modeling (lower panel). Inset: the mod- +eling results with symmetric inter-flagellar coupling. (F) Effective forcing strength ε(θ) as a +function of the inter-flagellar coupling asymmetry λc/λt. Points: measured from simulation; +lines: analytical approximation (Eq. (3)); dashed lines: εc respectively for the θc-flow, θa-flow, +and θt-flow (from top to bottom). (G) Reproducing the flow synchronization of wt cells under +calcium depletion (H) Reproducing results of ptx1. See Table. 1 for the modeling parameters. +27 + +A +B +trans +cis +(fe, pc) +12 +(ft, Pt) +111 +1 +ii +1 +() +fi, -0 +6 +0 +6 +Idh. +v (Hz) +Λt +ji, {-0.27 +i, t=0.85 +Et +Ec +iv, T=1.00 +(fr, Pf) +0 +0 +2 +4 +6 +8 +Induced flow (0=-45°) +t (s) +C +D +10 +10 +Ec,t (Hz) +(zH) +5 +Low noise level +Ec,t +High noise level +0 +0 +-10-5 +¥05101520 +25 +-10-5 +051015 20 25 +v (Hz) +v (Hz) +E +F +Exp, wt +3 +0 +(zH) +2 +0 +0 +Model,wt +m +Analytical +2e/t=4 +1 +6 +Monte-Carlo +Ac=入t +000 +Ec +0 +0 +-6 +6 +0 +0 +5 +10 +15 +20 +8 +入/M +v (Hz) +G +H +Exp,wt +Exp, ptx1 +EGTA +0 +0 +T +1 +L +[Model,wt +Model, ptx1 +EGTA +0 +0 +6 +0 +6 +-6 +0 +6 +v (Hz) +v (Hz)Supplementary materials for +The younger flagellum coordinates the beating in +C. reinhardtii +Da Wei1,3,Greta Quaranta2, Marie-Eve Aubin-Tam1†, Daniel S.W. Tam2∗ +1Department of Bionanoscience, Delft University of Technology, +2628CJ Delft, Netherlands. +2Laboratory for Aero and Hydrodynamics, Delft University of Technology, +2628CD Delft, Netherlands. +3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, +Chinese Academy of Sciences; Beijing 100190, China. +†Corresponding author. Email: m.e.aubin-tam@tudelft.nl; +∗Corresponding author. Email: d.s.w.tam@tudelft.nl. +1 +arXiv:2301.13278v1 [physics.bio-ph] 30 Jan 2023 + +S1 +Extracting coupling strength by fitting phase dynamics +In the work described in the manuscript, the flagellum-flow coupling strength ε in wt cells is +mainly extracted by the synchronization profile τ(ν) ≥50%. Meanwhile, in previous works [1, +2], fitting the distribution of phase dynamics is employed to extract ε. In the latter approach, the +idea is that the phase locking during synchronization leads to a peaked probability distribution +of ∆ϕ, whose width is affected by the effective noise Teff. The distribution, P(∆ϕ), can be +derived from the Adler equation Eq. (1) as: +P(∆ϕ) = +� ∆ϕ+2π +δct +exp(V (∆ϕ′) − V (∆ϕ) +Teff +)d∆ϕ′. +(S1) +Here V (∆ϕ) = ν∆ϕ + ε cos(∆ϕ) is a wash-board potential, Teff is the noise, and ∆ϕ is the +difference between the flagellar phase and the flow’s phase. +Here, we demonstrate that these two approaches are equivalent in extracting ε. For all +wt cells tested in the TRIS-minimal medium (N=11), their ε(θ) measured by the τ(ν) width +and extracted from fitting are plotted against each other, Fig. S1. All points center around the +identity line, showing the equivalence in obtaining ε by the two methods. For the ptx1 dataset, +ε are extracted from fitting the phase dynamics. +2 + +�������������� +����������������� +���������� +��������� +������������� +Figure S1: Equivalence of extracting coupling strength ε by different methods. Each point +represents one cell under either the θa-flow (green square), the θc-flow (red circle), or the θt- +flow (blue triangle). The x coordinate is the coupling strength ε measured by the half width of +synchronization profile τ(ν) ≥ 50%; and the y coordinate is obtained by fitting the flagellar +phase dynamics. +3 + +S2 +Hydrodynamic computation for flow along 90 degree +Similar to Fig. 2 in the main text, we present the computed drag force and power for the +flow along 90◦. The solid lines and the shadings represent the median and the interquartile +range of FFlow and PFlow over the flow-synchronized beats, respectively. Force magnitudes +are scaled by F0 = 6πµRU0 = 9.9 pN, which is the Stokes drag on a typical free-swimming +cell (radius R = 5 µm, swim velocity U0 = 110 µm/s); while the viscous powers are scaled by +P0 = F0U0 = 6πµRU 2 +0 = 1.1 fW. Here µ = 0.95 mPa·s is the dynamic viscosity of water at +22 oC. Quantitatively, the mean force is 0.37F0 and 0.34F0 (Fig. S2 top panel) while the mean +power is -0.2P0 and -0.4P0 (Fig. S2 bottom panel), for the cis and the trans respectively. +Figure S2: Computed hydrodynamic loads on the flagella. Computation results of the drag +force (upper panel) and the force’s rate of work (lower panel) on the cis (red) and the trans +(blue) flagellum during synchronized cycles, when the cell is subjected to the flow with θ = 90◦. +Scaling factors F0=9.9 pN and P0=1.1 fW. +4 + +FFlow/Fo +1 +0.5 +0 + cis loads +10 +PFlow/Po +- trans loads +0 +-10 +0 +元/2 +2元3元/22元 +Flagellar phase (rad)S3 +The model +The external flow and the two flagella are described by three coupled ordinary differential equa- +tions (ODEs). Phase dynamics of these equations are examined by Monte-Carlo simulation. +The temporal resolution of simulation (dt) is 1 ms, which corresponds to the experimental +frame rates (801 Hz). +� +� +� +� +� +� +� +� +� +� +� +� +� +dϕf +dt = 2πff +(S2a) +dϕc +dt = 2πfc − 2πλt sin(ϕc − ϕt) − 2πεc sin(ϕc − ϕf) + ζc(t) +(S2b) +dϕt +dt = 2πft − 2πλc sin(ϕt − ϕc) − 2πεt sin(ϕt − ϕf) + ζt(t). +(S2c) +The cis, the trans, and the external flow are described as oscillators, whose intrinsic fre- +quencies are fc,t,f and phases ϕc,t,f, respectively. The flow is assumed to be noise free and the +two flagella are assumed to have the same level of noise (ζc = ζt). The noises are assumed to +be Gaussian, ⟨ζc,t(τ + t)ζc,t(τ)⟩ = 2T c,t +eff δ(t). +S3.1 +Flagellar synchronization +Setting εc and εt to 0, the interaction between the two flagella in the absence of the flow is +modeled by: +� +� +� +� +� +dϕt +dt = 2πfc − 2πλt sin(ϕc − ϕt) + ζc(t) +(S3a) +dϕc +dt = 2πft − 2πλc sin(ϕt − ϕc) + ζt(t). +(S3b) +When the two flagella are able to beat synchronously, dϕc +dt = dϕt +dt = f0, we can obtain the +analytical expression of f0 by adding up λc×Eq. (S3a) and λt×Eq. (S3b): +f0 = λtft + λcfc +λc + λt +. +(S4) +5 + +Meanwhile, the steady-state phase difference δct = ϕc −ϕt is obtained by subtracting Eq. (S3a) +from Eq. (S3b): +sin(δct) = fc − ft +λc + λt += νct +λtot +. +(S5) +It is therefore obvious that the two flagella can only beat at the same frequency (dϕc/dt = +dϕt/dt = f0) if |νct/λtot| ≤ 1. +S3.2 +Interaction between three oscillators +Now we put the flow back into the picture. According to experimental observations, the two +flagella mostly beat synchronously, we therefore focus on this case and first simplify the equa- +tions. By adding up λc×Eq. (S2b) and λt×Eq. (S2c), and substituting ϕc,t as ϕ0 = ϕc −δct/2 = +ϕt + δct/2, we obtain: +dϕ0 +dt = 2πf0−2π λcεc +λc + λt +sin +� +ϕ0 − ϕf − δct +2 +� +−2π +λtεt +λc + λt +sin +� +ϕ0 − ϕf + δct +2 +� ++λtζt + λcζc +λc + λt +. +(S6) +Given different choices of coupling constants (λc,t, εc,t), this equation would generate com- +plex phase dynamics - as we shall see in the following sections. We first limit the discussion to +small δct - as it is observed in our experiment as well as in [3]. The model’s asymptotic behavior +at δct ≈ 0 is: +dϕ0 +dt = 2πf0 − 2πε sin(ϕ0 − ϕf) + ζ0(t), +(S7) +where +f0 = λtft + λcfc +εtc + λt +, ε = λtεt + λcεc +λc + λt +, ζ0 = λtζt + λcζc +λc + λt +. +(S8) +In this strong-coupling limit (δct ≈ 0, or equivalently, λtot ≫ νct), the coupled flagella +behaves as a single oscillator whose beating frequency f0 will not be interfered by the external +flow. The analytical form well captures the system’s behavior, as shown by Fig. 5F. Next we +explore the model’s behaviors when λtot − νct is comparable with ε. +6 + +��� +��� +��� +��� +��� +��� +������������ +f� +f� +��������cis��������� +��������trans��������� +��������trans ����cis +��� +��� +��� +������ +������ +������ +Figure S3: Determine the lower limit of λtot. The time fractions of the cis (a) and the trans +flagellum (b) synchronized by the flow. (c) The time fraction of where cis and trans are syn- +chronized. Arrows points towards increasing (λtot − ν)/ε. +S3.3 +Lower limit of inter-flagellar coupling +The value (λtot − νct)/ε determines if the flow can disrupt the synchronization between cis +and trans. We assume νct = 20 Hz[4, 5, 6, 3] and focus on synchronization of the θa-flow. +We plot the synchronization time fractions with increasing λtot in Fig. S3. When it satisfies +(λtot − νct)/ε ≥ 2, external flows cease to affect the flagellar synchronization observably. As +the strongest flow (21U0) applied experimentally corresponds to ε ≈ 10 Hz, altogether, we +conclude that λtot ≳ νct + 2εmax = 40 Hz. In the main text, we set λtot = 60 = 3νct Hz, which +satisfies this relation and matches the observation that the phase lag between the flagella (δct) is +small. +7 + +S4 +Flagellar noise of the ptx1 mutant +Here we show an as-yet uncharacterized strong noise present in the synchronous beating of the +mutant ptx1. The in-phase (IP) mode of ptx1 cells and the breaststroke beating of the wt cells +are similar in waveform and frequency [7, 8]. However, the former has a much stronger noise. +Figure S4: Stronger frequency fluctuation of the IP mode of ptx1 cells. (a-e) Representative +probability distributions of the beating frequency of a wt (a) and four ptx1 cells (b-e) over 30 +seconds. Probability distributions of the IP (purple) and AP mode (yellow) are respectively nor- +malized for better visualization. The time fractions of the AP mode are noted in each panel. (f) +The wt and ptx1 cells represented by its mean beating frequency ⟨f0⟩ and the standard deviation +of the beating frequencies over time σ(f0). +The strong noises show obviously in fluctuations of IP beating frequencies [8]. +In Fig. S4, we display the distribution of beating frequency of a representative wt cell (panel +a) and four representative ptx1 cells (panels b-e). The broad peaks of the IP (purple) and AP +(yellow) beating of ptx1 sharply contrast the narrow peak of wt. We quantify the frequency +fluctuations of all the cells in the main text (N=11 for wt and N=14 for ptx1), Fig. S4f. The +cells are represented by its mean beating frequency over time ⟨f0⟩ and the frequency’s standard +deviation σ(f0). Clearly, the breaststroke beating of wt, the IP, and the AP mode of ptx1 each +forms a cluster. The wt cluster is at (⟨f0⟩, σ(f0)) = (50.5 ± 2.6, 0.8 ± 0.3) Hz (mean± 1 std. +the over cell population); and it is evidently less dispersed than both the IP and the AP mode +8 + +0.1 +(a) +(f) +wt +wildtype +ptxl, IP mode +0 +40 +50 +60 +70 +80 +ptxl, AP mode +0.06 +4 +AP: 6.9% +(b) +ptx1 +0 +(zH) (°J)o +0.04 +AP: 10.2%. +(c) +PDF +- +0 +2 +0.06 +AP: 39.7% +(d) +口 +0 +口 +0.04 +AP: 20.4% +口 +(e) +0 +0 +40 +50 +60 +70 +80 +40 +50 +60 +70 +80 +Frequency (Hz) +(fo) (Hz)of ptx1, which are at (47.4 ± 3.1, 3.4 ± 0.9) Hz and (67.6 ± 2.1, 1.9 ± 0.7) Hz, respectively. +Under the assumption of a white (Gaussian) noise, σ(f0) is proportional to the noise level ζ, +and thus scales with √Teff. Consider that σ(f0) for ptx1 is 3-5 folds larger than that of wt, +we therefore conclude that the noise level in ptx1 is an order of magnitude larger than wt, +T ptx1 +eff +/T wt +eff ∼ O(10). +Figure S5: Effect of a low-noise cis in stabilizing the beating of the trans (a) Fluctuations in +beating frequency (σ(f0)) under different coupling schemes and flagellar noises. Other model +parameters are the same as used in the main text. The red and blue shaded area represent the +experimentally observed range for ptx1 and wt cells, respectively, with short bars marking the +mean values. (b) the rate of slip under the conditions. Error bars correspond to 1 std. over N=9 +repetitions. +The stronger noise in ptx1 can be attributed to two sources, namely, the loss of a stable +cis and the loss of the unilateral coupling, Fig. S5. We perform Monte-Carlo simulations of +the coupled beating of cis and trans under three conditions: (1) a stable cis (T c +eff = T 0 +eff = +1.57 rad/s2) coupled with the trans unilaterally (λc = 4λt), (2) a stable cis coupled with the +trans bilaterally (λc = λt), and (3) an equally noisy cis (T c +eff = T t +eff) bilaterally coupled with +trans, see the blue, yellow, and red data in Fig. S5 respectively. It is obvious that, when the trans +is coupled to a stable cis, varying its noise over an order of magnitude only leads to a ∼ 20% +stronger frequency fluctuation (the blue line in Fig. S5(a)). On the contrary, lacking either +the unilateral coupling or the low-noised cis would increase the fluctuation for 200% (yellow +9 + += = += += +0.1 +(a) +(b) +3 +ptx1 +0.08 +Slip rate (Hz) +o(fo) (Hz) +0.06 +2 +0.04 +1 +0.02 +im +0 +0 +100 +101 +10° +101 +Teff / Teff +Teff / Teffline) or 300% (red line). Qualitatively, simulation results are in agreement with experimental +measurements assuming that T t +eff/T c +eff ∼ O(10), see the red and blue shaded areas in Fig. S5(a). +Moreover, a low-noise cis is already sufficient to prevent slips from interrupting the synchrony +between cis and trans, even for bilateral coupling. In Fig. S5(b), as long as the cis-noise remains +low, slips will be sparse (< 0.01 Hz). Together, these simulation results highlight the stabilizing +effect of a low-noise cis flagellum, and illustrates the contribution of unilateral coupling in +further enhancing the stabilization. +References +[1] Polin, M., Tuval, I., Drescher, K., Gollub, J. P. & Goldstein, R. E. Chlamydomonas swims +with two “gears” in a eukaryotic version of run-and-tumble locomotion. Science 325, 487– +490 (2009). +[2] Quaranta, G., Aubin-Tam, M.-E. & Tam, D. Hydrodynamics versus intracellular coupling +in the synchronization of eukaryotic flagella. Physical Review Letters 115, 238101 (2015). +[3] Wan, K. Y., Leptos, K. C. & Goldstein, R. E. Lag, lock, sync, slip: the many phases of +coupled flagella. Journal of The Royal Society Interface 11, 20131160 (2014). +[4] Kamiya, R. & Hasegawa, E. Intrinsic difference in beat frequency between the two flagella +of Chlamydomonas reinhardtii. Experimental Cell Research 173, 299–304 (1987). +[5] Kamiya, R. Analysis of cell vibration for assessing axonemal motility in Chlamydomonas. +Methods 22, 383–387 (2000). +[6] Okita, N., Isogai, N., Hirono, M., Kamiya, R. & Yoshimura, K. Phototactic activity in +Chlamydomonas ’non-phototactic’ mutants deficient in Ca2+-dependent control of flagellar +dominance or in inner-arm dynein. Journal of Cell Science 118, 529–537 (2005). +10 + +[7] Horst, J. & Witman, G. B. Ptx1, a nonphototactic mutant of Chlamydomonas, lacks control +of flagellar dominance. The Journal of Cell Biology 120, 733–741 (1993). +[8] Leptos, K. C. et al. Antiphase synchronization in a flagellar-dominance mutant of Chlamy- +domonas. Physical Review Letters 111, 1–5 (2013). +11 + diff --git a/DNFQT4oBgHgl3EQfPTY7/content/tmp_files/load_file.txt b/DNFQT4oBgHgl3EQfPTY7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c7dc098c7eae6393e1005d7ddd938984f9f51a8e --- /dev/null +++ b/DNFQT4oBgHgl3EQfPTY7/content/tmp_files/load_file.txt @@ -0,0 +1,1200 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf,len=1199 +page_content='The younger flagellum coordinates the beating in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' reinhardtii Da Wei1,3,Greta Quaranta2, Marie-Eve Aubin-Tam1†, Daniel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Tam2∗ 1Department of Bionanoscience, Delft University of Technology, 2628CJ Delft, Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 2Laboratory for Aero and Hydrodynamics, Delft University of Technology, 2628CD Delft, Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Beijing 100190, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Email: m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='aubin-tam@tudelft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='nl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Email: d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='tam@tudelft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='nl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='13278v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='bio-ph] 30 Jan 2023 Abstract Eukaryotes swim with coordinated flagellar (ciliary) beating and steer by fine-tuning the coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The model organism for studying flagellate motility, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' reinhardtii (CR), employs synchronous, breast-stroke-like flagellar beating to swim, and it modulates the beating amplitudes differentially to steer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' This strategy hinges on both inherent flagellar asymmetries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' different response to chemical messengers) and such asymmetries be- ing effectively coordinated in the synchronous beating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In CR, the synchrony of beating is known to be supported by a mechanical connection between flagella, however, how flagellar asymmetries persist in the synchrony remains elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' For example, it has been speculated for decades that one flagellum leads the beating, as its dynamic properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' frequency, waveform, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=') appear to be copied by the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In this study, we combine experi- ments, computations, and modeling efforts to elucidate the roles played by each flagellum in synchronous beating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' With a non-invasive technique to selectively load each flagellum, we show that the coordinated beating essentially responds to only load exerted on the cis flagellum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' and that such asymmetry in response derives from a unilateral coupling between the two flagella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Our results highlight a distinct role for each flagellum in coordination and have implication for biflagellates’ tactic behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' One-Sentence Summary: The younger flagellum of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' reinhardtii coordinates the synchronous beating and couples to external forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 2 Introduction The ability to swim towards desirable environments and away from hazardous ones is funda- mental to the survival of many microorganisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' These so-called tactic behaviors are exhib- ited by many motile microorganisms ranging from bacteria (1, 2) to larger flagellates and cili- ates (3–5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Different microorganisms have developed specific strategies for steering, depending on the tactic behavior and on their specific sensory and motility repertoire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' For example, bacte- ria modulate the tumbling rate (1) while flagellates and ciliates modulate the waveform (6–9), amplitude (10, 11) and frequency of their flagellar/ciliary (4, 12) beating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The goal of these active modulations of the motility is to achieve a spatially asymmetric generation of propulsive force to steer the organism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' reinhardtii (CR), the model organism for studies of flagellar motility, achieves tactic nav- igation by a fine-tuned differential modulation on its two flagella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Studying this organism offers great opportunities to look into how flagella coordinate with each other and how such coordi- nation helps facilitate targeted steering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' CR has a symmetric cell body and two near-identical flagella inherited from the common ancestors of land plants and animals (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' It swims by beating its two flagella synchronously and is capable of photo- and chemotaxis (10, 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' For this biflagellated organism, effective steering hinges on both flagellar asymmetry and flagellar coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' On the one hand, the two flagella must be asymmetric to respond differentially to stimuli (10,15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' on the other hand, the differential responses must be coordinated by the cell such that the beating would remain synchronized to guarantee effective swimming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Understand- ing this remarkable feat requires knowledge about both flagellar asymmetry and coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The two flagella are known to be asymmetric in several, possibly associated, aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' First of all, they differ in developmental age (16, 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The flagellum closer to the eyespot, the cis(- eyespot) flagellum, is always younger than the other one, the trans(-eyespot) flagellum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' This is because the cis is organized by a basal body (BB) that develops from a pre-matured one in the mother cell;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' and this younger BB also organizes the flagellar root (D4 rootlet) that dictates the eyespot formation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Second, the two flagella have asymmetric protein composition (19– 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' For example, the trans flagellum is richer in CAH6, a protein possibly involved in CO2 sensing (14,20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Finally, the flagella have different dynamic properties (22–24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Their beating is modulated differentially by second messengers such as calcium (22,23) and cAMP (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' When beating alone, the trans beats at a frequency 30%-40% higher than the cis (23,26–28);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' the trans also displays an attenuated waveform (29) and a much stronger noise (29,30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3 Remarkably, despite these inherent asymmetries, CR cells establish robust synchronization between the flagella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Such coordination enables efficient swimming and steering of the cells and takes basis on the fibrous connections between flagellar bases (31,32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Intriguingly, in the coordinated beating, both flagella display dynamic properties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', flagellar waveform, beating frequency (∼50 Hz), and frequency fluctuation, that are more similar to those of the cis flag- ellum (26, 28–30, 33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' This has led to a long-standing hypothesis that ”the cis somehow tunes the trans flagellum” (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' This implies that the symmetric flagellar beating (”breast-stroke”) observed is the result of interactions between two flagella playing differential roles in coordi- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' How does the basal coupling make this possible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Recent theoretical efforts show that the basal coupling can give rise to different synchronization modes (34–36);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' and that flagellar dynamics, such as beating frequency, may simply emerge from the interplay between mechan- ics of basal coupling and bio-activity (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Yet, most theoretical efforts examining flagellar synchronization have assumed two identical flagella, limiting the results’ implication for the realistic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Moreover, little experiments directly probe the flagella’s differential roles during synchronous beating (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Therefore, flagellar coordination in this model organism remains un- clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' To clarify the picture experimentally, one needs to selectively force each flagellum, and characterize the dynamics of the flagellar response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In this study, we address this challenge and devise a non-invasive approach to apply external forces selectively on the cis- or the trans-flagella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Oscillatory background flows are imposed along an angle with respect to the cell’s symmetry axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Such flows result in controlled hydro- dynamic forces, which are markedly different on the two flagella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' With experiments, hydrody- namic computations, and modeling, we show definitively that the two flagella are unilaterally coupled, such that the younger flagellum (cis) coordinates the beating, whereas the elder one simply copies the dynamic properties of the younger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' This also means that only external forces on the cis may mechanically fine-tune the coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We also study the effect of calcium in the cis’ leading role as calcium is deeply involved in flagellar asymmetry and hence photo- tactic steering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In addition, a well-known mutant that lacks flagellar dominance (ptx1) (23,38) is examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Results show that the coordinating role of cis does not need environmental free calcium, whereas it does require the genes lost in ptx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Our results discern the differential roles of CR’s flagella, highlight an advanced function of the inter-flagellar mechanical coupling, and have implications for biflagellates’ tactic motility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 4 Experimental scheme for selective loading We set out to establish a non-invasive experimental technique that exerts differential loads on the flagella of CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Following the study by Quaranta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (31), we induce oscillatory background flows to exert hydrodynamic forcing to flagella of captured cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' With programmed oscillations of the piezoelectric stage, the amplitude, frequency, and direction of the background flows are all controlled, enabling selective loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' To quantitatively estimate the selectivity of the flows along different angles (θ), we compute the flagellar loads under the flows along θ = −45◦, 0◦, and 45◦, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Computations based on boundary element methods (BEM) and slender-body theory (SBT) give the real-time drag force F on each flagellum and the power P exerted by the viscous forces on each flagellum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' For given realistic flagellar shapes, we compare computed loads with and without external flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' From these we isolate the loads from the induced flows FFlow and PFlow (Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Loads on each flagellum under flows of θ = 0◦, −45◦, 45◦ are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Upper panels display the magnitude of the drag force FFlow = |FFlow|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' while lower panels show viscous power PFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Force magnitudes are scaled by F0 = 6πµRU0 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='9 pN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' while the powers by P0 = F0U0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='1 fW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' F0 is the Stokes drag on a typical free-swimming cell (radius R = 5 µm, speed U0 = 110 µm/s, water viscosity µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='95 mPa·s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Evidently, along θ = 0◦, flows load the flagella equally (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 2A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' However, at θ = −45◦, flows load the cis flagellum ∼ 2 times larger than the trans (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 2B, F c Flow ≈ 2F t Flow);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' whereas flows at θ = 45◦ do the opposite (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 2C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The selectivity also manifests in (the absolute values of) PFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We do notice that flows along θ = +45◦ are able to synchronize the flagella with PFlow < 0, meaning that the flagella are working against the flows, and this shall be discussed in later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Hereon forward, we refer to θc-flows, flows for which θ = −45◦ and the cis-flagellum is selectively loaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Likewise, θt-flows denote flows on θ = +45◦ that selectively load the trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' θa-flows denote the axial flow along θ = 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We next introduce how we quantify the flows’ effective forcing strength (ε) on the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Phase dynamics of flagellar beating is extracted from videography (31,39,40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Recordings are masked and thresholded to highlight the flagella (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 1B-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Then the mean pixel values over time within two sampling windows (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 1D) are converted to observable-invariant flagellar phases (41), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 1E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Throughout this study, as cis and trans always beat synchronously (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 1E inset), their phases ϕc,t are used interchangeably as the flagellar phase ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The flagellar phase 5 dynamics under external periodic forcing is described by Adler equation (42–44): d∆ϕ dt = −2πν − 2πε sin(∆ϕ) + ζ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (1) ∆ϕ = ϕ − 2πfft is the phase difference between the beating and the forcing, with ff the forcing frequency, and ε the forcing strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The detuning ν = ff − f0 is the frequency mismatch between the beating (f0) and forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' ζ(t) represents a white noise that satisfies ⟨ζ(τ + t)ζ(τ)⟩ = 2Teffδ(t), with Teff an effective temperature and δ(t) the Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' When the forcing strength outweighs the detuning (ε > |ν|), synchronization with the flow (d∆ϕ/dt = 0) emerges, see the plateaus marked black in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 1F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We characterize synchro- nization with τ = tsync/ttot, where tsync is the total time of flow synchronization and ttot the flow duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 1F presents the phase dynamics which are representative and range from: no synchronization (τ=0, i), unstable synchronization (0 < τ < 1, ii-iii), and stable synchro- nization (τ=1, iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In this study, the frequency range in ν for which τ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='5 is used to measure ε (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 1F inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' This method is equivalent to previous fitting-based methods (28, 31), see SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Asymmetric susceptibility to flow synchronization Now we examine cell responses to flows of various amplitudes and along different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' First we explore flow synchronization over a broad range of amplitudes and frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' θa- flows with frequencies ff ∈ [40, 75] Hz and amplitudes U ∈ [390, 2340] µm/s are imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The scanned range covers reported intrinsic frequencies of both the cis and trans flagellum (22,24, 26, 27);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' while the amplitude reaches the maximum instantaneous speed of a beating flagellum (∼ 2000 µm/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3A displays the resultant flow-synchronized time fractions τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Up until the strongest flow amplitude, the large forces cannot disrupt the synchronized flagellar beating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In addition, synchronization is never established around frequencies other than f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' This shows that the inter-flagella coupling is much stronger than the maximum amplitude of forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Next we examine the synchronization with the θc-flows and θt-flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Flows of a fixed amplitude (∼ 7U0) but varying frequencies around f0 are applied to each captured cell (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' With these, the flow-synchronized time fraction τ as a function of the detuning (ν) and flow direction (θc,a,t) is recorded and helps quantify the flows’ effective forcing ε(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Comparing τ(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' θc) to τ(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' θt), with τ(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' θa) as reference, we find that θc-flows are the most effective in synchronizing the beating (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We illustrate this point with the profiles of an 6 exemplary cell (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3B inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' First, although both the θc-flow (red) and the θt-flow (blue) can synchronize the cell at small detunings (|ν| <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='5Hz), the θc-flow maintains the synchro- nization for the whole time ( τ(θc) =1), while the θt-flow for a slightly smaller time fraction ( τ(θc) ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' This is due to phase-slips (step-like changes in ∆ϕ(t) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 1F) between flag- ella and the flow, and means that the θt-flow synchronization is less stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Additionally, for intermediate detuning (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='5 Hz< |ν| <4 Hz), τ(θc) is always larger than τ(θt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In some cases, the θc-flow synchronizes the cell fully whereas the θt-flow fails completely (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', at ν = −2 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Together, these results imply that a flow of given amplitude synchronizes flagellar beating more effectively if it selectively loads the cis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We repeat the experiments with cells from multiple cultures, captured on different pipettes, and with different eyespot orientations (∼50% heading rightward in the imaging plane) to rule out possible influence from the setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' τ(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' θ) of N=11 wt cells tested in the TRIS-minimal medium (pH=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='0) are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3B (labeled as ”TRIS”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' On average, ε(θc) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='9 Hz and is 70% larger than ε(θt) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='7 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' It bears emphasis that ε(θc) > ε(θt) holds true for every single cell tested (11/11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3C, we show this by representing each cell as a point in the ε(θc) - ε(θt) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' A point being below the first bisector line ( ε(θc) = ε(θt) ) indicates that ε(θc) > ε(θt) for this cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' All cells cluster clearly below the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' This asymmetry manifest equivalently through τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3D, each point represents the time fractions of the same cell synchronized by the θc-flow and the θt-flow at the same frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Most points (>90%) are below the first bisector line, meaning that τ(θc) > τ(θt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Altogether, all results show that selectively loading the cis flagellum establishes synchronization with the flow more effectively, pointing to cis and trans playing differential roles in the coordinated beating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We next study whether this newly observed cis-trans asymmetry is affected by calcium depletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Calcium is a critical second messenger for modulating flagellates motility and is deeply involved in phototaxis (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The depletion of the free environmental calcium is known to degrade flagellar synchronization and exacerbate flagellar asymmetry (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Here we focus on whether calcium depletion affects the asymmetry ε(θc) > ε(θt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We deplete environmental calcium by EGTA-chelation, following the protocol in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Similar to previous reports (22, 47), the number of freely swimming cells drops significantly in EGTA-containing medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' However, the remaining cells beat synchronously for hours after capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' For these beating cells, calcium depletion is first confirmed by characterizing their deflagellation behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Indeed, calcium depletion is reported to inhibit deflagellation (28, 48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In experiments with standard calcium concentration, all cells deflagellated under pipette suction (20/20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' For experiments 7 conducted in calcium depleting EGTA-containing medium, we observe deflagellation to occur in none but one cell (1/19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' After confirming the calcium depletion, we perform the same sets of flow synchronization experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3B show the median synchronization profiles τ(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' θ) (N=6 cells, labeled as ”EGTA”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The flagellar asymmetry is unaffected, see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Note that ε(θc) > ε(θt) again applies for every single cell tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The mean values of ε drop slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' However, the different effectiveness between θc-flows and θt-flows, ε(θc) − ε(θt) , is not af- fected, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3E inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Finally, we determine how the forcing strength of the flow depends on the hydrodynamic forces exerted by the flow on the flagella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We compute the hydrodynamic beat-averaged loads, F Flow = � 2π 0 FFlowdϕ/2π, P Flow = � 2π 0 PFlowdϕ/2π, induced by the flow on the trans and on the cis flagella, see the horizontal lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' These loads are computed for the θc-flow, θt-flow, θa-flow and we also include experiments and computations performed with flows along θ = 90◦ (circles), see SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3F and G represent ε as a function of the loads on the cis and trans flagellum respectively, with each symbol representing one of the four different flow directions, see the drawings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We find that the effective forcing strength scales with the time-averaged drag on the cis, ε ∼ F c Flow, while we find no such correlation between ε and F t Flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The linear relation between ε and F c Flow has an intercept near zero (ε|F c Flow=0 ≈ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Given the total forces on both flagella (F c Flow + F t Flow) for these flows remains almost constant (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='74-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='79F0), the zero-intercept implies that for a hypothetical flow that exerts no load on the cis but solely forces the trans, it will not be able to synchronize the cell at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' This suggests a negligible contribution of the forcing on the trans in establishing synchronization with flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The asymmetry is lost in ptx1 mutants Furthermore, we examine the flagellar dominance mutant ptx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In this mutant, both flagella re- spond similarly to changes of calcium concentrations (38) and have similar beating frequencies when demembranated and reactivated (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Ptx1 mutants have two modes of coordinated beating, namely, the in-phase (IP) synchro- nization and the anti-phase (AP) synchronization (29, 49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' First, we apply θa-flow in the same frequency and amplitude ranges as for wt cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We find that the IP mode around f0 ≈ 50 Hz is the only mode that can be synchronized by external flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We focus on this mode and report τ as τ = tsync/tIP for this mutant, where tIP is the total time of IP-beating under the applied 8 flows, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 4A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Synchronization profiles τ(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' θ) of ptx1 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The median profiles are of similar width and height, indistinguishable from each other, and hence indicate a loss of asymmetric susceptibility to flow synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The loss is further confirmed by the extracted ε(θ) (31) and τ(θ) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 4C-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Cells and synchronization attempts are distributed evenly across the first bisector lines (7/14 cells are below ε(θc) = ε(θt) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 4C, and ∼50% points are below τ(θc) = τ(θt) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 4D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Altogether, all results show consistently that the asymmetry is lost in ptx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Modeling Framework To investigate the implications of our experimental results on the coupling between flagella and their dynamics, we develop a model for the system (SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='S3), representing flagella and external flows as oscillators with directional couplings: � � � � � ˙ϕf = 2πff ˙ϕc = 2π[fc − λt sin(ϕc-ϕt) − εc sin(ϕc-ϕf)] + ζc(t) ˙ϕt = 2π[ft − λc sin(ϕt-ϕc) − εt sin(ϕt-ϕf)] + ζt(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (2) ϕf,c,t(t) respectively represent the phase of the flow, the cis, and the trans flagellum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' ff,c,t represents the inherent frequency of the forcing (flow), the cis, and the trans respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The phase dynamics of each flagellum is modulated by its interactions with the other flagellum as well as the background flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Take the cis ( ˙ϕc) for example, the effect of the trans and the forcing on the cis are respectively accounted for by the λt-term and the εc-term, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In other words, λt and εc measure the sensitivity of the actual cis-frequency to the phase differences between oscillators (ϕc − ϕt,f), see the arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 5A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Lastly, ζc,t represent the white noise of the cis and trans flagellum respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In the following parts, without loss of generality, the noise are assumed equally strong and uncorrelated (⟨ζ2 c ⟩ = ⟨ζ2 t ⟩, or T c eff = T t eff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Nuanced phase dynamics under differential noise levels can be found in SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (2) can be readily reduced to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (1), which allows us to write the experimentally mea- sured values (f0, ε(θ), Teff) analytically with εc,t, λc,t, and ζc,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The asymptotic behavior of the 9 model under the condition ϕc ≈ ϕt ≈ ϕf are (SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='S3): � � � � � f0 = αfc + (1 − α)ft, Teff = α2T c eff + (1 − α)2T t eff, ε(θ) = αεc(θ) + (1 − α)εt(θ), (3) with α = λc/(λc + λt) representing the dominance of cis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' It is then clear that when α ≈ 1, the coordinated beating will display dynamic properties of the cis flagellum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 5A illustrates an exemplary modeling scheme describing flagellar beating subjected to θc-flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The direction and thickness of arrows represent coupling direction and strength respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The selective loading on the cis is represented by εc > εt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' while λc > λt reflects that the cis has a more dominant role in the coordinated beating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We run Monte-Carlo simulation with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (2) using customized MATLAB scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Coordinated beating under symmetric forcing We first model the flow synchronization induced by θa-flow (symmetric flagellar loads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In this case, ε(θ) = αεc(θ) + (1 − α)εt(θ) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (3)) reduces to ε = εc,t and is independent of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We set εc,t as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='4 Hz to match the measured ε(θa) =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='4 Hz (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' At similar detunings as in the experimental results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 1F, our Monte-Carlo simulations reproduces the phase dynamics with: (i) no flow synchronization, (ii-iii) unstable synchroniza- tion, and (iv) stable synchronization (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 5B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Repeating the simulations for varying forcing strength ε (= εc,t) and frequency ff yields Arnold tongue diagrams in agreement with those reported from our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The Arnold Tongue for wt in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3A and ptx1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 4A are reproduced with simulations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 5C and D respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The only parameter value changed between Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 5C and D is the level of noise (T c,t eff ), which is increased by an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The differences in phase dynamics between wt and ptx1, when subjected to sym- metric external loading, are therefore accounted by solely varying the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Coordinated beating under selective loading We next model flow synchronization by the θc-flows and the θt-flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The selective forcing (εc ̸= εt) allows the effect of flagellar dominance (λc ̸= λt) to manifest in the effective forcing strength ε(θ) and hence in the synchronization profiles τ(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' θ), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 5E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Similar to our exper- imental observations, θc-flow synchronizes the coordinated beating over the broadest range of ν (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' largest ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' This is directly attributed to the dominance λc > λt: by setting λc = λt, 10 the differences between τ(θc) and τ(θt) disappear even under selective loading (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 5E inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 5F details how the asymmetry of inter-flagellar coupling (λc/λt) affects the asymmetry between τ(θc) and τ(θt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The open symbols represent ε(θ) measured from modeled τ(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' θ) and the lines represent Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The difference between ε(θc) and ε(θt) increases with λc/λt, and they each saturates to reflect only the forcing on the cis (εc, the grey dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' With fc = 45 Hz, ft = 65 Hz (23, 26), and f0 ≈ 50 Hz, we deduce from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (3) that λc = 4λt for wt cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' For wt cells under calcium depletion, experimental results are reproduced with a lower total forcing strength (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 5G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' εc + εt is set to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='08 Hz (15% lower) to reflect the 7% − 20% decrease in ε(θ) induced by calcium depletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The ptx1 results are reproduced with a stronger noise (T c,t eff = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='42 rad2/s) and a symmetric inter-flagellar coupling λc/λt = 1, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 5H and Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Both changes are necessary for reproducing the synchronization profiles of ptx1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 5H: while the stronger noise lowers the maximal values of τ(θ, ν), setting λc/λt = 4 would still result in τ(θc) > τ(θt) in the central range (|ν| ≲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='4 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Finally, it is noteworthy that the noise in ptx1 increases not only because a higher noise value for individual flagella, but also because the cis-trans coupling has become symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' As shown by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (3), the unilateral coupling promotes not only the cis- frequency in the synchrony but also the cis-noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Given T c eff ≪ T t eff and λc = 4λt, we confirm with simulations that the cis stabilizes the beating frequency of the trans and decreases its beating noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The simulations are in good agreement with experimental noise measurements, see SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='S4 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Discussion The two flagella of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' reinhardtii have long been known to have inherently different dynamic properties such as frequency, waveform, level of active noise, and responses to second messen- gers (23,25,26,29,30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Intriguingly, when connected by basal fibers and beating synchronously, they both adopt the kinematics of the cis-(eyespot) flagellum, which led to the assumption that the flagella may have differential roles in coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In this work, we test this hypothesis by employing oscillatory flows applied from an angle with respect to the cells’ symmetry axis and thus exert biased loads on one flagellum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Without an exception, in wt cells, θc-flows, the ones that selectively load the cis flagellum, are always more effective in synchronizing the flagellar beating than the θt-flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' This is shown by the larger effective forcing strengths ( ε(θc) > ε(θt) , Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3B-C) and larger synchronized time 11 fractions ( τ(θc) > τ(θt) , Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Mapping the measured forcing strength ε(θ) as a function of the loads, we find empirically that ε ∝ F c Flow (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3F) and that trans-loads appear to mat- ter negligibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' These observations all indicate that the cis-loads determine whether an external forcing can synchronize the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Moreover, this point is further highlighted by an unexpected finding: when θt-flows are applied, the trans flagellum always beats against the external flow (P t Flow < 0) and the only stabilizing factor for flow synchronization is the cis flagellum working along with the flow during the recovery stroke (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 2C lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' These observations defini- tively prove that the two flagella have differential roles in the coordination and interestingly imply that flagella are coupled to external flow only through the cis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' To have a mechanistic understanding of this finding, we model the system with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In the model, selective hydrodynamic loading and flagellar dominance in the coordinated beating are respectively represented by εc ̸= εt and λc ̸= λt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Setting out from the model, we obtain closed-form expressions for observables such as f0 and ε (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (3)), which illustrate how flag- ellar dominance and selective loading affect the coordinated flagellar beating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Moreover, with Monte-Carlo simulation, we clarified the interplay between flows and flagella (SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='S3), and reproduces all experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' With the model, we show that a ”dominance” of the cis (λc > λt) is sufficient to explain why the coordinated flagellar beating bears the frequency and the noise level of the cis flag- ellum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In the model, such dominance means that the cis-phase is much less sensitive to the trans-phase than the other way around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We then reproduce the phase dynamics of flow synchro- nization at varying detunings (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 5B), amplitudes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 5C), and noise (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 5D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Exploiting the observation that the coordination between flagella cannot be broken by external flows up to the strongest ones tested (εmax ∼ 10 Hz, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3A), we quantify the lower limit of the total basal coupling, λc + λt, to be approximately 40 Hz (deduced in SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='S3), which is an order magnitude larger than the hydrodynamic inter-flagellar coupling (31,50–52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The modulation of flagellar dominance mediates tactic behaviors (22, 23, 38, 47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Calcium is hypothesized to be underlying the modulation of dominance, as it causes the connecting fiber between flagella to contract (53), modulates the cis- and trans activity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' beating am- plitude) differentially (22), and calcium influx comprises the initial step of CR’s photo- (54) and mechanoresponses (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We therefore investigate flagellar coupling in the context of tactic steering by depleting the environmental free calcium and hence inhibiting signals of calcium influxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Cells are first acclimated to calcium depletion, and then tested with the directional flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Our results show that the cis dominance does not require the involvement of free envi- 12 ronmental calcium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Calcium depletion merely induces an overall drop in the forcing strength perceived by the cell ε(θ) (7% − 20%), which is captured by reducing εc + εt for 15% (mean drop) in the model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 5G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Together, our results indicate that the leading role of cis, is an inherent property, that does not require active influx of external calcium, and possibly reflects an intrinsic mechanical asymmetry of the cellular mesh that anchors the two flagella into the cell body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In ptx1 cells, a lack of flagellar dominance (λc = λt) and a stronger noise level help repro- duce our experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Previous studies suggested that both flagella of ptx1 are similar to the wildtype trans (23), and that the noise levels of this mutant’s synchronous beating are much greater than those of wt (29) (see also SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' If both flagella and their anchoring roots indeed have the composition of the wildtype trans, such symmetry would predict λc = λt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' This symmetric coupling renders the noise of ptx1 Teff = T t eff (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (3)), which is about an order of magnitude larger than the noise of wt Teff ≈ T c eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The comparison between ptx1 and wt highlights an intriguing advantage of the observed unilateral coupling (λc ≫ λt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' that is, it strongly suppresses the high noise of the trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Consid- ering that the trans is richer in CAH6 protein and this protein’s possible role in inorganic carbon sensing (14,20), the potential sensing role of the trans is worth noticing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Assuming the strong noise present in the trans originates from the biochemical processes related to sensing, then the unilateral coupling effectively prevents such noise from perturbing the cell’s synchronous beating and effective swimming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In this way, the asymmetric coupling may combine the benefit of having a stable cis as the driver while equipping a noisy trans as a sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Material and methods Cell culture CR wildtype (wt) strain cc125 (mt+) and flagellar dominance mutant ptx1 cc2894 (mt+) are cultured in TRIS-minimal medium (pH=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='0) with sterile air bubbling, in a 14h/10h day-night cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Experiments are performed on the 4th day after inoculating the liquid culture, when the culture is still in the exponential growth phase and has a concentration of ∼ 2 × 105 cells/ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Before experiments, cells are collected and resuspended in fresh TRIS-minimal (pH=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 13 Calcium depletion In calcium depletion assays, cells are cultured in the same fashion as mentioned above but washed and resuspended in fresh TRIS-minimal medium + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='5 mM EGTA (pH=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Free calcium concentration is estimated to drop from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='33 mM in the TRIS-minimal medium, to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='01 µM in the altered medium (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Experiments start at least one hour after the resuspension in order to acclimate the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Experimental setup Single cells of CR are studied following a protocol similar to the one described in (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Cell suspensions are filled into a customized flow chamber with an opening on one side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The air- water interface on that side is pinned on all edges and is sealed with silicone oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' A micropipette held by micromanipulator (SYS-HS6, WPI) enters the chamber and captures single cells by as- piration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The manipulator and the captured cell remain stationary in the lab frame of reference, while the flow chamber and the fluid therein are oscillated by a piezoelectric stage (Nano-Drive, Mad City Labs), such that external flows are applied to the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Frequencies and amplitudes of the oscillations are individually calibrated by tracking micro-beads in the chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Bright field microscopy is performed on an inverted microscope (Nikon Eclipse Ti-U, 60× water immersion objective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Videos are recorded with a sCMOS camera (LaVision PCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='edge) at 600-1000 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Measurement scheme The flagellar beating of each tested cell is recorded before, during, and after the application of the flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We measure the cell’s average beating frequency f0 over 2 s (∼100 beats).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' For ptx1 cells, f0 is reported for the in-phase (IP) synchronous beating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Unless otherwise stated, directional flows (θ = 0, ±45◦) are of the same amplitude (780±50 µm/s, mean±std), similar to those used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Flow frequencies ff are scanned over [f0 − 7, f0 + 7] Hz for each group of directional flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Computation of the flagellar loads To quantify the hydrodynamic forces on the flagella, we first track realistic flagellar deforma- tion from videos wherein background flows are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Then we employ a hybrid method combining boundary element method (BEM) and slender-body theory (40, 55) to compute the 14 drag forces exerted on each flagellum and the forces’ rates of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In this approach, each flag- ellum is represented as a slender-body (55) with 26 discrete points along its centerline and the time-dependent velocity of each of the 26 points is calculated by its displacement across frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The cell body and the pipette used to capture the cell are represented as one entity with a com- pleted double layer boundary integral equation (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Stresslet are distributed on cell-pipette’s surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' while stokeslet and rotlet of the completion flow are distributed along cell-pipette’s centerline (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The no-slip boundary condition on the cell-pipette surface is satisfied at col- location points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Lastly, stokeslets are distributed along the centerlines of the flagella, so that no-slip boundary conditions are met on their surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Integrating the distribution of stokeslets f(s) over a flagellar shape, one obtains the total drag force F = � f(s)ds is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Similarly, the force’s rate of work is computed as P = � f(s) · U(s)ds, where U(s) is the velocity of the flagellum at the position s along the centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The computations shown in this study are based on videos of a representative cell which originally beats at ∼50 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The cell is fully synchronized by flows along different directions (θ = 0◦, ±45◦ and 90◦) at 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='2 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In the computations, the applied flows are set to have an amplitude of 780 µm/s to reflect the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Computations begin with the onset of the background flows (notified experimentally by a flashlight event), and last for ∼30 beats (500 frames sampled at 801 fps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Additionally, we confirm the results of θt-flow-synchronization, that both flagella spend large fractions of time beating against the flows, with other cells and with θt-flows at other frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Isolate loads of external flows The total loads (F and P) computed consist of two parts, one from the flow created by the two flagella themselves and the other from the applied flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In the low Reynolds number regime, the loads of the two parts add up directly (linearity): F = FSelf + FFlow, and P = PSelf + PFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' To isolate FFlow and PFlow, we compute F′ = FSelf and P ′ = PSelf by running the computation again but without the external flows, and obtain FFlow = F − F′ and PFlow = P − P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Modeling parameters We assume the flagellar intrinsic frequencies fc and ft to be 45 Hz and 65 Hz respectively (23, 26, 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' On this basis, λc : λt is assumed to be 4:1 to account for the observed f0 (∼ 50 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' εc : εt is set as 2:1, 1:1, and 1:2 for the θc-flows, the θa-flows, and the θt-flows respectively, 15 see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 2A-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Additionally, εc + εt is assumed to be constant to reflect the fact that F c Flow + F t Flow approximately does not vary with flow directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We take a typical value of T c,t eff = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='57 rad2/s (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The sum of inter-flagellar coupling λtot = λc + λt is set to be large enough, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', λtot = 3νct with νct = |ft − fc|, to account for the fact that: 1) the coordinated beating is approximated in-phase, and 2) up until the strongest flow applied, the coordinated beating cannot be broken (quantitative evaluation is detailed in SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' To model wt cells under calcium depletion, we decrease εc + εt by 15% - which is the mean decrease in the observed ε(θc) , ε(θa) , and ε(θt) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' For ptx1 cells, we assume a symmetric inter-flagellar coupling (λc = λt) and a stronger noise level (SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The parameters are summarized in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Table 1: Modeling parameters variable symbol (unit) TRIS EGTA ptx1 Intrinsic freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (23,26) fc, ft (Hz) 45,65 45,65 45,65 Basal coupling∗ λc + λt (Hz) 60 60 60 cis dominance (23,38) λc : λt (-) 4:1 4:1 1:1 Flow detuning ν (Hz) [-10,10] [-10,10] [-10,10] Total forcing (51) εc + εt (Hz) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='08 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='8 Noise∗ (31) T c,t eff (rad2/s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='57 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='57 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='42 ∗ detailed in SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='S3 16 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Berg, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Brown, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Chemotaxis in escherichia coli analysed by three-dimensional tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Nature 239, 500–504 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Smriga, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Fernandez, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Mitchell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Stocker, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Chemotaxis toward phytoplank- ton drives organic matter partitioning among marine bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences 113, 1576–1581 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Hegemann, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Berthold, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Chapter 13 - sensory photoreceptors and light control of flagellar activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In Harris, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Stern, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Witman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=') The Chlamydomonas Sourcebook (Second Edition), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3, 395–429 (Academic Press, London, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Ueki, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Matsunaga, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Inouye, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Hallmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' How 5000 independent rowers coor- dinate their strokes in order to row into the sunlight: Phototaxis in the multicellular green alga volvox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' BMC Biology 8, 103 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Stehnach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Waisbord, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Walkama, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Guasto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Viscophobic turning dictates microalgae transport in viscosity gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Nature Physics 17, 926–930 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Brokaw, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Josslin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Bobrow, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Calcium ion regulation of flagellar beat symmetry in reactivated sea urchin spermatozoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Biochemical and Biophysical Research Communica- tions 58, 795–800 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Gong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The steering gaits of sperm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Philosophical Transactions of the Royal Society B: Biological Sciences 375, 20190149 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Gadˆelha, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Hern´andez-Herrera, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Montoya, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Darszon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Corkidi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Human sperm uses asymmetric and anisotropic flagellar controls to regulate swimming symmetry and cell steering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Science Advances 6, eaba5168 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Bennett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Golestanian, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' A steering mechanism for phototaxis in Chlamydomonas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Journal of The Royal Society Interface 12, 20141164 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' R¨uffer, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Nultsch, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Flagellar photoresponses of Chlamydomonas cells held on mi- cropipettes: II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Change in Flagellar Beat Pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Cell Motility and the Cytoskeleton 18, 269–278 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 17 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Ueki, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Wakabayashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Dynein-mediated photobehavioral responses in Chlamy- domonas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In Dyneins, 368–385 (Elsevier Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', 2017), second edition edn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Naitoh, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Kaneko, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Reactivated triton-extracted models of paramecium: Modification of ciliary movement by calcium ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Science 176, 523–524 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Merchant, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The Chlamydomonas genome reveals the evolution of key animal and plant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Science 318, 245–250 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Choi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Kwak, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Sung, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Sim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Quantitative analysis of the chemotaxis of a green alga, Chlamydomonas reinhardtii, to bicarbonate using diffusion- based microfluidic device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Biomicrofluidics 10, 014121–014121 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' R¨uffer, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Nultsch, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Flagellar photoresponses of Chlamydomonas cells held on mi- cropipettes: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' change in flagellar beat frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Cell Motility and the Cytoskeleton 15, 162–167 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Holmes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Dutcher, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Cellular asymmetry in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' reinhardtii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Journal of Cell Science 94, 273–285 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Dutcher, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & O’Toole, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The basal bodies of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' reinhardtii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Cilia 5, 1–7 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Mittelmeier, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Boyd, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Lamb, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Dieckmann, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Asymmetric properties of the Chlamydomonas reinhardtii cytoskeleton direct rhodopsin photoreceptor localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Journal of Cell Biology 193, 741–753 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Sakakibara, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Mitchell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Kamiya, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' A Chlamydomonas outer arm dynein mutant missing the alpha heavy chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The Journal of cell biology 113, 615–622 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Mackinder, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' A spatial interactome reveals the protein organization of the algal co2-concentrating mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Cell 171, 133–147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='e14 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Yu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Liu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Venkatachalam, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Hopkinson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Lechtreck, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The bbsome restricts entry of tagged carbonic anhydrase 6 into the cis-flagellum of chlamydomonas reinhardtii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' PLOS ONE 15, 1–21 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Kamiya, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Witman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Submicromolar levels of calcium control the balance of beating between the two flagella in demembranated models of Chlamydomonas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Journal of Cell Biology 98, 97–107 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 18 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Okita, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Isogai, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Hirono, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Kamiya, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Yoshimura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Phototactic activity in Chlamydomonas ’non-phototactic’ mutants deficient in Ca2+-dependent control of flagellar dominance or in inner-arm dynein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Journal of Cell Science 118, 529–537 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Takada, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Kamiya, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Beat frequency difference between the two flagella of Chlamy- domonas depends on the attachment site of outer dynein arms on the outer-doublet micro- tubules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Cell Motility and the Cytoskeleton 36, 68–75 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Saegusa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Yoshimura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' cAMP controls the balance of the propulsive forces generated by the two flagella of Chlamydomonas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Cytoskeleton 72, 412–421 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Kamiya, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Hasegawa, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Intrinsic difference in beat frequency between the two flagella of Chlamydomonas reinhardtii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Experimental Cell Research 173, 299–304 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' R¨uffer, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Nultsch, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Comparison of the beating of cis- and trans-flagella of Chlamy- domonas cells held on micropipettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Cell Motility and the Cytoskeleton 7, 87–93 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Wan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Leptos, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Goldstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Lag, lock, sync, slip: the many phases of coupled flagella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Journal of The Royal Society Interface 11, 20131160 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Leptos, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Antiphase synchronization in a flagellar-dominance mutant of Chlamy- domonas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Physical Review Letters 111, 1–5 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Wan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Coordination of eukaryotic cilia and flagella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Essays In Biochemistry 62, 829– 838 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Quaranta, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Aubin-Tam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Tam, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Hydrodynamics versus intracellular coupling in the synchronization of eukaryotic flagella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Physical Review Letters 115, 238101 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Wan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Goldstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Coordinated beating of algal flagella is mediated by basal coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences 113, E2784–E2793 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' R¨uffer, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Nultsch, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' High-speed cinematographic analysis of the movement of Chlamydomonas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Cell Motility 5, 251–263 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Klindt, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Ruloff, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Wagner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Friedrich, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In-phase and anti-phase flagellar synchronization by waveform compliance and basal coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' New Journal of Physics 19, 113052 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 19 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Transitions in synchronization states of model cilia through basal connection coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Journal of The Royal Society Interface 15 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Guo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Man, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Wan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Kanso, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Intracellular coupling modulates biflagellar synchrony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Journal of The Royal Society Interface 18, 20200660 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Wan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Goldstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Rhythmicity, recurrence, and recovery of flagellar beating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 113, 238103 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Horst, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Witman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Ptx1, a nonphototactic mutant of Chlamydomonas, lacks control of flagellar dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The Journal of Cell Biology 120, 733–741 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Wei, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Dehnavi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Aubin-Tam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Tam, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Is the zero reynolds number approx- imation valid for ciliary flows?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Physical Review Letters 122, 124502 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Wei, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Dehnavi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Aubin-Tam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Tam, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Measurements of the unsteady flow field around beating cilia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Journal of Fluid Mechanics 915, A70 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Kralemann, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Cimponeriu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Rosenblum, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Pikovsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Mrowka, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Phase dy- namics of coupled oscillators reconstructed from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' E 77, 066205 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Pikovsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Rosenblum, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Kurths, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Synchronization: A Universal Concept in Non- linear Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Cambridge Nonlinear Science Series (Cambridge University Press, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Polin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Tuval, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Drescher, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Gollub, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Goldstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Chlamydomonas swims with two “gears” in a eukaryotic version of run-and-tumble locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Science 325, 487– 490 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Friedrich, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Hydrodynamic synchronization of flagellar oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The European Physi- cal Journal Special Topics 225, 2353–2368 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Yoshimura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Stimulus Perception and Membrane Excitation in Unicellular Alga Chlamy- domonas, 79–91 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Wakabayashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Ide, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Kamiya, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Calcium-dependent flagellar motility activation in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' reinhardtii in response to mechanical agitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Cell Motility and the Cytoskeleton 66, 736–742 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 20 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Pazour, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Agrin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Leszyk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Witman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Proteomic analysis of a eukaryotic cilium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The Journal of Cell Biology 170, 103–113 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Quarmby, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Hartzell, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Two distinct, calcium-mediated, signal transduction pathways can trigger deflagellation in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' reinhardtii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The Journal of Cell Biology 124, 807–815 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' R¨uffer, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Nultsch, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Flagellar coordination in Chlamydomonas cells held on mi- cropipettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Cell Motility and the Cytoskeleton 41, 297–307 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Brumley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Wan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Polin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Goldstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Flagellar synchronization through direct hydrodynamic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' eLife 3, e02750 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Klindt, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Ruloff, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Wagner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Friedrich, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Load response of the flagellar beat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Physical Review Letters 258101, 1–5 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Pellicciotta, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Entrainment of mammalian motile cilia in the brain with hydrody- namic forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences 117, 8315–8325 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Hayashi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Yagi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Yoshimura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Kamiya, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Real-time observation of Ca2+-induced basal body reorientation in Chlamydomonas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Cell Motility and the Cytoskeleton 41, 49–56 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Harz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Hegemann, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Rhodopsin-regulated calcium currents in Chlamydomonas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Na- ture 351, 489–491 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Keller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Rubinow, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Slender-body theory for slow viscous flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Journal of Fluid Mechanics 75, 705–714 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Power, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Miranda, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Second kind integral equation formulation of stokes’ flows past a particle of arbitrary shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' SIAM Journal on Applied Mathematics 47, 689–698 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Keaveny, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Shelley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Applying a second-kind boundary integral equation for surface tractions in stokes flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Journal of Computational Physics 230, 2141 – 2159 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Kamiya, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Analysis of cell vibration for assessing axonemal motility in Chlamydomonas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Methods 22, 383–387 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 21 Acknowledgments The authors thank Roland Kieffer for technical support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' thanks Ritsu Kamiya for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The authors acknowledge support by the European Research Council (ERC starting grants no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 716712 and no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 101042612).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Author Contributions D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' performed experiments, computations, designed the model, and drafted the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' performed early experiments and obtained preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' conceived the study, supervised the project and critically revised the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Competing interests Authors declare that they have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Supplementary materials Supplementary Text Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S1 to S5 References (23,26,28,29,31,38,43,58) 22 Figure 1: Experimental workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (A) Captured CR cells are subjected to sinusoidal flows of frequency ff along given angles (θ) in the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Flows along θ = −45◦, 0◦, 45◦ of same amplitude (780±50 µm/s, mean±std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=') are used and termed as shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (B-E) Extracting flagellar phase ϕc and ϕt by image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Raw images (B) are thresholded and contrast-adjusted to highlight the flagella (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Mean pixel values within the user-defined interrogation windows (red and blue circles) capture the raw phases of beating (D), which are then converted to observable- independent phases (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Inset: phase difference ϕc − ϕt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (F) Flagella-flow phase dynamics at decreasing detuning ν = ff − f0 with f0 the cell’s beating frequency without external flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Traces i to iv are taken at detunings marked in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Plateaus marked black represent flow synchronization, whose time fractions τ = tsync/ttot are noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' ttot is the total time of recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Inset: the flow synchronization profile, τ(ν), reports the effective forcing strength 2ε by its width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 23 A Oa- flow,0 B eyespot Ot- flow, 45 c -flow,-45° A (s/ur) 780μm/s D D t () E c pt (2元) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='5S 0 4444442 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='5 0 4 8 12 16 t (beat) F 12 IV ii () - ) i, T=0 T 2 8 ii ii, T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='18 1 0 6 0 ii, T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='80 v (Hz) 4 iv, T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='00 0 0 2 4 6 8 10 t (s)Figure 2: External flagellar loads when beating is synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Force magnitude (upper pan- els) and power (lower panels) exerted by external flows of θ = 0◦ (A, θa-flow), −45◦ (B, θc-flow), and +45◦ (C, θt-flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The medians (solid lines) and interquartile ranges (shadings) are computed over ∼20 synchronized beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Dashed horizontal lines: loads averaged over a synchronized beat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Force magnitudes and powers are scaled by F0=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='9 pN and P0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='1 fW respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Flagellar phase corresponds to the displayed shapes in the middle x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 24 I cis loads trans loads = Median Interquartile :.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' : Beat-averaged A B FFlow/Fo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='5 0 H /Po 10 0 10 0元/2元3元/22元 0元/2元3元/2 2元 0元/2元3元/2 2元 Flagellar phase (rad) Flagellar phase (rad) Flagellar phase (rad)Figure 3: Flow synchronization of wt cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (A) Arnold tongue of a representative cell tested with θa-flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The contour is interpolated from N=132 measurements (6 equidistant amplitudes × 22 equidistant frequencies), and color-coded by the entrained time fraction τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (B) The syn- chronization profiles τ(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' θ) of a representative wt cell (inset), the median profile of the TRIS group wt cells (N=11, solid lines) and the EGTA group (N=6, dashed lines), with either θc- flows (red), θa-flows (yellow) or θt-flows (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Shaded areas are the interquartile ranges for the TRIS group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (C) Tested wt cells represented on the ε(θc) − ε(θt) plane (TRIS group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Solid line: the first bisector line (y = x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (D) Comparing τ(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' θc) and τ(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' θt) for each cell at each ap- plied frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' N=132 pairs of experiments are represented on the τ(θc) − τ(θt) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' More than 90% of them are below the first bisector line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (E) The coupling strengths ε(θ) of the TRIS group (black) and the EGTA group (gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Bars and error bars: mean and 1 std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Inset: δε = ε(θc) − ε(θt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' NS: not significant, p>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='05, Kruskal-Wallis test, One-Way ANOVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Relations between the forcing strength ε and the loads on the cis (F) and the trans flagellum (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Markers represent different flow angles, see the drawings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 25 20 C 4 0 (zH) (10)3 2 8 0 0 10-5 0 510 15 2025 0 2 4 6 v (Hz) ε(0c) (Hz) B D A single celi 4 0 4 Median over population T=1 TRIS 0 EGTA : 2 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='0 Hz T 0 t(Gc) (-) 1 E 6 (zH) 3NS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='3 Hz E 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='8 Hz m 0 8 4 0 4 8 v (Hz) TRIS EGTA cis loads trans loads F 4 G 4 2 2 口 m 3 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='5 0 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='5 0 5 FFlow/Fo Plow/Po FFlow/Fo Pflow/PoFigure 4: The asymmetric susceptibility to flow synchronization is lost in the flagellar domi- nance mutant ptx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (A) Arnold tongue of a representative ptx1 cell tested with θa-flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The contour is interpolated from N=132 measurements (6 equidistant amplitudes × 22 equidistant frequencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Color bar: the entrained time fraction τ = tsync/tIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (B) Flow synchronization profiles τ(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' θ) of N=14 ptx1 cells, tested with θc-flows (red), θa-flows (yellow) and θt-flows (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (C) ε(θc) and ε(θt) of the tested cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The first bisector line (solid): y = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (D) τ(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' θc,t) for each cell at each applied frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' N=154 points are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 26 A 20 6 () n/n 口 10 (zH) 4 口 口 口 8 口 @2 口 口 10 5 0 5 10 15 20 25 口 3 v (Hz) 0 B Median Interquartile 0 2 4 6 (0c) (Hz) 0 (-) (0)1 T 0 1 0 0 8 4 0 4 8 0 1 v (Hz) T(c) (-)Figure 5: Modeling the asymmetric flow synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (A) Modeling scheme describing a cell beating under directional flow (θc-flow as an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Arrows represent the directional coupling coefficients with line thickness representing the relative strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' For example, λc points from cis to trans, representing how the latter (ϕc) is sensitive to the former (ϕt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' mean- while, the arrow of λc being thicker than λt means that ϕt is much more sensitive to ϕc than the other way around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (B) Modeled phase dynamics of flow synchronization under θa-flows, analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 1F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Reproducing the Arnold tongue diagrams at the noise level of wt (C) and ptx1 (D), analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3A and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 4A respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (E) Flow synchronization profiles τ(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' θ) obtained experimentally (upper panel) and by modeling (lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Inset: the mod- eling results with symmetric inter-flagellar coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (F) Effective forcing strength ε(θ) as a function of the inter-flagellar coupling asymmetry λc/λt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Points: measured from simulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' lines: analytical approximation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (3));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' dashed lines: εc respectively for the θc-flow, θa-flow, and θt-flow (from top to bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (G) Reproducing the flow synchronization of wt cells under calcium depletion (H) Reproducing results of ptx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' See Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 1 for the modeling parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 27 A B trans cis (fe, pc) 12 (ft, Pt) 111 1 ii 1 () fi, -0 6 0 6 Idh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' v (Hz) Λt ji, {-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='27 i, t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='85 Et Ec iv, T=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='00 (fr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Pf) 0 0 2 4 6 8 Induced flow (0=-45°) t (s) C D 10 10 Ec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='t (Hz) (zH) 5 Low noise level Ec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='t High noise level 0 0 10-5 ¥05101520 25 10-5 051015 20 25 v (Hz) v (Hz) E F Exp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' wt 3 0 (zH) 2 0 0 Model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='wt m Analytical 2e/t=4 1 6 Monte-Carlo Ac=入t 000 Ec 0 0 6 6 0 0 5 10 15 20 8 入/M v (Hz) G H Exp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='wt Exp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' ptx1 EGTA 0 0 T 1 L [Model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='wt Model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' ptx1 EGTA 0 0 6 0 6 6 0 6 v (Hz) v (Hz)Supplementary materials for The younger flagellum coordinates the beating in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' reinhardtii Da Wei1,3,Greta Quaranta2, Marie-Eve Aubin-Tam1†, Daniel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Tam2∗ 1Department of Bionanoscience, Delft University of Technology, 2628CJ Delft, Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 2Laboratory for Aero and Hydrodynamics, Delft University of Technology, 2628CD Delft, Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Beijing 100190, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Email: m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='aubin-tam@tudelft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='nl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Email: d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='tam@tudelft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='nl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='13278v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='bio-ph] 30 Jan 2023 S1 Extracting coupling strength by fitting phase dynamics In the work described in the manuscript, the flagellum-flow coupling strength ε in wt cells is mainly extracted by the synchronization profile τ(ν) ≥50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Meanwhile, in previous works [1, 2], fitting the distribution of phase dynamics is employed to extract ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In the latter approach, the idea is that the phase locking during synchronization leads to a peaked probability distribution of ∆ϕ, whose width is affected by the effective noise Teff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The distribution, P(∆ϕ), can be derived from the Adler equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (1) as: P(∆ϕ) = � ∆ϕ+2π δct exp(V (∆ϕ′) − V (∆ϕ) Teff )d∆ϕ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (S1) Here V (∆ϕ) = ν∆ϕ + ε cos(∆ϕ) is a wash-board potential, Teff is the noise, and ∆ϕ is the difference between the flagellar phase and the flow’s phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Here, we demonstrate that these two approaches are equivalent in extracting ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' For all wt cells tested in the TRIS-minimal medium (N=11), their ε(θ) measured by the τ(ν) width and extracted from fitting are plotted against each other, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' All points center around the identity line, showing the equivalence in obtaining ε by the two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' For the ptx1 dataset, ε are extracted from fitting the phase dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 2 �������������� ����������������� ���������� ��������� ������������� Figure S1: Equivalence of extracting coupling strength ε by different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Each point represents one cell under either the θa-flow (green square), the θc-flow (red circle), or the θt- flow (blue triangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The x coordinate is the coupling strength ε measured by the half width of synchronization profile τ(ν) ≥ 50%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' and the y coordinate is obtained by fitting the flagellar phase dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 3 S2 Hydrodynamic computation for flow along 90 degree Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 2 in the main text, we present the computed drag force and power for the flow along 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The solid lines and the shadings represent the median and the interquartile range of FFlow and PFlow over the flow-synchronized beats, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Force magnitudes are scaled by F0 = 6πµRU0 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='9 pN, which is the Stokes drag on a typical free-swimming cell (radius R = 5 µm, swim velocity U0 = 110 µm/s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' while the viscous powers are scaled by P0 = F0U0 = 6πµRU 2 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='1 fW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Here µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='95 mPa·s is the dynamic viscosity of water at 22 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Quantitatively, the mean force is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='37F0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='34F0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S2 top panel) while the mean power is -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='2P0 and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='4P0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S2 bottom panel), for the cis and the trans respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Figure S2: Computed hydrodynamic loads on the flagella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Computation results of the drag force (upper panel) and the force’s rate of work (lower panel) on the cis (red) and the trans (blue) flagellum during synchronized cycles, when the cell is subjected to the flow with θ = 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Scaling factors F0=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='9 pN and P0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='1 fW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 4 FFlow/Fo 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='5 0 cis loads 10 PFlow/Po trans loads 0 10 0 元/2 2元3元/22元 Flagellar phase (rad)S3 The model The external flow and the two flagella are described by three coupled ordinary differential equa- tions (ODEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Phase dynamics of these equations are examined by Monte-Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The temporal resolution of simulation (dt) is 1 ms, which corresponds to the experimental frame rates (801 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' � � � � � � � � � � � � � dϕf dt = 2πff (S2a) dϕc dt = 2πfc − 2πλt sin(ϕc − ϕt) − 2πεc sin(ϕc − ϕf) + ζc(t) (S2b) dϕt dt = 2πft − 2πλc sin(ϕt − ϕc) − 2πεt sin(ϕt − ϕf) + ζt(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (S2c) The cis, the trans, and the external flow are described as oscillators, whose intrinsic fre- quencies are fc,t,f and phases ϕc,t,f, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The flow is assumed to be noise free and the two flagella are assumed to have the same level of noise (ζc = ζt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The noises are assumed to be Gaussian, ⟨ζc,t(τ + t)ζc,t(τ)⟩ = 2T c,t eff δ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='1 Flagellar synchronization Setting εc and εt to 0, the interaction between the two flagella in the absence of the flow is modeled by: � � � � � dϕt dt = 2πfc − 2πλt sin(ϕc − ϕt) + ζc(t) (S3a) dϕc dt = 2πft − 2πλc sin(ϕt − ϕc) + ζt(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (S3b) When the two flagella are able to beat synchronously, dϕc dt = dϕt dt = f0, we can obtain the analytical expression of f0 by adding up λc×Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (S3a) and λt×Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (S3b): f0 = λtft + λcfc λc + λt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (S4) 5 Meanwhile, the steady-state phase difference δct = ϕc −ϕt is obtained by subtracting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (S3a) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (S3b): sin(δct) = fc − ft λc + λt = νct λtot .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (S5) It is therefore obvious that the two flagella can only beat at the same frequency (dϕc/dt = dϕt/dt = f0) if |νct/λtot| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='2 Interaction between three oscillators Now we put the flow back into the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' According to experimental observations, the two flagella mostly beat synchronously, we therefore focus on this case and first simplify the equa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' By adding up λc×Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (S2b) and λt×Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (S2c), and substituting ϕc,t as ϕ0 = ϕc −δct/2 = ϕt + δct/2, we obtain: dϕ0 dt = 2πf0−2π λcεc λc + λt sin � ϕ0 − ϕf − δct 2 � −2π λtεt λc + λt sin � ϕ0 − ϕf + δct 2 � +λtζt + λcζc λc + λt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (S6) Given different choices of coupling constants (λc,t, εc,t), this equation would generate com- plex phase dynamics - as we shall see in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We first limit the discussion to small δct - as it is observed in our experiment as well as in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The model’s asymptotic behavior at δct ≈ 0 is: dϕ0 dt = 2πf0 − 2πε sin(ϕ0 − ϕf) + ζ0(t), (S7) where f0 = λtft + λcfc εtc + λt , ε = λtεt + λcεc λc + λt , ζ0 = λtζt + λcζc λc + λt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (S8) In this strong-coupling limit (δct ≈ 0, or equivalently, λtot ≫ νct), the coupled flagella behaves as a single oscillator whose beating frequency f0 will not be interfered by the external flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The analytical form well captures the system’s behavior, as shown by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 5F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Next we explore the model’s behaviors when λtot − νct is comparable with ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 6 ��� ��� ��� ��� ��� ��� ������������ f� f� ��������cis��������� ��������trans��������� ��������trans ����cis ��� ��� ��� ������ ������ ������ Figure S3: Determine the lower limit of λtot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The time fractions of the cis (a) and the trans flagellum (b) synchronized by the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (c) The time fraction of where cis and trans are syn- chronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Arrows points towards increasing (λtot − ν)/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='3 Lower limit of inter-flagellar coupling The value (λtot − νct)/ε determines if the flow can disrupt the synchronization between cis and trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We assume νct = 20 Hz[4, 5, 6, 3] and focus on synchronization of the θa-flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We plot the synchronization time fractions with increasing λtot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' When it satisfies (λtot − νct)/ε ≥ 2, external flows cease to affect the flagellar synchronization observably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' As the strongest flow (21U0) applied experimentally corresponds to ε ≈ 10 Hz, altogether, we conclude that λtot ≳ νct + 2εmax = 40 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In the main text, we set λtot = 60 = 3νct Hz, which satisfies this relation and matches the observation that the phase lag between the flagella (δct) is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 7 S4 Flagellar noise of the ptx1 mutant Here we show an as-yet uncharacterized strong noise present in the synchronous beating of the mutant ptx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The in-phase (IP) mode of ptx1 cells and the breaststroke beating of the wt cells are similar in waveform and frequency [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' However, the former has a much stronger noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Figure S4: Stronger frequency fluctuation of the IP mode of ptx1 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (a-e) Representative probability distributions of the beating frequency of a wt (a) and four ptx1 cells (b-e) over 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Probability distributions of the IP (purple) and AP mode (yellow) are respectively nor- malized for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The time fractions of the AP mode are noted in each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (f) The wt and ptx1 cells represented by its mean beating frequency ⟨f0⟩ and the standard deviation of the beating frequencies over time σ(f0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The strong noises show obviously in fluctuations of IP beating frequencies [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S4, we display the distribution of beating frequency of a representative wt cell (panel a) and four representative ptx1 cells (panels b-e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The broad peaks of the IP (purple) and AP (yellow) beating of ptx1 sharply contrast the narrow peak of wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We quantify the frequency fluctuations of all the cells in the main text (N=11 for wt and N=14 for ptx1), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S4f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The cells are represented by its mean beating frequency over time ⟨f0⟩ and the frequency’s standard deviation σ(f0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Clearly, the breaststroke beating of wt, the IP, and the AP mode of ptx1 each forms a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The wt cluster is at (⟨f0⟩, σ(f0)) = (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='3) Hz (mean± 1 std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' the over cell population);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' and it is evidently less dispersed than both the IP and the AP mode 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='1 (a) (f) wt wildtype ptxl, IP mode 0 40 50 60 70 80 ptxl, AP mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='06 4 AP: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='9% (b) ptx1 0 (zH) (°J)o 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='04 AP: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (c) PDF 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='06 AP: 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='7% (d) 口 0 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='04 AP: 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='4% 口 (e) 0 0 40 50 60 70 80 40 50 60 70 80 Frequency (Hz) (fo) (Hz)of ptx1, which are at (47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='9) Hz and (67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='7) Hz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Under the assumption of a white (Gaussian) noise, σ(f0) is proportional to the noise level ζ, and thus scales with √Teff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Consider that σ(f0) for ptx1 is 3-5 folds larger than that of wt, we therefore conclude that the noise level in ptx1 is an order of magnitude larger than wt, T ptx1 eff /T wt eff ∼ O(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Figure S5: Effect of a low-noise cis in stabilizing the beating of the trans (a) Fluctuations in beating frequency (σ(f0)) under different coupling schemes and flagellar noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Other model parameters are the same as used in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The red and blue shaded area represent the experimentally observed range for ptx1 and wt cells, respectively, with short bars marking the mean values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' (b) the rate of slip under the conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Error bars correspond to 1 std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' over N=9 repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The stronger noise in ptx1 can be attributed to two sources, namely, the loss of a stable cis and the loss of the unilateral coupling, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' We perform Monte-Carlo simulations of the coupled beating of cis and trans under three conditions: (1) a stable cis (T c eff = T 0 eff = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='57 rad/s2) coupled with the trans unilaterally (λc = 4λt), (2) a stable cis coupled with the trans bilaterally (λc = λt), and (3) an equally noisy cis (T c eff = T t eff) bilaterally coupled with trans, see the blue, yellow, and red data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' It is obvious that, when the trans is coupled to a stable cis, varying its noise over an order of magnitude only leads to a ∼ 20% stronger frequency fluctuation (the blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S5(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' On the contrary, lacking either the unilateral coupling or the low-noised cis would increase the fluctuation for 200% (yellow 9 = = = = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='1 (a) (b) 3 ptx1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='08 Slip rate (Hz) o(fo) (Hz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='06 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='04 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='02 im 0 0 100 101 10° 101 Teff / Teff Teff / Teffline) or 300% (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Qualitatively, simulation results are in agreement with experimental measurements assuming that T t eff/T c eff ∼ O(10), see the red and blue shaded areas in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Moreover, a low-noise cis is already sufficient to prevent slips from interrupting the synchrony between cis and trans, even for bilateral coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' S5(b), as long as the cis-noise remains low, slips will be sparse (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='01 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Together, these simulation results highlight the stabilizing effect of a low-noise cis flagellum, and illustrates the contribution of unilateral coupling in further enhancing the stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' References [1] Polin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Tuval, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Drescher, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Gollub, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Goldstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Chlamydomonas swims with two “gears” in a eukaryotic version of run-and-tumble locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Science 325, 487– 490 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' [2] Quaranta, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Aubin-Tam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Tam, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Hydrodynamics versus intracellular coupling in the synchronization of eukaryotic flagella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Physical Review Letters 115, 238101 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' [3] Wan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Leptos, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Goldstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Lag, lock, sync, slip: the many phases of coupled flagella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Journal of The Royal Society Interface 11, 20131160 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' [4] Kamiya, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Hasegawa, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Intrinsic difference in beat frequency between the two flagella of Chlamydomonas reinhardtii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Experimental Cell Research 173, 299–304 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' [5] Kamiya, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Analysis of cell vibration for assessing axonemal motility in Chlamydomonas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Methods 22, 383–387 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' [6] Okita, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Isogai, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Hirono, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=', Kamiya, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Yoshimura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Phototactic activity in Chlamydomonas ’non-phototactic’ mutants deficient in Ca2+-dependent control of flagellar dominance or in inner-arm dynein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Journal of Cell Science 118, 529–537 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 10 [7] Horst, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' & Witman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Ptx1, a nonphototactic mutant of Chlamydomonas, lacks control of flagellar dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' The Journal of Cell Biology 120, 733–741 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' [8] Leptos, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Antiphase synchronization in a flagellar-dominance mutant of Chlamy- domonas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' Physical Review Letters 111, 1–5 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} +page_content=' 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf'} diff --git a/EdE0T4oBgHgl3EQfywKq/content/tmp_files/2301.02664v1.pdf.txt b/EdE0T4oBgHgl3EQfywKq/content/tmp_files/2301.02664v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6f229009ab549e78518ca4825762bcea712aaaaf --- /dev/null +++ b/EdE0T4oBgHgl3EQfywKq/content/tmp_files/2301.02664v1.pdf.txt @@ -0,0 +1,636 @@ +Lindbladian-Induced Alignment in Quantum +Measurements +R. Englman and A. Yahalom +Ariel University, Ariel 40700,Israel +January 10, 2023 +Keywords: Quantum measurement theory, Density matrix evolution, Quan- +tum state resolution, Lindblad operators, Quantum speed limit. +Abstract +An expression of the Lindbladian form is proposed that ensures an un- +ambiguous time-continuous reduction of the initial system-pointer wave- +packet to one in which the readings and the observable’s values are aligned, +formalized as the transition from an outer product to an inner product of +the system’s and apparatus’ density matrices. The jump operators are in +the basis of the observables, with uniquely determined parameters derived +from the measurement set-up (thereby differing from S. Weinberg’s Lind- +bladian resolution of wave-packet formalism) and conforming to Born’s +probability rules. The novelty lies in formalising the adaptability of the +surroundings (including the measuring device) to the mode of observa- +tion. Accordingly, the transition is of finite duration (in contrast to its +instantaneousness in the von Neumann’s formulation). This duration is +estimated for a simple half-spin-like model. +1 +Introduction +In the century-run of quantum physics (plus 4 years, if one marks its beginning +with the award of a Nobel Prize in 1918 to Max Planck for ”his discovery of +quanta”) a single shadow of non-sequitur has darkened its glorious achievements, +one that goes variously under the names of wave-function collapse, reduction of +the wave-packet, quantum measurement, einselection, etc. Aspects of the prob- +lem (or its articulations) were manifold, such as the breakdown of the predicted +time-development in accordance with the Schr¨odinger equation, the abruptness +of change in a measurement (”natura non facit saltum”, where art thou?), the +apparent non-applicability of quantum rules to macroscopic systems, imputed +arbitrariness of Born’s probability rules, the requirement of ”infinite regress” +1 +arXiv:2301.02664v1 [quant-ph] 6 Jan 2023 + +for the measuring apparatus and others. Numerous papers enlarged on these is- +sues [1, 2] and various proposals for resolution of the problem were put forward. +These include the observer’s cognition [3], stochastic effects [4], in particular +spontaneous localization [2, 5, 6], a many world scenario [7], non-linearity addi- +tion to the Schr¨odinger equation [8], Poincar´e recurrent state [9], gravitationally +induced collapse [10, 11, 12], etc. +Common to these works, and with the specific purpose of providing a +blue-print for measurements compatible with the Copenhagen formulation of +quantum theory, was the need to give expression to the coupling of the micro- +scopic system with its macroscopic environment. Standing apart from these and +belonging to the field of non-equilibrium thermodynamics and to the establish- +ment of equilibrium, a general form for this interaction was given by Lindblad +[13] and by Gorin , Kossakowski and Sudarshan [14], satisfying some necessary +conditions. Constructing a merger between the two separately oriented fields, +S. Weinberg recently proposed a Lindblad-operator mechanism for the collapse +of the density matrix (DM) in the course of a complete measurement [15]. No- +tably, the mechanism was linear in the state’s DM. The collapsed state (Eq. (1) +in [15]) comprises the set of projection operators of the measurable item; the +system’s Hamiltonian is described by a spectral decomposition onto the same +operators (Eq. +(16) in [15]) (although in the verbal discussion a more gen- +eral situation is considered): collapse is achieved ”independent[ly] of the details +of these [Lindblad] operators”. Decay between energy eigenstates had earlier +been treated by the Lindblad formalism (for a pedagogical presentation the +volume [16], Chapter 8 may be consulted) employing the interaction represen- +tation. However, this is not convenient for treating measurements of observables +that do not commute with the Hamiltonian. Detailed theories relate to the out- +come (”mapping”) of quantum operations, including measurements; the present +work describes the process of these happening.(For a pedagogical introduction +to stochasticity-induced wave-packet- reduction, obviating pointer reading, one +may refer to [17].) +2 +Overview of the Method and Terms +2.1 +The leading idea, also in review +While the concept of unity of observer and observation had already featured +in Bohr’s view: ”The answer that we get is built up from the combined interac- +tion of [the observer’s] state and the object of interrogation.” [18], this was not +given a formal expression in the Copenhagen interpretation. It was more em- +phatically asserted both by J. Bell: ”I meant that the ’apparatus’ should not be +severed from the rest of the world in boxes ...[19]” and A. Peres: ” A measure- +ment both creates and records a property of the system [20]”. This change in +2 + +the course of a measurement affects also the environment outside the observed +system ; in the words of A. Leggett ”...under these conditions the macroscopic +apparatus, and more generally any part of the macro-world which has suffered +changes in the course of the measurement process, does not end up in a state +with definite macroscopic properties at all,... [1]”. +The same line of thought appears to motivate S. Weinberg, who wrote in his +preamble to a 2016 Lindbladian formulation of the masurement process[15], that +”We will instead [of the original formulation of the Copenhagen interpretation, +(which we will not dwell on here)] adopt the popular modern +view that the +Copenhagen interpretation refers to open systems in which the transition is +driven by the ineraction of the microscopic system under study (which may +include an observer) chosen to bring the transition about.” (Our italics.) +These developments indicate the justification for a formulation in which the +effect of the apparatus is incorporated in the equation defining the evolution of +the system, rather than one in which the two entities are separate, barring an +interaction between them. +2.1.1 +”Alignment” +The process whereby the pointer readings become in correspondence with the +possible values of the observable. Formally, for I possible values, the combined +density matrix reduces from comprising I2 terms to one having I terms. (E.g., +equation (2.5) in [1].) +2.1.2 +”Dissipator” +Added term (in the form of sums of appropriately weighted jump-operators) +to the standard time dependent Schr¨odinger equation, inducing non-unitary +evolution in the system, accompanied by changes of its information entropy. +2.2 +Motivation for the choice of formalism +Thermalization of open systems can be described by a Lindbladian formal- +ism in which Gibbsian probabilities are so inserted as parameters, that the +”Dissipator” vanishes at these values of the density matrix. Replacement of +the Gibbsian probabilities by Born probabilities achieves alignment in a state +reduction and does so continuously. +Limitations: Born’s probability rules are assumed, not derived; the interac- +tion term is not traced to a microscopic mechanism. +The source of this interaction term, shown in Eqn. 6 below, incorporating +the coupling between the observed system and its surroundings (including the +3 + +measuring device) is an open question (also raised by a referee). In its appli- +cation to a thermalization process, the Lindbladian jump operators have been +derived, though with the aids of several approximations (e.g.,[21]), as well as, +more recently, for the dissipation in a Dicke system with a bosonic background +[22]. We are not in the position to provide such first principle derivation for +the Lindbladian jump-operators bringing about a transition and incorporat- +ing the Born rules. It seems to be specific to the type of measurement under +consideration and it is clear that just any jump operator, as in Weinberg’s +Lindbladian formulation will not do the job . Likely, one would need to in- +clude non-Markovian dynamics, so that the coupling to the device and eventual +pointer reading are two separate consecutive events. Inclusion of such dynamics +is outside the scope of the present work. +3 +Assumptions +We explore the time (t)-development of the combined density matrix ρ(t) of +the measured system (S) and of the reading (pointer, dial, etc.) on the mea- +suring apparatus (A) for a complete and discrete measurement , expressing the +underlying assumptions by three propositions. +Proposition 1. In accord with the long-time historical approach, the mea- +sured object S and the pointer of the measuring set-up A are treated on equal +footings as subject to microscopic quantum laws, and formally describable by +their respective Hamiltonians. Aware of the difficulties connected with an ”in- +finite regress”, the effects of the rest of the Universe on S+A are not included +in the formalism; instead, for a phenomenological, approximative description, a +Lindbladian term appears in the master equation. +Proposition 2. Prior to the measurement with A and S decoupled, and being +free of external influence for a long time, both are in energy quantum states, +pure or mixed. After the measurement, the state is not an energy eigenstate +and subsequently it will spread over to a superposition of energy eigenstates. +The fast decoherence case treated below in section 5 is akin to the Zeno effect +[23]. +Proposition 3. Only those states of the reading apparatus (e.g., the right +or left positions of a pointer) that may be in direct correspondence with the +measured states of the system (e.g., spin up or down) are given expression in +the formalism. (At a beginning, the case treated is one in which there is a one- +to-one correspondence between the states of the system and the readings of the +apparatus; a generalization is given subsequently.) A discussion in section +8 +touches on the epistemological status of the Lindbladian terms in a measurement +process. +4 + +4 +Analysis +Considering (for simplicity) a pure state for the system, its initial state-vector +written in the basis of the observed property |S, i > takes the form +ψS(t = 0) = +� +i=1,..,I +cS +i |S, i > +(1) +Born’s rule for the probability of observing the i-component is |cS +i ]2 ≡ pi, sum- +ming to unity. Likewise, for the apparatus readings j, numbering J, one has +the superposition with (complex and normalized) coefficients cA +j +ψA(t = 0) = +� +j=1,..,J +cA +j |A, j > +(2) +We start with the one-to-one correspondence situation, for which I = J, and the +reading j on A establishes uniquely the value i = j for the system’s measured +property. +For the combined state-vector the density operator has the outer-product +form (where the stars denote complex conjugates): +� +i,j,i′,j′ +|S, i > |A, j > cS∗ +i cA∗ +j cA +j′cS +i′ < A, j′| < S, i′| ≡ +� +i,j,i′,j′ +|i, j > Ciji′j′ < i′, j′| +(3) +the right hand side written in an obvious shortened notation, in which Ciji′j′ = +cS∗ +i cA∗ +j cA +j′cS +i′. After collapse, the density operator takes the aligned, single-sum +form +� +i +|S, i > |A, i > |cS +i |2 < S, i| < A, i| +(4) +It will be now shown that this is the time-asymptotic solution of the Lind- +bladian master equation properly parametrized. +We recall Lindblad’s equation for the time varying density of states operator +ρ ≡ ρ(t), as being of the following general form: +∂ρ +∂t = − i +¯h[H, ρ] + +� +n +γn⟨LnρL† +n − 1 +2(L† +nLnρ + ρL† +nLn)⟩ +(5) +The second term, here named the ”Lindblad term” [13, 14] though in different +contexts also referred to as the Dissipator [24], contains Ln’s that are Lindblad +jump-operators. We shall consistently work in the observable + pointer’s basis +(i.e., not in an energy basis). In this basis, neither the density operator ρ = ρ(t), +nor the A+S Hamiltonian H is diagonal at the beginning or in the course of +the development. But, as will be demonstrated, the Lindbladian formalism, by +a proper choice of its form, drives A+S to the desired diagonal form for the +combined observable +pointer basis. We postulate just one single term in the +previous n-sum, as well as off-diagonal forms, namely |i, j >< i′, j′|, (i, j ̸= +5 + +i′, j′), for the jump-operators in the observable basis, leading to the following +parametrized form of the Lindblad term +Lρ +≡ +ΓΩ +� +i′,j′̸=i,j +r(i, j) +r(i′, j′)⟨|i, j >< i′, j′|ρ|i′, j′ >< i, j| +− +1 +2(|i′, j′ >< i, j|i, j >< i′, j′|ρ + ρ|i′, j′ >< i, j|i, j >< i′, j′|⟩ +(6) +Here a circular frequency Ω is inserted, so as to make Γ , that quantifies the +strength of the system-environment coupling, dimensionless. One notes that in +the pre-factor appear the parameters r(i, j), r(i′, j′)(i, j, i′, j′ = 1, ..., I) whose +significance will be clear by deriving the matrix elements of the above operator. +These are +Lρi,j,i′,j′ += +δi,i′δj,j′r(i, j) +� +k,l +r−1(k, l)ρk,l,k,l +− +1 +2[r−1(i, j) + r−1(i′, j′)]ρi,j,i′,j′ +� +k,l +r(k, l) +(7) +It can be seen that the trace of the above vanishes and that each matrix element +vanishes upon the substitution +ρi,j,i′,j′ = δi,i′δj,j′r2(i, j) +(8) +While these properties hold for any arbitrary r(ij), the observable-pointer align- +ment is achieved by identifying the r parameters with the system’s superposition +coefficient: r(ij) = |cS +i |δi,j, or +r(i, j)2 = |cS +i |2 ≡ piδi,j +(9) +the last being the Born probabilities appearing in the collapsed state. As already +noted, this identification of probabilities relates to the well known procedure for +the Lindblad-induced thermalization of open systems, for which detailed balance +imposes the relation between the pre-factors γ(δE)/γ(−δE) = e−βδE/Z, the +latter being the canonical probabilities (with β = 1/kBT, kB the Boltzmann +constant, T the ambient temperature and Z the partition function [24, 25, 21]) +. +[It also seems fair to point out that also in the standard (Copenhagen, or von +Neumannian) description of the alignment stage, as appears in e.g. Eq.(2.5) of +[1], this development is summarily stated, without specification of the underlying +mechanism.] +5 +Fast Decoherence Limit +We now consider the case that the time development in the state is predom- +inantly due to the coupling to the environment, rather than to the unitary +6 + +change induced by the Hamiltonian, meaning that the second term on the right +hand side in Eq. 5 dominates the first. Quantitatively: Γ >> ||H||/¯hΩ. Ne- +glecting the commutator we now form matrix elements of the Lindblad term in +Eq. 5 in the observable+pointer basis. Because of the approximation made, the +off-diagonal matrix elements are decoupled from the diagonal ones. The master +equation of the off-diagonal terms reads (with a notation simplified by writing for +the index pairs i, j → r, i′, j′ → s and consequently for ρi,j,i′,j′ → ρrs ≡ ρrs(t) +dρrs +dt += −ΓΩ +�√pr + √ps +2 +ρrs +� +m +√pm +� +, r ̸= s +(10) +This shows that off-diagonal matrix elements decay exponentially in time (de- +cohere), maintaining their real character that they had initially. Had we kept +the (imaginary) commutator term, we would have found that the decay is mod- +ulated by the eigen-energies of the Hamiltonian. +For the diagonal matrix elements we find, +dρrr +dt += ΓΩ +� +√pr +� +m +ρmm +√pm +− ρrr +√pr +� +m +√pm +� +(11) +Again, it can be seen that the trace of the last expression vanishes, and so +does the right-hand side under the substitution ρrr → pr. With these taking the +values as in Eq. 9, one arrives at the aligned form (written out in the original, +system-pointer indexes) +ρ(t → ∞) = +� +i +|ψA +i > |ψS +i > |cS +i |2 < ψS +i | < ψA +i | +(12) +5.1 +Illustrative example for a two-way experiment +Exemplifying the foregoing for a two-valued system (such as a 1 +2-spin electron), +prepared as an eigenstate of a Zeeman-field with the magnetic field inclined at an +angle 2αS to the vertical, in conjunction with an apparatus pointer, represented +as being likewise in an eigenstate of a quasi-Zeeman field inclined at an angle 2αA +to the vertical. The eigenstates are linear superpositions of their z- spins; these +are the observables that are to be determined by the measurement. Initially, the +system and the pointer are in the superposition states as shown above in Eqs. +1 and 2 and whose superposition coefficients cS +i and cA +j now have the values, +sin / cos(αS) and sin / cos(αA), respectively. The DM in the observable basis +is now a 4x4 matrix, in which appear all the combinations of the products of +the above circular functions. As the outcome of the application of the Lindblad +operator in the rate equation, at long times the matrix becomes reduced to the +diagonal form discussed earlier. In these, cos2(αS) = p1 and sin2(αS) = p4 +belonging to the aligned observable lie on the diagonal and are non-zero; the +other two diagonal entries for the anti-aligned situations are zero. +Plotted in Figure 1 are computed DM eigenvalues as functions of time +(in red and blue), normalized to their respective Born probabilities, showing +7 + +-4 +-3 +-2 +-1 +0 +1 +2 +Log10time[invfrequn] +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +P1,P2,Decoh +Figure 1: Density matrix eigenvalues normalized to their asymptotic (pointer- +aligned) values for the two aligned terms in the illustrative example (in red and +blue), plotted against time in inverse circular frequency unit. In green is shown +a decohering off-diagonal matrix element. Lindblad coupling strength Γ = 5, α +angles .37 π and .65 π. +their asymptotic convergence. +In green, the typical decohering tendency of +an off-diagonal element is demonstrated. Figure 2 depicts the entropy S(t) = +− � +r Pr(t) log Pr(t) of the system and apparatus-pointer, (in which Pr(t) are +computed eigenvalues of the DM.) The non-monotonic behavior is characteristic +of of the Lindblad formalism, in which the environment’s entropy change is not +taken into account. +[In numerical work, based on forward integration, putting zeros for some +of the pi’s introduces singularities, eventually algebraically cancelling out, but +preventing flow of computation. Therefore, instead, one puts arbitrarily small +values for these and obtains for the aligned DM one that is arbitrarily close to, +but not exactly equal to the true one.] +6 +Eigenvalue analysis +An alternative to the numerical solution of the differential rate equation is eigen- +value analysis, already treated in [15], based on the Landbladian term being a +linear function of the diagonals in the density matrix. Thereby, the resulting +rate equations have solution of the form +ρnn(t) = +� +k +vn,keλkt +(13) +in which λk and vn,k are the diagonalized eigenvalues and eigenvectors of the +Lindbladian matrix diagonals in Eq. 11. Calculation shows that for the 4 x +8 + +-4 +-3 +-2 +-1 +0 +Log10time[invfrequn] +0.05 +0.10 +0.15 +0.20 +S +Figure 2: Entropy of the combined system plus apparatus. Noteworthy is the +initial peak common to the Lindblad formalism . +4 matrix considered above there are three negative eigenvalues and one zero +eigenvalue, which alone is of interest at the long term behavior. Belonging to +this eigenvalue, the (transposed) eigenvector is found to be {p1, p2, p3, p4} ≈ +{cS +1 , 0, 0, cS +4 }, as required for the alignment between the quantum states and the +reading in the measuring apparatus. +6.1 +Measurement speed +Figure 1 shows that alignment is achieved for the model with the chosen strength +parameter (Γ = 5) by a time of cca. 0.1/Ω. By varying the strength in the +computed model, we find a shortening of this time that is inversely proportional +to the strength. This is expected from the quantum speed limit (QSL) results +that border quantum transition times τ from below. +Essentially, QSL is the ratio of two norms [26, 27], that of the ”quantum +distance” [28] and of the speed of the state evolution. Formally +τ > ||ρ(t → ∞) − ρ(t = 0)|| +|| dρ(t) +dt || += ||ρ(t → ∞) − ρ(t = 0)|| +||[Lρ(t)]|| +(14) +Ways of calculating the norms vary, e.g., [29, 30]. Recently, for a system de- +veloping due to a Lindbalian operator, three contributions to the speed were +discerned [24]. To estimate ||ρ(t → ∞) − ρ(t = 0)|, we have used the ”Trace +Distance ”defined as +T(ρ, σ) = 1 +2Tr[ +� +(ρ − σ)] = 1 +2 +� +i +|µi| +(15) +[31], where µi are the eigenvalues of the matrix differences. The DM velocity, +as defined above , changes (decreases) with time, ultimately vanishing at the +9 + +fulfilment of alignment; we have taken the root-mean-square sum of the rate of +the diagonal matrix elements at initial times. These yield a very low limit of +τ > 1 +2 +1.41 +5 ∗ 69.03 = .0045/Ω +(16) +to be compared with the actually computed value, about 20 times longer. Better +(higher) limits of transition times may be generated by different ways of forming +the norm for the DM velocity (e.g. not at the beginning). +6.2 +Multiple Reading-System correspondence +A simple generalization of the foregoing applies when each (eigen-)value of the +observable is in correspondence with not just one reading of the pointer, but +with several (say, R) readings, all of the same significance for the outcome. Then +one simply inserts pi/R into the corresponding Lindblad term, in place of just +pi. In the more complex case, that not all readings have the same likelihood, +pi would have to be weighted by a probability factot, rather than by a constant +denominator. +7 +The Lindbladian, ”Who ordered this?” +Historically, Lindblad terms were introduced as the most general forms that +maintain complete positivity of the DM’s and preserve their trace [13, 14]. The +various derivations that have been presented (and among these a recent one by +[32]), involve several approximations for the coupling between the system and its +environment. Insomuch that the derivation involves also tracing over the degrees +of freedom of the environment, much detail of the latter is lost and of course +it is impossible to work backwards from the Lindbladian to the environment. +What is remarkable is that for special purposes the appropriate Lindbladian +operators take a very special, practically unique form. Such is the case for the +accepted description of thermalization [21, 24] by a Lindblad formalism. The +parametrization of the Lindblad term employed in the present work, though it +may appear arbitrary and particular for each case, is in fact identical to the +one used for thermalization subject to the relabelling of the Gibbsian thermal +distribution function as (the Born) probabilities (p1, p2, ...),with the proviso of +working in the observable, rather than in the energy basis. +(This contrasts +with the different approach in [15], which claims attainment of collapse for any +Lindbladian operator.) At the same time, it needs to be noted that the analog +of detailed balance is missing in wave-function collapse. +How come to have +such a specific Lindbladian, whose source may be any measurement device and +procedure? One is left to wonder about the possibility of a special meta-physical +status of the Lindblad terms, or query with Wheeler ”Who ordered this?” +10 + +8 +Conclusion +The well-known Lindbladian extension to the quantum theory of motion to +environmental effects is here adapted to establish the resolution of a wave-packet +in a measurement as a smooth process. This is enabled by an unambiguous +parametrization of the jump-operators describing the interaction of the broad +environment with the observed system, both regarded as quantal entities. +Above, in section 2.1, a brief historically oriented preview has been provided +for the distinct approach in this work, namely, one based on the wholeness of +the entities (observed system and observing device), through the (Lindbladian) +equation yielding the evolution of the system. +A main result emerging from the formalism, and capable of experimental +verification, is the finitely temporal variation of the system, and this in a de- +terministic way rather than just statistically, on the average, contrasting also +with the instantaneous collapse description by (e.g.) von Neumann. Such tem- +poral variation in continuous-thermalization processes has been proposed quite +recently [33, 34], also by employment of a Lindbladian formalism and within a +Markovian framework. +Experimentally, verification of the time dependence of the transition in any +particular measurement, implicit in our formulae, could be observed by re- +peated observation performed on the system subject to non-demolition tran- +sitions. These observations would be akin to the Zeno-effect measurement, such +as has been achieved in the form of quasi-periodic oscillation of the result for +a Superconducting flux cubit[35]. Further work is needed for quantifying the +information-entropy change in the environment [31, 36]. +Acknowledgement +The authors thank the referee for meticulous reading and insightful ques- +tioning of a previous version of this paper. +References +[1] A.J. Leggett, Macroscopic quantum systems and the quantum theory of +measurement. Suppl. Progress Theor. Phys. 69, 80 (1980) +[2] G.C. Ghirardi, P. Pearle and A. Rimini, Markov processes in Hilbert space +and continuous spontaneous localization of systems of identical particles. +Phys. Rev. A 42, 78 (1990) +[3] E.P. Wigner, Remarks on the mind-body question. In I. J. Good (ed.) ”The +Scientist Speculates”(Heinemann, London,1961) +11 + +[4] W.H. Louisell, Quantum Statistical Properties of Radiation. (Wiley, New +York, 1973) +[5] G.C. Ghirardi, A. Rimini and T.Weber, Unified dynamics for microscopic +and macroscopic systems, Phys. Rev. D 34, 470 (1986) +[6] P. Pearle, Combining stochastic, dynamical state reduction with sponta- +neous localization, Phys. Rev. A 39 2277-89 (1989) +[7] H. Everett, Relative state formulation of quantum mechanics, Rev. Mod. +Phys. 29 454-462 (1957) +[8] I. Byalinicki-Birula and J. Mycielski, Nonlinear wave mechanics, Annals +Phys. (NY) 108 62-83 (1976). +[9] G. Sewell, On the mathematical structure of quantum measurement theory, +Repts. Math. Phys. 56 271-90 (2005) +[10] L.Di´osi, Models for universal reduction of macroscopic quantum fuctua- +tions. Phys. Rev. A 40 165 (1989) +[11] R. Penrose, On gravity’s role in quantum state reduction. Gen. Relativ. +Gravit. 28, 581 (1996). +[12] S. Donadi, K. Piscicchia, C. Curceanu, L. Di´osi, M. Laubenstein and A. +Bassi, Underground test of gravity-related wave function collapse. Nat. Phys. +17 74 (2021) +[13] G. Lindblad, On the generators of quantum dynamical semigroups, Com- +mun. Math. Phys., 48 112-30 (1976). +[14] V. Gorini, A. Kossakowski and E.C.G. Sudarshan, Completely positive +dynamical semigroups of N-level systems, J. Math. Phys. 17 821-5 (1976). +[15] S. Weinberg, What happens in a measurement?, Phys. Rev. A 93, 032124 +1-6 (2016) +[16] M.A. Nielsen and I.L. Chuang, Quantun Computation and Quantum In- +formation (Cambridge University Press, Cambridge, U.K., 2010) +[17] K. Jacobs and D.A. Steck, A straightforward introduction to continu- +ous quantum measurement, Contemporary Physics, 47 (5) 279-303 (2006). +Arxiv.org/abs/quant-ph/0611067. +[18] N. Bohr, Atomic Physics and Human Knowledge, (Wiley, New York, 1958) +[19] J. Bell, Physics World, 3 33-4 (1990). +[20] A. Peres, Quantum Theory: Concepts and Methods, (Kluwer Academic, +Norwell MA, 1998) +12 + +[21] H.-P. Breuer and F. F. Petruccione, The Theory of Open Quantum Systems +(Oxford University Press, New York, 2002) +[22] S. B. J¨ager , T. Schmit, G. Morigi , M. J. Holland and R. Betzholz, Lindblad +Master Equations for Quantum Systems Coupled to Dissipative Bosonic +Modes, Phys. Rev. Lett. 129 063601 1-7 (2022) +[23] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, (2003). Quantum +dynamics of single trapped ions. Rev. Mod. Phys. 75 (1), 281–324 (2003). +Doi:10.1103/RevModPhys.75.281. +[24] K. Funo, N. Shiraishi and K. Saito, Speed limit for open quantum systems, +New J. Phys. 21 0300611 1-9 (2019). +[25] J.M. Horowitz and J.M.R. Parrondo, Entropy production along non- +equilibrium jump trajectories, New J. Phys. 15 220601 1-23 (2013). ) +[26] L. I. Mandelshtam and I. E. Tamm, The uncertainty relation between en- +ergy and time in nonrelativistic quantum mechanics. J. Phys. (USSR) 9, 249 +(1945). +[27] N. Margulus and L. B. Levitin, The maximum speed of dynamical evolu- +tion. Physica D 120, 188 (1998). +[28] S. L. Braunstein and C. M. Caves, Statistical distance and the geometry +of quantum states. Phys. Rev. Lett. 72 3439-44 (1994) +[29] S. Deffner and S. Campbell, Quantum speed limits: from Heisenberg’s +uncertainty principle to optimal quantum control. J. Phys. A: Math. Theor. +50 453001 (2017) +[30] J. Anandan and Y. Aharonov, Geometry of quantum evolution. Phys. Rev. +Lett. 65 1697-701 (1990) +[31] S.M. Barnett, Quantum Information,(University Press, Oxford, 2009), +Chapter 4. +[32] D. Manzano, A short introduction to the Lindblad master equation. AIP +Adv. 10 (2) 1063 (2020); doi.org/10.1063/1.5115323 +[33] K. Korzekwa and M.Lostaglio, Optimizing thermalization, Phys. Rev.Lett. +129 040602 1-7 (2022) +[34] M.Lostaglio and K. Korzekwa, Continuous thermomajorizaton and a com- +plete set of Markovian thermal processes. Phys. Rev, A 106 012426 1-18 +(2022) +[35] K. Kakuyanagi, T. Baba, Y. Matsuzaki, H. Nakano, S. Saito and K. Semba, +Observation of quantum Zeno effect in a superconducting flux qubit, New +J. Phys. 17 063035 1-10 (2015) +[36] M. Keyl, Fundamentals of quantum information theory, Physics Reports +369 (5) 431-58 (2002) +13 + diff --git a/EdE0T4oBgHgl3EQfywKq/content/tmp_files/load_file.txt b/EdE0T4oBgHgl3EQfywKq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cd0251fdd74b1fb4de7bf14a12d05d53e215fa6b --- /dev/null +++ b/EdE0T4oBgHgl3EQfywKq/content/tmp_files/load_file.txt @@ -0,0 +1,471 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf,len=470 +page_content='Lindbladian-Induced Alignment in Quantum Measurements R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Englman and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Yahalom Ariel University, Ariel 40700,Israel January 10, 2023 Keywords: Quantum measurement theory, Density matrix evolution, Quan- tum state resolution, Lindblad operators, Quantum speed limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Abstract An expression of the Lindbladian form is proposed that ensures an un- ambiguous time-continuous reduction of the initial system-pointer wave- packet to one in which the readings and the observable’s values are aligned, formalized as the transition from an outer product to an inner product of the system’s and apparatus’ density matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' The jump operators are in the basis of the observables, with uniquely determined parameters derived from the measurement set-up (thereby differing from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Weinberg’s Lind- bladian resolution of wave-packet formalism) and conforming to Born’s probability rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' The novelty lies in formalising the adaptability of the surroundings (including the measuring device) to the mode of observa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Accordingly, the transition is of finite duration (in contrast to its instantaneousness in the von Neumann’s formulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' This duration is estimated for a simple half-spin-like model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 1 Introduction In the century-run of quantum physics (plus 4 years, if one marks its beginning with the award of a Nobel Prize in 1918 to Max Planck for ”his discovery of quanta”) a single shadow of non-sequitur has darkened its glorious achievements, one that goes variously under the names of wave-function collapse, reduction of the wave-packet, quantum measurement, einselection, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Aspects of the prob- lem (or its articulations) were manifold, such as the breakdown of the predicted time-development in accordance with the Schr¨odinger equation, the abruptness of change in a measurement (”natura non facit saltum”, where art thou?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' ), the apparent non-applicability of quantum rules to macroscopic systems, imputed arbitrariness of Born’s probability rules, the requirement of ”infinite regress” 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='02664v1 [quant-ph] 6 Jan 2023 for the measuring apparatus and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Numerous papers enlarged on these is- sues [1, 2] and various proposals for resolution of the problem were put forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' These include the observer’s cognition [3], stochastic effects [4], in particular spontaneous localization [2, 5, 6], a many world scenario [7], non-linearity addi- tion to the Schr¨odinger equation [8], Poincar´e recurrent state [9], gravitationally induced collapse [10, 11, 12], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Common to these works, and with the specific purpose of providing a blue-print for measurements compatible with the Copenhagen formulation of quantum theory, was the need to give expression to the coupling of the micro- scopic system with its macroscopic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Standing apart from these and belonging to the field of non-equilibrium thermodynamics and to the establish- ment of equilibrium, a general form for this interaction was given by Lindblad [13] and by Gorin , Kossakowski and Sudarshan [14], satisfying some necessary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Constructing a merger between the two separately oriented fields, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Weinberg recently proposed a Lindblad-operator mechanism for the collapse of the density matrix (DM) in the course of a complete measurement [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' No- tably, the mechanism was linear in the state’s DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' The collapsed state (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' (1) in [15]) comprises the set of projection operators of the measurable item;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' the system’s Hamiltonian is described by a spectral decomposition onto the same operators (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' (16) in [15]) (although in the verbal discussion a more gen- eral situation is considered): collapse is achieved ”independent[ly] of the details of these [Lindblad] operators”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Decay between energy eigenstates had earlier been treated by the Lindblad formalism (for a pedagogical presentation the volume [16], Chapter 8 may be consulted) employing the interaction represen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' However, this is not convenient for treating measurements of observables that do not commute with the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Detailed theories relate to the out- come (”mapping”) of quantum operations, including measurements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' the present work describes the process of these happening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' (For a pedagogical introduction to stochasticity-induced wave-packet- reduction, obviating pointer reading, one may refer to [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=') 2 Overview of the Method and Terms 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='1 The leading idea, also in review While the concept of unity of observer and observation had already featured in Bohr’s view: ”The answer that we get is built up from the combined interac- tion of [the observer’s] state and the object of interrogation.” [18], this was not given a formal expression in the Copenhagen interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' It was more em- phatically asserted both by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Bell: ”I meant that the ’apparatus’ should not be severed from the rest of the world in boxes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='[19]” and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Peres: ” A measure- ment both creates and records a property of the system [20]”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' This change in 2 the course of a measurement affects also the environment outside the observed system ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' in the words of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Leggett ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='under these conditions the macroscopic apparatus, and more generally any part of the macro-world which has suffered changes in the course of the measurement process, does not end up in a state with definite macroscopic properties at all,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' [1]”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' The same line of thought appears to motivate S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Weinberg, who wrote in his preamble to a 2016 Lindbladian formulation of the masurement process[15], that ”We will instead [of the original formulation of the Copenhagen interpretation, (which we will not dwell on here)] adopt the popular modern view that the Copenhagen interpretation refers to open systems in which the transition is driven by the ineraction of the microscopic system under study (which may include an observer) chosen to bring the transition about.” (Our italics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=') These developments indicate the justification for a formulation in which the effect of the apparatus is incorporated in the equation defining the evolution of the system, rather than one in which the two entities are separate, barring an interaction between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='1 ”Alignment” The process whereby the pointer readings become in correspondence with the possible values of the observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Formally, for I possible values, the combined density matrix reduces from comprising I2 terms to one having I terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=', equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='5) in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='2 ”Dissipator” Added term (in the form of sums of appropriately weighted jump-operators) to the standard time dependent Schr¨odinger equation, inducing non-unitary evolution in the system, accompanied by changes of its information entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='2 Motivation for the choice of formalism Thermalization of open systems can be described by a Lindbladian formal- ism in which Gibbsian probabilities are so inserted as parameters, that the ”Dissipator” vanishes at these values of the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Replacement of the Gibbsian probabilities by Born probabilities achieves alignment in a state reduction and does so continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Limitations: Born’s probability rules are assumed, not derived;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' the interac- tion term is not traced to a microscopic mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' The source of this interaction term, shown in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 6 below, incorporating the coupling between the observed system and its surroundings (including the 3 measuring device) is an open question (also raised by a referee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' In its appli- cation to a thermalization process, the Lindbladian jump operators have been derived, though with the aids of several approximations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=',[21]), as well as, more recently, for the dissipation in a Dicke system with a bosonic background [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' We are not in the position to provide such first principle derivation for the Lindbladian jump-operators bringing about a transition and incorporat- ing the Born rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' It seems to be specific to the type of measurement under consideration and it is clear that just any jump operator, as in Weinberg’s Lindbladian formulation will not do the job .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Likely, one would need to in- clude non-Markovian dynamics, so that the coupling to the device and eventual pointer reading are two separate consecutive events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Inclusion of such dynamics is outside the scope of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 3 Assumptions We explore the time (t)-development of the combined density matrix ρ(t) of the measured system (S) and of the reading (pointer, dial, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=') on the mea- suring apparatus (A) for a complete and discrete measurement , expressing the underlying assumptions by three propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' In accord with the long-time historical approach, the mea- sured object S and the pointer of the measuring set-up A are treated on equal footings as subject to microscopic quantum laws, and formally describable by their respective Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Aware of the difficulties connected with an ”in- finite regress”, the effects of the rest of the Universe on S+A are not included in the formalism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' instead, for a phenomenological, approximative description, a Lindbladian term appears in the master equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Prior to the measurement with A and S decoupled, and being free of external influence for a long time, both are in energy quantum states, pure or mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' After the measurement, the state is not an energy eigenstate and subsequently it will spread over to a superposition of energy eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' The fast decoherence case treated below in section 5 is akin to the Zeno effect [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Only those states of the reading apparatus (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=', the right or left positions of a pointer) that may be in direct correspondence with the measured states of the system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=', spin up or down) are given expression in the formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' (At a beginning, the case treated is one in which there is a one- to-one correspondence between the states of the system and the readings of the apparatus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' a generalization is given subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=') A discussion in section 8 touches on the epistemological status of the Lindbladian terms in a measurement process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 4 4 Analysis Considering (for simplicity) a pure state for the system, its initial state-vector written in the basis of the observed property |S, i > takes the form ψS(t = 0) = � i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='.,I cS i |S, i > (1) Born’s rule for the probability of observing the i-component is |cS i ]2 ≡ pi, sum- ming to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Likewise, for the apparatus readings j, numbering J, one has the superposition with (complex and normalized) coefficients cA j ψA(t = 0) = � j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='.,J cA j |A, j > (2) We start with the one-to-one correspondence situation, for which I = J, and the reading j on A establishes uniquely the value i = j for the system’s measured property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' For the combined state-vector the density operator has the outer-product form (where the stars denote complex conjugates): � i,j,i′,j′ |S, i > |A, j > cS∗ i cA∗ j cA j′cS i′ < A, j′| < S, i′| ≡ � i,j,i′,j′ |i, j > Ciji′j′ < i′, j′| (3) the right hand side written in an obvious shortened notation, in which Ciji′j′ = cS∗ i cA∗ j cA j′cS i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' After collapse, the density operator takes the aligned, single-sum form � i |S, i > |A, i > |cS i |2 < S, i| < A, i| (4) It will be now shown that this is the time-asymptotic solution of the Lind- bladian master equation properly parametrized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' We recall Lindblad’s equation for the time varying density of states operator ρ ≡ ρ(t), as being of the following general form: ∂ρ ∂t = − i ¯h[H, ρ] + � n γn⟨LnρL† n − 1 2(L† nLnρ + ρL† nLn)⟩ (5) The second term, here named the ”Lindblad term” [13, 14] though in different contexts also referred to as the Dissipator [24], contains Ln’s that are Lindblad jump-operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' We shall consistently work in the observable + pointer’s basis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=', not in an energy basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' In this basis, neither the density operator ρ = ρ(t), nor the A+S Hamiltonian H is diagonal at the beginning or in the course of the development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' But, as will be demonstrated, the Lindbladian formalism, by a proper choice of its form, drives A+S to the desired diagonal form for the combined observable +pointer basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' We postulate just one single term in the previous n-sum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' as well as off-diagonal forms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' namely |i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j >< i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j′|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j ̸= 5 i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' for the jump-operators in the observable basis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' leading to the following parametrized form of the Lindblad term Lρ ≡ ΓΩ � i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='j′̸=i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='j r(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j) r(i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j′)⟨|i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j >< i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j′|ρ|i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j′ >< i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j| − 1 2(|i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j′ >< i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j|i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j >< i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j′|ρ + ρ|i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j′ >< i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j|i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j >< i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j′|⟩ (6) Here a circular frequency Ω is inserted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' so as to make Γ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' that quantifies the strength of the system-environment coupling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' One notes that in the pre-factor appear the parameters r(i, j), r(i′, j′)(i, j, i′, j′ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=', I) whose significance will be clear by deriving the matrix elements of the above operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' These are Lρi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='j′ = δi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='i′δj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='j′r(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j) � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='l r−1(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' l)ρk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='l − 1 2[r−1(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j) + r−1(i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j′)]ρi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='j′ � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='l r(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' l) (7) It can be seen that the trace of the above vanishes and that each matrix element vanishes upon the substitution ρi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='j′ = δi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='i′δj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='j′r2(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j) (8) While these properties hold for any arbitrary r(ij),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' the observable-pointer align- ment is achieved by identifying the r parameters with the system’s superposition coefficient: r(ij) = |cS i |δi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' or r(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' j)2 = |cS i |2 ≡ piδi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='j (9) the last being the Born probabilities appearing in the collapsed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' As already noted, this identification of probabilities relates to the well known procedure for the Lindblad-induced thermalization of open systems, for which detailed balance imposes the relation between the pre-factors γ(δE)/γ(−δE) = e−βδE/Z, the latter being the canonical probabilities (with β = 1/kBT, kB the Boltzmann constant, T the ambient temperature and Z the partition function [24, 25, 21]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' [It also seems fair to point out that also in the standard (Copenhagen, or von Neumannian) description of the alignment stage, as appears in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='5) of [1], this development is summarily stated, without specification of the underlying mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='] 5 Fast Decoherence Limit We now consider the case that the time development in the state is predom- inantly due to the coupling to the environment, rather than to the unitary 6 change induced by the Hamiltonian, meaning that the second term on the right hand side in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 5 dominates the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Quantitatively: Γ >> ||H||/¯hΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Ne- glecting the commutator we now form matrix elements of the Lindblad term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 5 in the observable+pointer basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Because of the approximation made, the off-diagonal matrix elements are decoupled from the diagonal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' The master equation of the off-diagonal terms reads (with a notation simplified by writing for the index pairs i, j → r, i′, j′ → s and consequently for ρi,j,i′,j′ → ρrs ≡ ρrs(t) dρrs dt = −ΓΩ �√pr + √ps 2 ρrs � m √pm � , r ̸= s (10) This shows that off-diagonal matrix elements decay exponentially in time (de- cohere), maintaining their real character that they had initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Had we kept the (imaginary) commutator term, we would have found that the decay is mod- ulated by the eigen-energies of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' For the diagonal matrix elements we find, dρrr dt = ΓΩ � √pr � m ρmm √pm − ρrr √pr � m √pm � (11) Again, it can be seen that the trace of the last expression vanishes, and so does the right-hand side under the substitution ρrr → pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' With these taking the values as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 9, one arrives at the aligned form (written out in the original, system-pointer indexes) ρ(t → ∞) = � i |ψA i > |ψS i > |cS i |2 < ψS i | < ψA i | (12) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='1 Illustrative example for a two-way experiment Exemplifying the foregoing for a two-valued system (such as a 1 2-spin electron), prepared as an eigenstate of a Zeeman-field with the magnetic field inclined at an angle 2αS to the vertical, in conjunction with an apparatus pointer, represented as being likewise in an eigenstate of a quasi-Zeeman field inclined at an angle 2αA to the vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' The eigenstates are linear superpositions of their z- spins;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' these are the observables that are to be determined by the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Initially, the system and the pointer are in the superposition states as shown above in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 1 and 2 and whose superposition coefficients cS i and cA j now have the values, sin / cos(αS) and sin / cos(αA), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' The DM in the observable basis is now a 4x4 matrix, in which appear all the combinations of the products of the above circular functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' As the outcome of the application of the Lindblad operator in the rate equation, at long times the matrix becomes reduced to the diagonal form discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' In these, cos2(αS) = p1 and sin2(αS) = p4 belonging to the aligned observable lie on the diagonal and are non-zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' the other two diagonal entries for the anti-aligned situations are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Plotted in Figure 1 are computed DM eigenvalues as functions of time (in red and blue), normalized to their respective Born probabilities, showing 7 4 3 2 1 0 1 2 Log10time[invfrequn] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='4 P1,P2,Decoh Figure 1: Density matrix eigenvalues normalized to their asymptotic (pointer- aligned) values for the two aligned terms in the illustrative example (in red and blue), plotted against time in inverse circular frequency unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' In green is shown a decohering off-diagonal matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Lindblad coupling strength Γ = 5, α angles .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='37 π and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='65 π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' their asymptotic convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' In green, the typical decohering tendency of an off-diagonal element is demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Figure 2 depicts the entropy S(t) = − � r Pr(t) log Pr(t) of the system and apparatus-pointer, (in which Pr(t) are computed eigenvalues of the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=') The non-monotonic behavior is characteristic of of the Lindblad formalism, in which the environment’s entropy change is not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' [In numerical work, based on forward integration, putting zeros for some of the pi’s introduces singularities, eventually algebraically cancelling out, but preventing flow of computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Therefore, instead, one puts arbitrarily small values for these and obtains for the aligned DM one that is arbitrarily close to, but not exactly equal to the true one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='] 6 Eigenvalue analysis An alternative to the numerical solution of the differential rate equation is eigen- value analysis, already treated in [15], based on the Landbladian term being a linear function of the diagonals in the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Thereby, the resulting rate equations have solution of the form ρnn(t) = � k vn,keλkt (13) in which λk and vn,k are the diagonalized eigenvalues and eigenvectors of the Lindbladian matrix diagonals in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Calculation shows that for the 4 x 8 4 3 2 1 0 Log10time[invfrequn] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='20 S Figure 2: Entropy of the combined system plus apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Noteworthy is the initial peak common to the Lindblad formalism .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 4 matrix considered above there are three negative eigenvalues and one zero eigenvalue, which alone is of interest at the long term behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Belonging to this eigenvalue, the (transposed) eigenvector is found to be {p1, p2, p3, p4} ≈ {cS 1 , 0, 0, cS 4 }, as required for the alignment between the quantum states and the reading in the measuring apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='1 Measurement speed Figure 1 shows that alignment is achieved for the model with the chosen strength parameter (Γ = 5) by a time of cca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='1/Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' By varying the strength in the computed model, we find a shortening of this time that is inversely proportional to the strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' This is expected from the quantum speed limit (QSL) results that border quantum transition times τ from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Essentially, QSL is the ratio of two norms [26, 27], that of the ”quantum distance” [28] and of the speed of the state evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Formally τ > ||ρ(t → ∞) − ρ(t = 0)|| || dρ(t) dt || = ||ρ(t → ∞) − ρ(t = 0)|| ||[Lρ(t)]|| (14) Ways of calculating the norms vary, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=', [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Recently, for a system de- veloping due to a Lindbalian operator, three contributions to the speed were discerned [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' To estimate ||ρ(t → ∞) − ρ(t = 0)|, we have used the ”Trace Distance ”defined as T(ρ, σ) = 1 2Tr[ � (ρ − σ)] = 1 2 � i |µi| (15) [31], where µi are the eigenvalues of the matrix differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' The DM velocity, as defined above , changes (decreases) with time, ultimately vanishing at the 9 fulfilment of alignment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' we have taken the root-mean-square sum of the rate of the diagonal matrix elements at initial times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' These yield a very low limit of τ > 1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='41 5 ∗ 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='03 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='0045/Ω (16) to be compared with the actually computed value, about 20 times longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Better (higher) limits of transition times may be generated by different ways of forming the norm for the DM velocity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' not at the beginning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='2 Multiple Reading-System correspondence A simple generalization of the foregoing applies when each (eigen-)value of the observable is in correspondence with not just one reading of the pointer, but with several (say, R) readings, all of the same significance for the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Then one simply inserts pi/R into the corresponding Lindblad term, in place of just pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' In the more complex case, that not all readings have the same likelihood, pi would have to be weighted by a probability factot, rather than by a constant denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 7 The Lindbladian, ”Who ordered this?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Historically, Lindblad terms were introduced as the most general forms that maintain complete positivity of the DM’s and preserve their trace [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' The various derivations that have been presented (and among these a recent one by [32]), involve several approximations for the coupling between the system and its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Insomuch that the derivation involves also tracing over the degrees of freedom of the environment, much detail of the latter is lost and of course it is impossible to work backwards from the Lindbladian to the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' What is remarkable is that for special purposes the appropriate Lindbladian operators take a very special, practically unique form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Such is the case for the accepted description of thermalization [21, 24] by a Lindblad formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' The parametrization of the Lindblad term employed in the present work, though it may appear arbitrary and particular for each case, is in fact identical to the one used for thermalization subject to the relabelling of the Gibbsian thermal distribution function as (the Born) probabilities (p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='),with the proviso of working in the observable, rather than in the energy basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' (This contrasts with the different approach in [15], which claims attainment of collapse for any Lindbladian operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=') At the same time, it needs to be noted that the analog of detailed balance is missing in wave-function collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' How come to have such a specific Lindbladian, whose source may be any measurement device and procedure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' One is left to wonder about the possibility of a special meta-physical status of the Lindblad terms, or query with Wheeler ”Who ordered this?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 10 8 Conclusion The well-known Lindbladian extension to the quantum theory of motion to environmental effects is here adapted to establish the resolution of a wave-packet in a measurement as a smooth process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' This is enabled by an unambiguous parametrization of the jump-operators describing the interaction of the broad environment with the observed system, both regarded as quantal entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Above, in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='1, a brief historically oriented preview has been provided for the distinct approach in this work, namely, one based on the wholeness of the entities (observed system and observing device), through the (Lindbladian) equation yielding the evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' A main result emerging from the formalism, and capable of experimental verification, is the finitely temporal variation of the system, and this in a de- terministic way rather than just statistically, on the average, contrasting also with the instantaneous collapse description by (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=') von Neumann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Such tem- poral variation in continuous-thermalization processes has been proposed quite recently [33, 34], also by employment of a Lindbladian formalism and within a Markovian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Experimentally, verification of the time dependence of the transition in any particular measurement, implicit in our formulae, could be observed by re- peated observation performed on the system subject to non-demolition tran- sitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' These observations would be akin to the Zeno-effect measurement, such as has been achieved in the form of quasi-periodic oscillation of the result for a Superconducting flux cubit[35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Further work is needed for quantifying the information-entropy change in the environment [31, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Acknowledgement The authors thank the referee for meticulous reading and insightful ques- tioning of a previous version of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Leggett, Macroscopic quantum systems and the quantum theory of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Progress Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 69, 80 (1980) [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Ghirardi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Pearle and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Rimini, Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' A 42, 78 (1990) [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Wigner, Remarks on the mind-body question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' In I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Good (ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=') ”The Scientist Speculates”(Heinemann, London,1961) 11 [4] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Louisell, Quantum Statistical Properties of Radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' (Wiley, New York, 1973) [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Ghirardi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Rimini and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='Weber, Unified dynamics for microscopic and macroscopic systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' D 34, 470 (1986) [6] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Pearle, Combining stochastic, dynamical state reduction with sponta- neous localization, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' A 39 2277-89 (1989) [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Everett, Relative state formulation of quantum mechanics, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 29 454-462 (1957) [8] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Byalinicki-Birula and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Mycielski, Nonlinear wave mechanics, Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' (NY) 108 62-83 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Sewell, On the mathematical structure of quantum measurement theory, Repts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 56 271-90 (2005) [10] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='Di´osi, Models for universal reduction of macroscopic quantum fuctua- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' A 40 165 (1989) [11] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Penrose, On gravity’s role in quantum state reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Relativ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Gravit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 28, 581 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Donadi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Piscicchia, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Curceanu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Di´osi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Laubenstein and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Bassi, Underground test of gravity-related wave function collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 17 74 (2021) [13] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Lindblad, On the generators of quantum dynamical semigroups, Com- mun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=', 48 112-30 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' [14] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Gorini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Kossakowski and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Sudarshan, Completely positive dynamical semigroups of N-level systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 17 821-5 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Weinberg, What happens in a measurement?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' A 93, 032124 1-6 (2016) [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Nielsen and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Chuang, Quantun Computation and Quantum In- formation (Cambridge University Press, Cambridge, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=', 2010) [17] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Jacobs and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Steck, A straightforward introduction to continu- ous quantum measurement, Contemporary Physics, 47 (5) 279-303 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='org/abs/quant-ph/0611067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' [18] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Bohr, Atomic Physics and Human Knowledge, (Wiley, New York, 1958) [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Bell, Physics World, 3 33-4 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Peres, Quantum Theory: Concepts and Methods, (Kluwer Academic, Norwell MA, 1998) 12 [21] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Breuer and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Petruccione, The Theory of Open Quantum Systems (Oxford University Press, New York, 2002) [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' J¨ager , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Schmit, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Morigi , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Holland and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Betzholz, Lindblad Master Equations for Quantum Systems Coupled to Dissipative Bosonic Modes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 129 063601 1-7 (2022) [23] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Leibfried, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Blatt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Monroe, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Wineland, (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Quantum dynamics of single trapped ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 75 (1), 281–324 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='1103/RevModPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' [24] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Funo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Shiraishi and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Saito, Speed limit for open quantum systems, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 21 0300611 1-9 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Horowitz and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Parrondo, Entropy production along non- equilibrium jump trajectories, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 15 220601 1-23 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' ) [26] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Mandelshtam and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Tamm, The uncertainty relation between en- ergy and time in nonrelativistic quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' (USSR) 9, 249 (1945).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' [27] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Margulus and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Levitin, The maximum speed of dynamical evolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Physica D 120, 188 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' [28] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Braunstein and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Caves, Statistical distance and the geometry of quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 72 3439-44 (1994) [29] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Deffner and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Campbell, Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 50 453001 (2017) [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Anandan and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Aharonov, Geometry of quantum evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 65 1697-701 (1990) [31] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Barnett, Quantum Information,(University Press, Oxford, 2009), Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' [32] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Manzano, A short introduction to the Lindblad master equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' AIP Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 10 (2) 1063 (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='5115323 [33] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Korzekwa and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='Lostaglio, Optimizing thermalization, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 129 040602 1-7 (2022) [34] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content='Lostaglio and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Korzekwa, Continuous thermomajorizaton and a com- plete set of Markovian thermal processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Rev, A 106 012426 1-18 (2022) [35] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Kakuyanagi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Baba, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Matsuzaki, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Nakano, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Saito and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Semba, Observation of quantum Zeno effect in a superconducting flux qubit, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' 17 063035 1-10 (2015) [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} +page_content=' Keyl, Fundamentals of quantum information theory, Physics Reports 369 (5) 431-58 (2002) 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'} diff --git a/EdE1T4oBgHgl3EQf-gYV/content/tmp_files/2301.03568v1.pdf.txt b/EdE1T4oBgHgl3EQf-gYV/content/tmp_files/2301.03568v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a10393019857467e259d324f3bb7cacc17f13543 --- /dev/null +++ b/EdE1T4oBgHgl3EQf-gYV/content/tmp_files/2301.03568v1.pdf.txt @@ -0,0 +1,1867 @@ +Quasi-equilibrium configurations of binary systems of dark matter admixed neutron +stars +Hannes R. R¨uter +,1 Violetta Sagun +,1 Wolfgang Tichy +,2 and Tim Dietrich +3, 4 +1CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal +2Department of Physics, Florida Atlantic University, Boca Raton, FL 33431, USA +3Institut f¨ur Physik und Astronomie, Universit¨at Potsdam, Haus 28, Karl-Liebknecht-Str. 24/25, Potsdam, Germany +4Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M¨uhlenberg 1, Potsdam 14476, Germany +(Dated: January 10, 2023) +Using an adapted version of the SGRID code, we construct for the first time consistent quasi- +equilibrium configurations for a binary system consisting of two neutron stars in which each is +admixed with dark matter. The stars are modelled as a system of two non-interacting fluids min- +imally coupled to gravity. For the fluid representing baryonic matter the SLy equation of state is +used, whereas the second fluid, which corresponds to dark matter, is described using the equation +of state of a degenerate Fermi gas. We consider two different scenarios for the distribution of the +dark matter. In the first scenario the dark matter is confined to the core of the star, whereas in the +second scenario the dark matter extends beyond the surface of the baryonic matter, forming a halo +around the baryonic star. The presence of dark matter alters the star’s reaction to the companion’s +tidal forces, which we investigate in terms of the coordinate deformation and mass shedding pa- +rameters. The constructed quasi-equilibrium configurations mark the first step towards consistent +numerical-relativity simulations of dark matter admixed neutron star binaries. +I. +INTRODUCTION +In the present era of gravitational wave (GW) astron- +omy, the internal properties of compact stars can be +probed during their mergers. Using numerical-relativity +(NR) simulations of the last stages of a binary coales- +cence, it is possible to relate observational GW data to +properties of the source. While these simulations have +undergone significant improvements in the past, the im- +pact of dark matter (DM) on the binary neutron star +(NS) dynamics has not yet been investigated in detail +and is not taken into account in standard GW analyses. +In fact, considering a coalescence of compact objects to +occur in pure vacuum, could be an oversimplification that +may lead to incorrect conclusions. +Due to their high compactness, NSs can trap and ac- +cumulate DM in their interior throughout the star’s evo- +lution. DM alters the compact star’s properties, e. g., its +mass, its radius, its tidal deformability, its energy density +and speed of sound profiles [1–15]. Its effect depends on +the relative fraction of DM and on the exact equation of +state (EoS) for the DM and baryonic matter (BM). For an +extended discussion of the impact of DM on compact star +properties and its smoking gun signals, see Refs. [16–18]. +While the effect of DM on isolated NSs can be probed +through electromagnetic observations, GW observations +of binary systems of DM admixed compact stars open up +a new observational window and the possibility to probe +a density and temperature range larger that of isolated +stars. To push forward our understanding of the imprint +of DM, we construct quasi-equilibrium configurations of +DM admixed NS binary system and study the impact of +DM focusing on quantities pertaining to binary system, +such as the orbital binding energy and the tidal deforma- +tions. +It is worth noting that not only NSs, but also black +holes could be embedded into DM. A step towards un- +derstanding the impact of DM on black hole mergers was +made in [19], where the behaviour of wave DM around +equal mass black hole binaries was studied in numerical +simulations. Furthermore, GW signals from binary coa- +lescences contain information of the binaries surrounding +medium [20]. +The effect of DM on the inspiral and post-merger +phases of DM admixed NSs has been studied by a few +groups. A first study by Ellis et al. [21] used a simple +mechanical model, and showed that a DM core can lead +to the appearance of additional peaks in the post-merger +GW spectrum. In [22] NR simulations of equal-mass bi- +naries consisting of BM admixed with a bosonic Klein- +Gordon field were performed. For a DM mass fraction of +10%, a redistribution of fermionic matter by the bosonic +cores was found, followed by the formation of a one-arm +spiral instability. Another approach approximating com- +pact dark component as test particles was studied in [23]. +The simulations show the DM component to remain grav- +itationally bound after the merger of BM and orbit the +center of the remnant with an orbital separation of a few +km. The DM core and a host star are likely to spin at +different rotational frequencies just after the merger due +to the absence of non-gravitational interaction. Further +on, they may synchronise via the gravitational angular +momentum transfer, including tidal effects [24]. +Up to our knowledge, the first two-fluid NR simulations +describing binaries of DM admixed NSs were performed +by Emma et al. [25] for a mixture of BM and mirror DM +only interacting via the gravitational field. The results +demonstrate that these systems tend to have a longer in- +spiral phase with increasing amount of DM, which could +be associated to the lower deformability of DM admixed +NSs. These simulations however, did not start from ini- +tial data satisfying the Hamiltonian and momentum con- +arXiv:2301.03568v1 [gr-qc] 9 Jan 2023 + +2 +straints [26–28] and the fluids did not start in an equilib- +rium configuration. Instead the initial data was approx- +imated by superimposing TOV-like solutions of isolated +DM admixed NSs. In this work we construct consistent, +constraint-solved, quasi-equilibrium conditions for a two- +fluid system of BM and DM. +One possible scenario for the formation of DM admixed +NSs is the capture of DM particles during the lifetime of +the star, from a progenitor to the equilibrated NS stages. +The core of a NS is very dense and hence the chance of +a DM particle experiencing scattering is relatively high. +In this scattering process the particle transfers its kinetic +energy to the star, becoming gravitationally bound [29– +31]. +This process is more efficient towards the Galac- +tic center, where the density of DM is many orders of +magnitude greater than in the galaxy’s arms [32, 33]. A +conservative estimate of DM capture in the most cen- +tral part of the Galaxy shows that stars accumulate up +to 0.01% of DM during the main sequence and equili- +brated NS stages combined [8]. However, also higher DM +factions inside compact stars can be achieved through +other scenarios, e.g., DM production during a supernova +explosion, accretion of DM clumps formed at the early +stage of the Universe, or initial star formation on a pre- +existing DM seed or local DM rich environments [34, 35]. +If DM is symmetric, it cannot reach a high fraction due +to self-annihilation, producing an electromagnetic or neu- +trino signal [36]. The latter scenario could lead to addi- +tional heating of isolated NSs as well as post-merger rem- +nants [37, 38], modification of kinematic properties [39]. +Moreover, production of light DM particles, e.g., axions, +in nucleon bremsstrahlung or in Cooper pair breaking +and formation processes in the NS interior [40–43], could +speed up the thermal evolution of a star by contributing +an additional cooling channel. +We consider DM to be either concentrated in a core or +extending beyond the surface of BM, forming a DM halo +around it. As a first step, we consider non-interacting, +fermonic DM with spin 1 +2. The single star properties of +this DM candidate have been discussed in Ref. [8]. The +baryonic component is modelled through a piecewiese- +polytropic fit [44] of the SLy EoS [45] that reproduces +nuclear matter ground state properties, fulfils heaviest +pulsars measurements [46, 47], X-ray observations by +NICER [48–52], and tidal deformability constraints from +GW170817 [53] and GW190425 [54] binary NS mergers. +The two components interact only through gravity, and +therefore do not repel each other, but overlap due to the +absence of non-gravitational interaction. This assump- +tion is in very good agreement with the observations of +the Bullet Cluster [55, 56] and direct DM searches [57], +which show that the DM-BM cross section to be many +orders of magnitude lower than the typical nuclear one, +σDM−BM ≈ 10−45 cm2 ≪ σBM ∼ 10−24 cm2. +By varying the particle mass and relative fraction of +DM, we obtain either a core configuration with a ra- +dius of the DM component less or equal to the baryonic +one, RD ≤ RB, or a halo with RD > RB [58]. +For +both scenarios, we construct initial configurations em- +ploying SGRID [59, 60]. Many other codes exist for the +construction of quasi-equilibrium NS binary systems, no- +tably the spectral codes LORENE [61, 62], Spells [63], +FUKA [64, 65], Elliptica [66], and the finite difference +based code COCAL [67, 68]. Up to our knowledge, these +codes are only able to solve systems consisting of a sin- +gle fluid. +Here we construct for the first time quasi- +equilibrium binary configurations with two fluids. +The formalism and results are presented in geometric +units in which the gravitational constant G = 1 and the +speed of light c = 1. In these units, lengths are given +as multiples of the solar mass, M⊙. For the conversion +to SI units a spatial length must be multiplied by L0 = +1476.6250 m/M⊙ and a time by T0 = 4.9254909 × 10−6 +s/M⊙. Where appropriate we also use MeV to specify en- +ergy and mass of particles, as well as SI units. Through- +out the paper, Greek letter indices denote four dimen- +sional, spacetime indices, whereas Latin indices denote +three-dimensional, spatial indices. +The paper is organized as follows. +In Section II we +summarize the two-fluid formalism and DM distribu- +tion regimes. Its implementation to the SGRID code is +described in Section III. In Section IV we analyse the +convergence properties of the constructed configurations, +quantify the difference in the velocities of the two flu- +ids and investigate some physical properties of the quasi- +equilibrium configuration over a sequence of separations. +Section V summarizes the results and discusses future +perspectives. +II. +FORMALISM +We describe the matter as a system of two non- +interacting perfect fluids only indirectly coupled through +the gravitational field. This model is well justified, be- +cause the interaction between standard model BM and +DM is weak. Utilisation of the perfect fluid model for DM +is also justified, as the mean free path and the scattering +time scale of DM particles can be small compared to the +characteristic time scales of the binary. In the following, +we estimate the mean free path and scattering time in +a semi-classical approach for a degenerate Fermi gas of +particles with the mass of 170 MeV (≈ 3 × 10−28 kg). +The Fermi gas consists of non-interacting fermions, for +which a self-scattering cross section σDM formally does +not exist. Instead, we use the value of the upper limit +obtained from observations of merging galaxies, which +yield σDM/m(DM) +p +< 1.25 cm2/g, with m(DM) +p +the mass +of the DM particles [56, 69]. In this work we construct +configurations with a particle density n(DM) of 0.7 fm−3 +in the center of the star. Together with the upper limit +for σDM this yields a mean free path λ = 1/(n(DM)σDM) +of 3.7 × 10−17 m, much smaller than the typical length +scale of a NS, which is on the order of 104 m. The scatter- +ing time scale can be estimated using the Fermi velocity, +which reaches values up to 0.8 c in the centre of the star. + +3 +Finally, using the value of the mean free path, this yields +a scattering time of tc = λ/vDM = 1.5 × 10−25 s, much +smaller than for example the orbital period of the binary, +which in our configurations is a small as 3 × 10−4 s. At +the surface of the stars DM reaches the free streaming +limit and the perfect fluid limit breaks down, but there +the density is so small, that the impact on the gravita- +tional field is low and hence the matter in this region can +be neglected. +For non-interacting fluids, the energy-momentum ten- +sor can be split into the two individual fluid components +given by: +T (s) +µν = (e(s) + p(s))u(s) +µ u(s) +ν ++ p(s)gµν , +(1) +where e is the proper energy density, p is the pressure, uµ +is the four velocity of the fluid and the label (s) denotes +the particles species, which is either BM or DM. The +Einstein field equations are then given by +Rµν + 1 +2gµνR = 8π(T (BM) +µν ++ T (DM) +µν +) +(2) +and, because the two particle species do not interact, +each fluid satisfies the equations of motion of a single +fluid. Consequently, each fluid satisfies energy momen- +tum conservation separately: ∇µT (s) +µν = 0. +For each fluid, we also define the rest mass density ρ(s) +0 , +which is computed from the number density n(s) by +ρ(s) +0 += m(s) +p n(s) , +(3) +with m(s) +p +being the mass of the particles. Furthermore, +the specific enthalpy is then given by +h(s) = e(s) + p(s) +ρ(s) +0 +. +(4) +To make the equations tractable, the spacetime metric +gµν is decomposed into a temporal and a spatial part by +introducing the spatial metric γij, the lapse α, and the +shift βi [27, 70, 71]. The line element in this 3+1 split +reads +ds2 = −α dt2 + γij (βidt + dxi)(βjdt + dxj) . +(5) +The extrinsic curvature Kij is related to the time deriva- +tive of γij, by the formula +Kij = − 1 +2α(∂tγij − Diβj − Djβi) , +(6) +where Di denotes the covariant derivative compatible +with the spatial metric γij. +We construct the partial differential equations govern- +ing quasi-equilibrium by following the derivation in [72], +which is trivially applied to a system of non-interacting +fluids. To generate quasi-equilibrium configurations, we +solve equations for velocity potentials φ(s), which are de- +fined through the following split of the four-velocity +γi +µu(s)µ = +1 +h(s) (Diφ(s) + w(s)i) , +(7) +where w(s)i is a divergence free vector, i.e., Diw(s)i = 0, +describing the rotational part of the fluid. Following the +derivation of [72], we fix the time derivatives of the fields +by assuming the existence of an approximate Killing vec- +tor ξ and a set of quasi-equilibrium conditions for the two +fluids +Lξe(s) ≈ 0 , +(8) +Lξp(s) ≈ 0 , +(9) +γi +µLξ(∇µφ(s)) ≈ 0 , +(10) +γi +µL +∇φ(s) +h(s)u(s)0 w(s) +µ +≈ 0 . +(11) +We omit further details of the derivation, since for non- +interacting fluids everything can be directly carried over +to the individual fluid components, and we state only +the resulting partial differential equation for the velocity +potentials φ(s): +Di +� +ρ(s) +0 α +h(s) (Diφ(s) + w(s)i) − ρ(s) +0 αu(s)0(βi + ξi) +� += 0 , +(12) +where the boost factor u(s)0 is given by +u(s)0 = +� +h(s)2 + (Diφ(s) + w(s) +i )(Diφ(s) + w(s)i) +αh(s) +, +(13) +and the specific enthalpy is given by the expression +h(s) = +� +L(s)2 − (Diφ(s) + w(s) +i )(Diφ(s) + w(s)i) , (14) +with +L(s)2 = +b(s) + +� +b(s)2 − 4α4((Diφ(s) + w(s) +i )w(s)i)2 +2α2 +(15) +and +b(s) = ((ξi+βi)Diφ(s)−C(s))2+2α2(Diφ(s)+w(s) +i )w(s)i . +(16) +The variable C(s) is a constant, which can vary for each +star and controls the mass of the fluid component. +For the approximate Killing vector ξi we make the fol- +lowing ansatz: +ξi = Ω(−y, x − xCM, 0) + vr +D (ri − ri +CM) , +(17) +where Ω is the instantaneous orbital frequency, D is the +separation between the star centres, vr is the radial ve- +locity, and xCM is the x-coordinate of the centre of mass. + +4 +At apsis the orbital frequency together with the sepa- +ration of the stars control the orbital parameters like ec- +centricity and length of the semi-major axis. Away from +apsis there is a non-vanishing radial component of the +velocity to be taken into account. In cases like the “circu- +lar” inspiral there is no apsis, but there is an always non- +vanishing radially inward directed velocity component. +The configurations presented in this work are constructed +within the quasi-circular approximation for which the ra- +dial component is neglected, vr = 0. +We set the value of Ω to its value at second +Post-Newtonian order in Arnowitt-Deser-Misner (ADM) +gauge [73–75] using the sum of the rest masses of the +two fluids as the mass estimates of the stars, which are +computed by +m(s) +0i = +� +Vi +ρ(s) +i u(s)0α +� +det(γjk)d3x , +(18) +where Vi is the spatial volume over which the i-th star +extends. The value of xCM is then given by +xCM = (m(BM) +01 ++ m(DM) +01 +)xc1 + (m(BM) +02 ++ m(DM) +02 +)xc2 +m(BM) +01 ++ m(DM) +01 ++ m(BM) +02 ++ m(DM) +02 +, +(19) +where xc1/2 are the x-coordinates of the centres of the +stars. +In this work, we present results for equal-mass +configurations only, i. e., xCM = 0. +Besides the continuity equation (Eq. (12)) governing +the fluid velocity potentials φ(s), the metric must be fixed +in a way satisfying the ADM constraints. To this end we +choose a conformally flat ansatz for the spatial metric, +i. e., γij = ψ4¯γij, with γij = δij and ∂tγij = 0, and con- +struct the data on maximally sliced hypersurfaces, i. e., +the trace of the extrinsic curvature vanishes: K = 0 and +∂tK = 0. +The free metric components are the lapse, +shift, and conformal factor ψ and their governing equa- +tions are formulated in terms of the extended conformal +thin sandwich equations (XCTS) [27, 28]. Together with +Eq. (12), the data is constrained by a set of seven coupled +partial differential equations, which are solved iteratively +one-by-one in a self-consistent manner. +III. +SGRID +We have adapted the pseudo-spectral SGRID code [59, +60] to generate quasi-equilibrium configurations for two +fluid systems. +We use the same iteration scheme that +is used in [60] for single-fluid NSs. We sketch the iter- +ation scheme in the following with an emphasis on the +adaptions and changes made. +1. To ensure the convergence of the solver, it is nec- +essary to provide an initial guess sufficiently close +to the true solution. This initial guess is chosen as +a superposition of two boosted TOV-like two fluid +stars of a given mass. To generate solutions with +particular rest masses for the fluid components, +one has to find the central pressures for which the +masses are realized. Since we are dealing with two +fluids, this is a two-dimensional root finding prob- +lem. In our tests, we found that using the Newton- +Raphson method is not always reliable, because the +masses are not a monotonous function of the central +pressures, hence, a Newton-Raphson solver easily +gets caught in a local extremum of the mass func- +tion. Instead, we employ a series of bisections on +the central pressure of one fluid component while +keeping the central pressure of the other fluid fixed. +The series of bisections iterates between the two +fluid components in a self-consistent manner until +the fluid masses are sufficiently close to the target +parameters. +2. If the residuals of Eq. (12) are larger than 10% +of the combined residuals of the XCTS equations, +we solve Eq. (12) and set the new φ(s) to be the +average of the old solution φ(s) +old and the just ob- +tained solution φ(s) +ell , using the following weights +φ(s) = 0.8φ(s) +old + 0.2φ(s) +ell . +3. We proceed by solving the XCTS equations and +update α, β, and ψ in the same way, averaging the +old and new solution. +4. We do not adjust the values of Ω and xCM as in [60]. +The value of Ω would be fixed within an eccentric- +ity reduction scheme. xCM is left at its Newtonian +value, Eq. (19). +5. We adjust the constants C(s), such that the rest +masses of each component and in each star match +our desired target masses. We then update the val- +ues of h(s) keeping it fixed until the end of the next +iteration. +6. If the sum of the residuals is below a certain toler- +ance or a prescribed maximum number of iterations +is reached, the iteration ends here and is concluded +with a final solving of the XCTS equations. +7. The system of partial differential equations does +not fix the position of the stars and, hence, they will +slowly drift if not kept under control. To keep the +stars in place, the center of the stars are driven back +to the desired position. For single fluids, the center +is usually defined in an unambiguous way as the +point of maximum density. For two fluids the defi- +nition is ambiguous, because the tidal deformations +due to the companion star are different for each +fluid component and, consequently, the maximum +densities are at different points. In most cases, how- +ever, the two maximum points will still be close. +The results shown in this work are obtained by +choosing the point with the maximum of the to- +tal proper energy density, e(tot) = e(BM) + e(DM), +as the center of the stars. We have chosen e(tot), + +5 +1 +1.05 1.1 1.15 1.2 1.25 1.3 +1 +1.05 +1.1 +1.15 +1.2 +-25 +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +25 +X +-10 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +10 +Y +-25 +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +25 +X +-10 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +10 +Y +-25 +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +25 +X +-10 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +10 +Y +-25 +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +25 +X +-10 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +10 +Y +FIG. 1. +Specific enthalpy in the z = 0 plane for a config- +uration with DM halo. In the upper halves only the specific +enthalpy of DM is shown, whereas in the lower halves the +BM component lies on top of it. The black lines indicate the +boundaries of the spectral elements. Each NS is comprised of +a central cubical element and six cubed sphere elements (of +which only four intersect the z = 0 plane). The separation +between the NS centres amounts to 32 M⊙ (47.3 km). +in particular, because it is a covariant scalar and it +is the major quantity determining the gravitational +potential, hence giving an estimate for the center of +mass of the star. To drive the center of mass back, +the values of h(s) are transformed by +h(s),new = h(s) + ∆ri∂ih(s) , +(20) +where ∆ri = ri +current − ri +desired. +8. Continue with step 2. +The SGRID code uses surface-fitted coordinates to re- +duce the Runge phenomenon at the surface of the star. +Each time we update the specific enthalpy h(s) (step 5 +in the iteration), we adapt the grid such that the bound- +aries of spectral elements coincide with the new surface +of the outer fluid. That means we only construct con- +figurations in which the surfaces of the two fluids do not +intersect, which would in principle be possible given the +different deformabilities of the fluids. Furthermore, we do +not construct domains that are adapted to the surface of +the inner fluid. +Therefore, at the surface of the inner +fluid one can expect to observe the Runge phenomenon +and a slight degradation of the convergence in the trun- +cation error. Fig. 1 shows a visualisation of the deformed +spectral elements inside the NS and the distribution of +matter in terms of the specific enthalpy. +To close the system, the EoS is required to relate e(s), +p(s), ρ(s) +0 , and h(s). For the EoS, SGRID reads in either +parameters of piecewise polytropes or EoS tables. EoS +tables are interpolated in a thermodynamically consis- +tent manner [76] using a cubic Hermite interpolation. To +find the thermodynamic quantities for a given specific +enthalpy a Newton-Raphson root finder is used. At low +densities we use a polytrope that is matched at the lowest +density of the table. +IV. +RESULTS +A. +Parameters of Constructed Configurations +We consider two different two-fluid configurations, one +in which DM is confined to the core of the NS, the dark +core configuration, and one in which DM extends beyond +the surface of the BM, so that the NS has a DM halo, +which we will refer to as the dark halo configuration. +Furthermore we compare to configurations consisting of +BM only: the single fluid configuration. +We describe BM by a piecewise-polytropic fit [44] to +the SLy EoS [45]. As a model of DM, we investigate the +degenerate, relativistic Fermi gas of spin- 1 +2 particles at +zero temperature, for which the EoS is read in as tabu- +lated data. EoSs at zero temperature are sufficient for +our calculations, because the Fermi energy of the sys- +tem is much higher than its temperature. The typical +temperature T0 of NS cores is of the order of 106 − 108 +K [77, 78]. We assume that DM has the same tempera- +ture as the BM, because the captured DM particles keep +scattering with baryons, rarely but often enough to ther- +malise with the BM component. A core temperature of +approximately 108 K is much lower than the Fermi en- +ergy of BM. This is also true for the Fermi gas EoS we +consider, e. g. in the dark halo case the Fermi energy of +DM reaches 403 MeV in the center of the star, an energy +smaller than that of the BM, but still much larger than +the temperature of the star, kBT0 ≈ 0.009 MeV. +For the dark core configuration the DM particles have a +mass of 1000 MeV and DM provides 5% of the NSs’ total +rest mass. +Fermionic DM particles with mass of 1000 +MeV present an interesting case, because they resemble +nucleons. +In the dark halo case we model DM by particles with +a mass of 170 MeV, for which the fluid is less dense and +hence easily forms a halo. Furthermore in the dark halo +configuration DM only contributes 0.5% of the total rest +mass. The choice of these values for the particle masses +is motivated by the results of [8], where it was shown that +for the DM particle masses below 174 MeV DM admixed +NSs are in agreement with astrophysical observations of +the heaviest NSs for arbitrary relative fraction of DM. +Moreover, the chosen mass of 170 MeV and the fraction +of 0.5% leads to a relatively small halo of approximately +twice the radius of the BM component, which is easy to +model. When the size of the halos is big enough so that +they touch each other, it is no longer possible to fit the +element surfaces to the outer fluid of a star. Hence, we +are discarding configurations with separations at which +the two halos merge. In all configurations the individual +NSs have the same total rest mass, i. e., the combined +rest mass of BM and DM is 1.4M⊙. In all setups, the +NSs have equal masses and are irrotational, i. e., they +have zero spins. + +6 +B. +Convergence +To validate the code, we check the convergence of the +Hamiltonian constraint for a dark halo configuration of +NSs with a separation of 44 M⊙ (65.0 km) on a quasi- +circular orbit. +Fig. 2 shows the magnitude of the Hamiltonian con- +straint H on the z = 0 plane. The constraint violations +are largest in the interior of the star, where they reach +values up to 4×10−5, whereas in the vacuum regions the +error drops to values below 10−9, but with some spikes +on the order of 10−7 at the element boundaries. A be- +haviour commonly seen for spectral codes. The Hamil- +tonian constraint is largest in the region where the inner +fluid is non-vanishing. In Fig. 2 one can observe a clear +transition on the surface of the baryonic fluid to lower +constraint violations in the DM halo. +Fig. 3 demonstrates the development of the volume- +normalised L2-norm of the Hamiltonian constraint for the +inner cube of one of the stars during the iterative solv- +ing process. The figure shows the behaviour for different +number of points n in each dimension, which is the same +for each spectral element. All curves show a saturation +in the norm of the Hamiltonian constraint towards the +end of the iteration process, which for all configurations +is stopped after 40 iterations. Furthermore, it is visible +that higher resolution leads to smaller violations of the +Hamiltonian constraint in the final solution. For compar- +ison Fig. 3 also shows the sequence for a corresponding +single fluid configuration with the same mass and sepa- +ration. After 40 iterations its Hamiltonian constraint is a +factor 10 smaller than the dark halo configurations and it +does not show any signs of saturation, i. e. it would prob- +ably reach even smaller constraint violations if iterated +further. +The convergence in the final solution is further inves- +tigated in Fig. 4, which shows its L2-norm of the Hamil- +tonian constraint with respect to the number of colloca- +tion points in the spectral elements. The figure shows +the constraint violation for the inner cube element and +for the cubed sphere facing towards the companion star, +which is also representative for all other cubed sphere el- +ements inside the NSs. The curves are almost straight +lines on the log-log-plot of Fig. 4, which is compatible +with a polynomial convergence of the constraints, i. e., +|H|L2 ∼ n−p, with p the order of convergence. +This +is the expected convergence behaviour for non-smooth +data, which we have due to the surface of the inner fluid. +Using the highest and lowest resolution we can estimate +the order of convergence in the inner cube element to be +p ≈ log22/10(|H|L2,n=10/|H|L2,n=22) ≈ 2.7. +To investigate the convergence of the actual solution +variables we interpolate the data from different resolu- +tions on a common set of points and compute norms +of the estimated errors on these points. +We interpo- +late the solution onto a 10 × 10 × 10-grid equidistant +in each direction, with coordinate components given by +ri ∈ {20m/9, m ∈ [0..9]}. This grid includes some points +1e-13 +1e-12 +1e-11 +1e-10 +1e-9 +1e-8 +1e-7 +1e-6 +1e-5 +1.00 +1.33 +1.05 1.1 1.15 1.2 1.25 +1.00 +1.22 +1.05 +1.1 +1.15 +FIG. 2. +Hamiltonian constraint in a dark halo configuration +in the z = 0 plane. In the lower half the specific enthalpy of +the two fluids is overlaid. +with pure vacuum, points with only one fluid present +and points with both fluids present. +The error in the +solution is estimated by taking the difference to the so- +lution with the highest resolution, i. e., the solution that +has 22 points in each dimension of the spectral elements. +In Fig. 5 we show the convergence of the 1-norm and +the maximum norm over the set of interpolated points +for the gxx component of the metric and the lapse α. +Both quantities do not show a monotonic decay of the er- +ror, but there is an overall trend of decaying error. This +somewhat broken convergence behaviour can again be +attributed to the presence of non-smooth fields on the +surface of the inner fluid. Fig. 6 shows the convergence +of the error in the specific enthalpy. +The DM in this +configuration is fitted to the element boundaries and its +specific enthalpy displays a relatively clear convergence +behaviour. +The BM fluid on the other hand shows a +very broken convergence and only very little improve- +ment from the lowest to the highest number of points. +The maximum norm of the error is actually growing for +the two largest number of points, whereas the 1-norm +of the error is also slightly broken, but with an overall +behaviour similar to that of gxx and α. +It should be noted, that it is not clear whether the +formalism of Sec. II actually possesses a unique solution. +The partial differential equation (12) is not strictly ellip- +tic on the fluid surface and hence the standard theorems +for the uniqueness of the solution can not be applied. In- +stead our algorithm might find a solution of many possi- +ble, which is another possible explanation for the slightly +broken convergence behaviour. +C. +Difference in the Fluid Velocities +It is worth pointing out that even if the BM and +DM fluid components are both irrotational, i. e., non- +spinning, the exact velocity profiles are not the same. +The reason for this does not lie in the notion of an irro- +tational fluid, but is caused by differences in the fluids’ +equations of motion. An irrotational fluid is defined by + +7 +0 +5 +10 +15 +20 +25 +30 +35 +40 +iteration +10 +-5 +10 +-4 +10 +-3 +10 +-2 +|H| +L +2 +/V +element +n += +22, single fluid +n += +12, dark halo +n += +14, dark halo +n += +18, dark halo +n += +22, dark halo +FIG. 3. +L2-norm over the inner cube in one of the stars, +normalised by the volume of the inner cube. The different +lines show configurations with different number of points n in +each dimension. +10 +12 +14 +16 +18 +20 +22 +numer of points in each dimension n +10 +-4 +|H| +L +2 +/V +element +inner cube +left cubed sphere +FIG. 4. +Normalised L2-norm of the Hamiltonian constraint +in a dark halo configuration for a different number of points +per dimension. The norm is normalised by the volume of the +spectral element. Note that the x-axis and y-axis are scaled +logarithmically. +10 +12 +14 +16 +18 +20 +numer of points in each dimension n +10 +-4 +10 +-3 +10 +-2 +10 +-1 +10 +0 +10 +1 +error norm of variable Y, ||Y +n +− +Y +22 +|| +g +xx +, 1-norm +g +xx +, maximum norm +α, 1-norm +α, maximum norm +FIG. 5. +Self-convergence of metric variables in dark halo +configurations. Black: error norm of the gxx component of +the metric. Blue: error norm of the lapse, α. We not that the +1-norm is not normalised by the number of points. +10 +12 +14 +16 +18 +20 +numer of points in each dimension n +10 +-4 +10 +-3 +10 +-2 +10 +-1 +10 +0 +||h +(s) +n +− +h +(s) +22 +|| +h +BM +, 1-norm +h +BM +, maximum norm +h +DM +, 1-norm +h +DM +, maximum norm +FIG. 6. +Self-convergence of the specific enthalpy in dark halo +configurations. +Black: error norm of the baryonic specific +enthalpy h(BM), which is the inner fluid. Blue: error norm of +the specific enthalpy of DM, h(DM). We not that the 1-norm +is not normalised by the number of points. +the vanishing of its kinematic vorticity tensor [79] +ωαβ := P µ +α P ν +β ∇[µuν] = 0 , +(21) +with P µ +α = δµ +α + uµuα. +This notion does not depend +on the thermodynamic properties of the fluid and hence +differences in the velocities can only be the result of the of +the equations of motion, that are used in the derivation of +the formlalism in Sec. II, i. e. the Euler equations [72, 80] +u(s)µ∇µ(h(s)u(s) +ν ++ ∇νh(s)) = 0 , +(22) +which follow from ∇µT (s) +µν = 0, and the continuity equa- +tion +∇µ(ρ(s) +0 u(s)µ) = 0 . +(23) +If for example the DM would have the same four-velocity +as the BM, it would still be irrotational, but might be +incompatible with the laws of energy-momentum or par- +ticle number conservation. +In nature the disparity in the fluid velocities is affected +by two counter-acting effects, particle scattering between +BM and DM on the one hand and physics determining +spin-down on the other hand. +In our formulation the +two fluids are modelled as non-interacting, but the BM- +DM scattering cross-section might be non-zero in nature, +which would drive the two fluids towards a common ve- +locity. This process is counter-acted by effects driving the +fluid into an irrotational state, as for example magnetic +braking for BM [81–83]. It is unclear whether a similar +effect exists for DM and whether it is dominant over the +effect of BM-DM scattering. By assuming vanishing of +the kinematic vorticity for the DM component, we as- +sume that such an effect exists and it is also dominating +over the scattering with BM. +We find that both fluids move with basically the same +velocity, with coinciding velocities in the star center, + +8 +12 +14 +16 +18 +20 +x +−0.15 +−0.10 +−0.05 +0.00 +0.05 +0.10 +relative difference, V +(s)x +1 +− +V +(DM)x +/V +(BM)x +, dark halo +1 +− +V +(DM)x +/V +(BM)x +, dark core +V +(BM)x +, dark halo +V +(DM)x +, dark core +FIG. 7. +Relative difference in the velocities for configurations +with a separation of 32 M⊙. The difference is shown along a +diagonal with the parametrization ri(s) = s(1, 1, 0)+ri +c, going +through the center of the star located at ri +c = (16M⊙, 0, 0). +V (BM)x (black, dash-dotted line) and V (DM)x (grey, dotted +line) show the x-component of the velocity of the respective +inner fluid. +but increasing difference towards the surface of the in- +ner fluid. We quantify this effect in terms of the residual +three-velocity V (s)i, in which the orbital movement given +by the Killing vector ξµ is split off, +V (s)i = u(s)i/u(s)0 − ξi . +(24) +Fig. 7 shows the x-component of V (s)i and the relative +difference of the fluid velocities for the region in which +both fluids are present. We present results for configu- +rations at a separation of 32 M⊙, a separation at which +the DM halos in the dark halo configurations are already +relatively close and deformed (Fig. 1). We find that dif- +ferences in the two fluids are smaller for larger separation, +which is intuitively understandable, because for large sep- +arations the system goes to the limit of isolated NSs in +which the fluid velocities coincide. +The data in Fig. 7 is shown along a diagonal through +the star parametrized in the following way: +ri(s) = +s(1, 1, 0)+ri +c, where ri +c is the center of the star. We choose +to present the data along this diagonal because the differ- +ence V (BM)i − V (DM)i has a quadrupolar structure with +nodes going through ri +c and being approximately parallel +to the x and y axes. Hence the difference is basically zero +on the x and y-axis, but very prominent along the spec- +ified diagonal. The relative difference between the resid- +ual velocities is below 0.2% near the center of the star +and reaches up to 10% on the surfaces of the inner fluids. +The difference between the velocities of the dark halo and +dark core configurations is relatively small, which can be +seen from the fact the curves of the velocities of the inner +fluids lie on top of each other. +D. +Binding Energy +NSs with a DM component are more tightly bound, +because the DM component adds gravitating mass, but +provides no additional repulsion to balance the gravita- +tional pressure [8]. The gravitational binding energy of +the particles is the difference of the ADM mass [27, 84, 85] +and the sum of the rest masses m(s) +0i of the components. +If all fluid particles would fall in from infinity, the true +ADM mass would equal the total rest mass. However, the +configurations that we construct do not contain GWs and +therefore they do not model the energy lost in gravita- +tional radiation. The difference in our ADM mass esti- +mate and the total rest mass is, therefore, a measure of +the particle binding energy: +Ebind,p = MADM − m(BM) +01 +− m(DM) +01 +− m(BM) +02 +− m(DM) +02 +. +(25) +Fig. 8 shows the particle binding energy as a function of +our estimate for the ADM angular momentum JADM. It +can be seen that dark core configurations are more tightly +bound than single fluid configurations. The dark halo +configurations seemingly coincide with the single fluid +case. This can be attributed to the relatively low DM +fraction of only 0.5% in these configurations. All config- +urations are more tightly bound for smaller JADM cor- +responding to smaller stellar separations. This is due to +the stronger orbital binding between the two stars. +Most of the binding energy is contained in the indi- +vidual stars and the contribution of the orbital binding +energy is universal in all configurations. The orbital bind- +ing energy Ebind,orb is the energy necessary for the two +NSs to escape to infinity. It can be computed using the +gravitational mass m(s) +i +of the components, by +Ebind,orb = MADM −m(BM) +1 +−m(DM) +1 +−m(BM) +2 +−m(DM) +2 +. +(26) +The gravitational masses m(s) +i +are obtained by solving a +TOV-like equation for isolated stars that have the same +rest masses. The gravitational mass m(s) +i +is smaller than +the rest mass m(s) +i0 , because it accounts for the binding +energy. Hence, Ebind,orb contains only contributions of +the binding energy that are due to the mutual binding +between the stars. Fig. 9 shows that the orbital binding +energy is mostly independent of the DM configuration. +The biggest effect is seen for the dark core configurations +for which the magnitude of Ebind,orb is about 2% smaller +than that of the other configurations. +E. +Deformation +To quantify the deformation of the stars we compute +the ratio of the diameters along the orbital radius and +along the polar axes. The diameter along the orbital ra- +dius is taken as ∆x, largest difference in the x-coordinates +of two points on the fluid surface. The polar diameter + +9 +6.00 +6.25 +6.50 +6.75 +7.00 +7.25 +7.50 +7.75 +J +ADM +/M +2 +⊙ +−0.300 +−0.295 +−0.290 +−0.285 +−0.280 +−0.275 +−0.270 +particle binding energy E +bind, +p +/M +⊙ +single fluid +dark halo +dark core +FIG. 8. +Particle binding energy Ebind,p as a function of the +ADM angular momentum. +6.00 +6.25 +6.50 +6.75 +7.00 +7.25 +7.50 +7.75 +J +ADM +/M +2 +⊙ +−0.0300 +−0.0275 +−0.0250 +−0.0225 +−0.0200 +−0.0175 +−0.0150 +orbital binding energy E +bind, +orb +/M +⊙ +single fluid +dark halo +dark core +FIG. 9. +Orbital binding energy Ebind,orb as a function of the +ADM angular momentum. +∆z, is the largest difference in the z-coordinate of two +points on the fluid surface. The tidal force of the com- +panion stretches the star in x-direction, whereas the poles +are slightly flattened. This measure of deformation is of +course coordinate-dependent, but it still provides some +physical insights. Fig. 10 shows the deformation ∆x/∆z +for each fluid surface. When the NSs are closer, the tidal +forces on the companion are stronger and hence the de- +formation is stronger. It can be observed that NSs with +a DM core are systematically less deformed than their +one-fluid counterparts. +The strong deformation in the dark halo case can also +be seen in Fig. 1, which shows a cut through the z = 0 +plane. +For a separation of 32 M⊙ (47.3 km) the de- +formation is clearly visible by eye. At a separation of +28 M⊙ (41.3 km) the deformation becomes already so +strong that the surfaces of the NSs touch and mass shed- +ding occurs. +The closeness to mass shedding can be quantified in +terms of the mass-shedding parameter χ, which was first +introduced in [86] and which we define as +χ(s) = +∂xh(s)|eq +∂zh(s)|pole,avg +, +(27) +25 +30 +35 +40 +45 +50 +55 +60 +65 +separation D +[km] +20 +25 +30 +35 +40 +45 +separation D/M +⊙ +1.000 +1.025 +1.050 +1.075 +1.100 +1.125 +1.150 +deformation ratio ∆x/∆z +BM, single fluid +BM, dark halo +BM, dark core +DM, dark halo +DM, dark core +FIG. 10. +Deformation ∆x/∆z of the fluid surfaces as func- +tion of the NS centres. The deformation is computed as the +ratio of the largest extents in x and z direction. Curves la- +beled BM show the deformation of the surface of the baryonic +fluid, whereas curves labeled DM show the deformation of the +DM surface. +where the label ”eq” denotes the point on the surface, +which is facing towards the other companion and for +which the x-coordinate is extremal. +The label ”pole” +denotes the surface points at which the z-coordinate is +extremal and where in Eq. (27) the label ”avg” indicates +that we have averaged over the values at the ”north and +south pole”. Note that for non-spinning stars the ”north” +and ”south pole” values only differ slightly due to round- +off error. In the mass shedding limit χ(s) will tend to +0. We evaluate the χ(s) for each fluid component indi- +vidually on the respective fluid surfaces. We show the +resulting χ(s) as a function of the distance of the centres +of the stars in Fig. 11. The DM fluid in the dark halo +scenario is easily deformable, which leads to a relatively +small mass shedding parameter of 0.9 already at a sep- +aration of 44 M⊙. We find that a separation of 28 M⊙ +leads to a configuration with touching star surfaces, from +which we conclude that mass shedding occurs somewhere +at a separation between 28 and 29 M⊙, which means the +system will transition relatively slowly to the mass shed- +ding regime over a time where the two NSs decrease their +separation by 16 M⊙. For the dark core configurations, +on the other hand, the transition to mass shedding is +rather sudden with χ reaching a value of 0.9 at sepa- +ration of approximately 23 M⊙ and the mass shedding +occurring for the baryonic fluid at a separation of 16 M⊙. +V. +CONCLUSION +We have extended the SGRID code to construct +constraint-solved, quasi-equlibrium configurations of bi- +naries of NSs consisting of two non-interacting fluids. +The second fluid represents DM that can comprise some +part of the matter of NS. In this study we have used the + +10 +25 +30 +35 +40 +45 +50 +55 +60 +65 +separation D +[km] +20 +25 +30 +35 +40 +45 +separation D/M +⊙ +0.4 +0.6 +0.8 +1.0 +mass shedding parameter χ +BM, single fluid +BM, dark halo +BM, dark core +DM, dark halo +DM, dark core +FIG. 11. +Mass shedding parameter χ as a function of the sep- +aration of the NS. Curves labeled BM show the deformation +of the surface of the baryonic fluid, whereas curves labeled +DM show the deformation of the DM surface. +EoS of a degenerate, relativistic Fermi gas with different +particle masses to model the DM fluid. +These quasi- +equlibrium configurations can be used as initial data for +NR inspiral simulations of DM admixed NS binaries. The +BAM code can already evolve mirror DM [25] and could +be easily extended to allow for general EoS for the DM +fluid. +Another possible application of the two fluid approach +are superfluid NS cores. At sufficiently high density BM +forms a state made of superfluid neutrons and supercon- +ducting protons, which can be described in a two fluid +approach. +However, the two fluids still interact with +each other due to the entrainment effect and the con- +dition of beta-equilibrium [87]. Solutions of isolated NS +with superfluid cores are constructed in [88, 89] taking +into account the interaction of the fluids. For a study of +superfluid and superconducting cores in binary NS the +formalism in this work could be extended using a similar +model for the interactions. In binary NS collisions the +temperature will rise above the critical temperature for +superfluidity and superconductivity, so that it becomes +necessary to include even a third fluid representing the +non-superfluid component. +We have tested the convergence of the constructed con- +figurations with respect to resolution. The Hamiltonian +constraint converges polynomially with an order of ≈ 2.7. +The lack of exponential convergence can be attributed to +the presence of the non-smooth transition of the density +at the surface of the inner fluid, which is not fitted to +the boundaries of the spectral elements. Self-convergence +tests for metric components and the specific enthalpies +show that the solution improves with increasing reso- +lution, but with a slightly broken convergence towards +higher resolution, which we again attribute to the sur- +face of the inner fluid. For future improvements to the +code it is a worthwhile consideration to implement a new +grid layout that allows fitting to the surface of a second +fluid +We have shown that the two fluids do not have the ex- +act same velocities, but that the difference in the resid- +ual velocities reaches up to 10% on the surface of the +inner fluids. The difference in the velocity profiles will +be even stronger if one assumes independent rotational +states for the components. In this work we only inves- +tigated only purely irrotational configurations, but our +formalism, in principle, allows for to construct configu- +rations with arbitrary spin for the individual stars and +fluid components. This is relevant in particular for the +DM component, which might only have insufficient mech- +anisms to lose angular momentum and hence could be in +a state of rapid rotation. +The presence of DM affects the compactness and de- +formability of NSs, which will change the merger dynam- +ics. We have shown that the presence of DM can delay +the point of mass-shedding to a later stage of the inspi- +ral, i. e., towards closer separations. This is in accordance +with the findings in numerical evolutions of two-fluid bi- +nary mergers [25]. In the case of a DM halo, mass shed- +ding could occur much earlier than for the baryonic com- +ponent. However the matter contained in the DM halo +is rather low and hence the impact of DM mass shedding +on the dynamics of the BM is potentially small, never- +theless, dynamical simulations are needed to verify this +assumption. +ACKNOWLEDGMENTS +This work was supported by funding from the FCT +– Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.P., within +the Project No. +EXPL/FIS-AST/0735/2021. +H.R.R. +and V.S. also acknowledge the support from the project +No. UIDB/04564/2020, and UIDP/04564/2020. W.T. +acknowledges funding from the National Science Foun- +dation under grant PHY-2136036. +[1] G. Narain, J. Schaffner-Bielich, and I. N. Mishustin, +Compact stars made of fermionic dark matter, Phys. Rev. +D 74, 063003 (2006), arXiv:astro-ph/0605724 [astro-ph]. +[2] P. Ciarcelluti and F. Sandin, Have neutron stars a +dark matter core?, Phys. Lett. B 695, 19 (2011), +arXiv:1005.0857 [astro-ph.HE]. +[3] K. M. Zurek, Asymmetric Dark Matter: Theories, Sig- +natures, and Constraints, Phys. Rept. 537, 91 (2014), +arXiv:1308.0338 [hep-ph]. +[4] I. Goldman, R. N. Mohapatra, S. Nussinov, D. Rosen- +baum, and V. Teplitz, Possible Implications of Asym- +metric Fermionic Dark Matter for Neutron Stars, Phys. + +11 +Lett. B 725, 200 (2013), arXiv:1305.6908 [astro-ph.CO]. +[5] M. I. Gresham, H. K. Lou, and K. M. Zurek, Nuclear +Structure of Bound States of Asymmetric Dark Matter, +Phys. Rev. D 96, 096012 (2017), arXiv:1707.02313 [hep- +ph]. +[6] J. Ellis, G. H¨utsi, K. Kannike, L. Marzola, M. Raidal, and +V. Vaskonen, Dark matter effects on neutron star prop- +erties, Phys. Rev. D 97, 123007 (2018), arXiv:1804.01418 +[astro-ph.CO]. +[7] A. E. Nelson, S. Reddy, and D. Zhou, Dark halos around +neutron stars and gravitational waves, J. Cosm. Astrop. +Phys. 2019, 012 (2019), arXiv:1803.03266 [hep-ph]. +[8] O. Ivanytskyi, V. Sagun, and I. Lopes, Neutron stars: +New constraints on asymmetric dark matter, Phys. Rev. +D 102, 063028 (2020). +[9] H. C. Das, A. Kumar, B. Kumar, S. K. Biswal, T. Nakat- +sukasa, A. Li, and S. K. Patra, Effects of dark matter on +the nuclear and neutron star matter, Mon. Not. R. As- +tron. Soc. 495, 4893 (2020), arXiv:2002.00594 [nucl-th]. +[10] A. Das, T. Malik, and A. C. Nayak, Dark matter ad- +mixed neutron star properties in the light of gravitational +wave observations: a two fluid approach, arxiv +(2020), +arXiv:2011.01318 [nucl-th]. +[11] V. Sagun, +E. Giangrandi, +O. Ivanytskyi, +I. Lopes, +and K. Bugaev, Constraints on the fermionic dark +matter from observations of neutron stars, Proc. Sc. +PANIC2021, +313 +(2022), +arXiv:2111.13289 +[astro- +ph.HE]. +[12] K.-L. Leung, M.-c. Chu, and L.-M. Lin, Tidal deforma- +bility of dark matter admixed neutron stars, Phys. Rev. +D 105, 123010 (2022). +[13] D. R. Karkevandi, S. Shakeri, V. Sagun, and O. Ivanyt- +skyi, Bosonic dark matter in neutron stars and its ef- +fect on gravitational wave signal, Physical Review D 105, +10.1103/physrevd.105.023001 (2022). +[14] V. Sagun, E. Giangrandi, O. Ivanytskyi, C. Providˆencia, +and T. Dietrich, How does dark matter affect com- +pact star properties and high density constraints of +strongly interacting matter, in 15th Conference on +Quark Confinement and the Hadron Spectrum (2022) +arXiv:2211.10510 [astro-ph.HE]. +[15] F. Di Giovanni, N. Sanchis-Gual, P. Cerd´a-Dur´an, and +J. A. Font, Can fermion-boson stars reconcile multimes- +senger observations of compact stars?, Phys. Rev. D 105, +063005 (2022), arXiv:2110.11997 [gr-qc]. +[16] A. Maselli, P. Pnigouras, N. G. Nielsen, C. Kouvaris, and +K. D. Kokkotas, Dark stars: gravitational and electro- +magnetic observables, Phys. Rev. D 96, 023005 (2017), +arXiv:1704.07286 [astro-ph.HE]. +[17] E. Giangrandi, V. Sagun, O. Ivanytskyi, C. Providˆencia, +and T. Dietrich, The effects of self-interacting bosonic +dark +matter +on +neutron +star +properties +(2022), +arXiv:2209.10905 [astro-ph.HE]. +[18] N. Rutherford, G. Raaijmakers, C. Prescod-Weinstein, +and A. Watts, Constraining bosonic asymmetric dark +matter with neutron star mass-radius measurements, +arXiv (2022), arXiv:2208.03282 [astro-ph.HE]. +[19] J. Bamber, J. C. Aurrekoetxea, K. Clough, and P. G. Fer- +reira, Black hole merger simulations in wave dark matter +environments (2022), arXiv:2210.09254 [gr-qc]. +[20] P. S. Cole, G. Bertone, A. Coogan, D. Gaggero, T. Kary- +das, B. J. Kavanagh, T. F. M. Spieksma, and G. M. +Tomaselli, Disks, spikes, and clouds: distinguishing en- +vironmental effects on BBH gravitational waveforms +(2022), arXiv:2211.01362 [gr-qc]. +[21] J. Ellis, A. Hektor, G. H¨utsi, K. Kannike, L. Marzola, +M. Raidal, and V. Vaskonen, Search for Dark Matter Ef- +fects on Gravitational Signals from Neutron Star Merg- +ers, Phys. Lett. B 781, 607 (2018), arXiv:1710.05540 +[astro-ph.CO]. +[22] M. Bezares, D. Vigan`o, and C. Palenzuela, Gravitational +wave signatures of dark matter cores in binary neutron +star mergers by using numerical simulations, Phys. Rev. +D 100, 044049 (2019), arXiv:1905.08551 [gr-qc]. +[23] A. Bauswein, G. Guo, J.-H. Lien, Y.-H. Lin, and M.- +R. Wu, Compact Dark Objects in Neutron Star Mergers +(2020), arXiv:2012.11908 [astro-ph.HE]. +[24] M. +Hippert, +E. +Dillingham, +H. +Tan, +D. +Curtin, +J. Noronha-Hostler, and N. Yunes, Dark Matter or Reg- +ular Matter in Neutron Stars? +How to tell the differ- +ence from the coalescence of compact objects (2022), +arXiv:2211.08590 [astro-ph.HE]. +[25] M. Emma, +F. Schianchi, +F. Pannarale, +V. Sagun, +and T. Dietrich, Numerical Simulations of Dark Mat- +ter Admixed Neutron Star Binaries, arXiv +(2022), +arXiv:2206.10887 [gr-qc]. +[26] E. Gourgoulhon, 3+1 Formalism in General Relativity +(Springer, Berlin, 2012) gr-qc/0703035. +[27] T. W. Baumgarte and S. L. Shapiro, Numerical Relativ- +ity: Solving Einstein’s Equations on the Computer (Cam- +bridge University Press, Cambridge, 2010). +[28] W. Tichy, The initial value problem as it relates to nu- +merical relativity, Reports on Progress in Physics 80, +026901 (2017). +[29] C. Kouvaris, The Dark Side of Neutron Stars, Adv. +High Energy Phys. 2013, 856196 (2013), arXiv:1308.3222 +[astro-ph.HE]. +[30] N. F. Bell, G. Busoni, S. Robles, and M. Virgato, Im- +proved Treatment of Dark Matter Capture in Neutron +Stars, JCAP 09, 028, arXiv:2004.14888 [hep-ph]. +[31] N. F. Bell, G. Busoni, T. F. Motta, S. Robles, A. W. +Thomas, and M. Virgato, Nucleon Structure and Strong +Interactions in Dark Matter Capture in Neutron Stars, +Phys. Rev. Lett. 127, 111803 (2021), arXiv:2012.08918 +[hep-ph]. +[32] C. Kouvaris and P. Tinyakov, Can neutron stars con- +strain dark matter?, Phys. Rev. D 82, 063531 (2010), +arXiv:1004.0586 [astro-ph.GA]. +[33] A. Del Popolo, M. Le Delliou, and M. Deliyergiyev, Neu- +tron Stars and Dark Matter, Universe 6, 222 (2020). +[34] G. Bertone and M. Fairbairn, Compact Stars as Dark +Matter +Probes, +Phys. +Rev. +D +77, +043515 +(2008), +arXiv:0709.1485 [astro-ph]. +[35] M. Baryakhtar et al., Dark Matter In Extreme As- +trophysical Environments, in 2022 Snowmass Summer +Study (2022) arXiv:2203.07984 [hep-ph]. +[36] C. Kouvaris, Wimp annihilation and cooling of neutron +stars, Phys. Rev. D 77, 023006 (2008), arXiv:0708.2362 +[astro-ph]. +[37] A. de Lavallaz and M. Fairbairn, Neutron stars as +dark matter probes, Phys. Rev. D 81, 123521 (2010), +arXiv:1004.0629 [astro-ph.GA]. +[38] K. Hamaguchi, N. Nagata, and K. Yanagi, Dark Matter +Heating vs. Rotochemical Heating in Old Neutron Stars, +Phys. Lett. B 795, 484 (2019), arXiv:1905.02991 [hep- +ph]. +[39] M. ´A. P´erez-Garc´ıa and J. Silk, Dark matter seeding and + +12 +the kinematics and rotation of neutron stars, Phys. Lett. +B 711, 6 (2012), arXiv:1111.2275 [astro-ph.CO]. +[40] A. Sedrakian, Axion cooling of neutron stars, Phys. Rev. +D 93, 065044 (2016). +[41] A. Sedrakian, Axion cooling of neutron stars. II. Be- +yond hadronic axions, Phys. Rev. D 99, 043011 (2019), +arXiv:1810.00190 [astro-ph.HE]. +[42] T. Dietrich and K. Clough, Cooling binary neutron +star remnants via nucleon-nucleon-axion bremsstrahlung, +Phys. Rev. D 100, 083005 (2019), arXiv:1909.01278 [gr- +qc]. +[43] M. Buschmann, C. Dessert, J. W. Foster, A. J. Long, +and B. R. Safdi, Upper Limit on the QCD Axion Mass +from Isolated Neutron Star Cooling, Phys. Rev. Lett. +128, 091102 (2022), arXiv:2111.09892 [hep-ph]. +[44] J. S. Read, B. D. Lackey, B. J. Owen, and J. L. Fried- +man, Constraints on a phenomenologically parameterized +neutron-star equation of state, Phys. Rev. D 79, 124032 +(2009), arXiv:0812.2163 [astro-ph]. +[45] F. Douchin and P. Haensel, A unified equation of state +of dense matter and neutron star structure, Astron. As- +trophys. 380, 151 (2001), arXiv:astro-ph/0111092. +[46] J. Antoniadis, P. C. C. Freire, N. Wex, T. M. Tauris, R. S. +Lynch, M. H. van Kerkwijk, M. Kramer, C. Bassa, V. S. +Dhillon, T. Driebe, J. W. T. Hessels, V. M. Kaspi, V. I. +Kondratiev, N. Langer, T. R. Marsh, M. A. McLaugh- +lin, T. T. Pennucci, S. M. Ransom, I. H. Stairs, J. van +Leeuwen, J. P. W. Verbiest, and D. G. Whelan, A Massive +Pulsar in a Compact Relativistic Binary, Science 340, +448 (2013), arXiv:1304.6875 [astro-ph.HE]. +[47] E. Fonseca et al., Refined Mass and Geometric Measure- +ments of the High-mass PSR J0740+6620, Astrophys. J. +Lett. 915, L12 (2021), arXiv:2104.00880 [astro-ph.HE]. +[48] M. C. Miller, F. K. Lamb, A. J. Dittmann, S. Bog- +danov, Z. Arzoumanian, K. C. Gendreau, S. Guillot, +A. K. Harding, W. C. G. Ho, J. M. Lattimer, R. M. Lud- +lam, S. Mahmoodifar, S. M. Morsink, P. S. Ray, T. E. +Strohmayer, K. S. Wood, T. Enoto, R. Foster, T. Oka- +jima, G. Prigozhin, and Y. Soong, Psr j0030+0451 mass +and radius from nicer data and implications for the prop- +erties of neutron star matter, The Astrophysical Journal +887, L24 (2019). +[49] T. E. Riley et al., A NICER View of PSR J0030+0451: +Millisecond Pulsar Parameter Estimation, Astrophys. J. +Lett. 887, L21 (2019), arXiv:1912.05702 [astro-ph.HE]. +[50] G. Raaijmakers et al., Constraining the dense matter +equation of state with joint analysis of NICER and +LIGO/Virgo measurements, Astrophys. J. Lett. 893, L21 +(2020), arXiv:1912.11031 [astro-ph.HE]. +[51] M. C. Miller et al., The Radius of PSR J0740+6620 from +NICER and XMM-Newton Data, Astrophys. J. Lett. +918, L28 (2021), arXiv:2105.06979 [astro-ph.HE]. +[52] T. E. Riley et al., A NICER View of the Massive Pul- +sar PSR J0740+6620 Informed by Radio Timing and +XMM-Newton Spectroscopy, Astrophys. J. Lett. 918, +L27 (2021), arXiv:2105.06980 [astro-ph.HE]. +[53] B. P. Abbott et al. (LIGO Scientific, Virgo), Properties +of the binary neutron star merger GW170817, Phys. Rev. +X 9, 011001 (2019), arXiv:1805.11579 [gr-qc]. +[54] B. P. Abbott et al. (LIGO Scientific, Virgo), GW190425: +Observation of a Compact Binary Coalescence with To- +tal Mass ∼ 3.4M⊙, Astrophys. J. Lett. 892, L3 (2020), +arXiv:2001.01761 [astro-ph.HE]. +[55] D. Clowe, M. Bradaˇc, A. H. Gonzalez, M. Markevitch, +S. W. Randall, C. Jones, and D. Zaritsky, A direct em- +pirical proof of the existence of dark matter, The Astro- +physical Journal 648, L109 (2006). +[56] S. W. Randall, M. Markevitch, D. Clowe, A. H. Gonza- +lez, and M. Bradac, Constraints on the Self-Interaction +Cross-Section of Dark Matter from Numerical Simula- +tions of the Merging Galaxy Cluster 1E 0657-56, Astro- +phys. J. 679, 1173 (2008), arXiv:0704.0261 [astro-ph]. +[57] J. +Billard +et +al., +Direct +detection +of +dark +mat- +ter—APPEC committee report*, Rept. Prog. Phys. 85, +056201 (2022), arXiv:2104.07634 [hep-ex]. +[58] V. Sagun, E. Giangrandi, O. Ivanytskyi, I. Lopes, and +K. A. Bugaev, Constraints on the fermionic dark matter +from observations of neutron stars, PoS PANIC2021, +313 (2022), arXiv:2111.13289 [astro-ph.HE]. +[59] W. Tichy, Constructing quasi-equilibrium initial data for +binary neutron stars with arbitrary spins, Phys. Rev. +D86, 064024 (2012), arXiv:1209.5336 [gr-qc]. +[60] W. +Tichy, +A. +Rashti, +T. +Dietrich, +R. +Dudi, +and +B. Br¨ugmann, Constructing binary neutron star ini- +tial data with high spins, high compactnesses, and +high mass ratios, Physical Review D 100, 10.1103/phys- +revd.100.124046 (2019), arXiv:1910.09690 [gr-qc]. +[61] E. Gourgoulhon, P. Grandcl´ement, J.-A. Marck, J. No- +vak, and K. Taniguchi, http://www.lorene.obspm.fr. +[62] E. +Gourgoulhon, +P. +Grandcl´ement, +K. +Taniguchi, +J. Marck, and S. Bonazzola, Quasiequilibrium sequences +of synchronized and irrotational binary neutron stars in +general relativity: Method and tests, Phys. Rev. D 63, +064029 (2001). +[63] N. Tacik, F. Foucart, H. P. Pfeiffer, R. Haas, S. Ossokine, +J. Kaplan, C. Muhlberger, M. D. Duez, L. E. Kidder, +M. A. Scheel, and B. Szil´agyi, Binary Neutron Stars with +Arbitrary Spins in Numerical Relativity, Phys. Rev. D92, +124012 (2015), arXiv:1508.06986 [gr-qc]. +[64] P. Grandcl´ement, S. Auliac, H. Roussille, and L. J. Pa- +penfort, https://kadath.obspm.fr/fuka/. +[65] L. J. Papenfort, S. D. Tootle, P. Grandcl´ement, E. R. +Most, and L. Rezzolla, New public code for initial data +of unequal-mass, spinning compact-object binaries, Phys. +Rev. D 104, 024057 (2021), arXiv:2103.09911 [gr-qc]. +[66] A. Rashti, F. M. Fabbri, B. Br¨ugmann, S. V. Chaurasia, +T. Dietrich, M. Ujevic, and W. Tichy, New pseudospec- +tral code for the construction of initial data, Phys. Rev. +D 105, 104027 (2022), arXiv:2109.14511 [gr-qc]. +[67] K. Ury¯u and A. Tsokaros, New code for equilibriums +and quasiequilibrium initial data of compact objects, +Phys.Rev. D85, 064014 (2012), arXiv:1108.3065 [gr-qc]. +[68] K. Ury¯u, S. Yoshida, E. Gourgoulhon, C. Markakis, +K. Fujisawa, A. Tsokaros, K. Taniguchi, and Y. Eriguchi, +New code for equilibriums and quasiequilibrium initial +data of compact objects. iv. rotating relativistic stars +with mixed poloidal and toroidal magnetic fields, Phys. +Rev. D 100, 123019 (2019), arXiv:1906.10393 [gr-qc]. +[69] D. Wittman, N. Golovich, and W. A. Dawson, The Mis- +measure of Mergers: Revised Limits on Self-interacting +Dark Matter in Merging Galaxy Clusters, Astrophys. J. +869, 104 (2018), arXiv:1701.05877 [astro-ph.CO]. +[70] G. B. Cook, Initial data for numerical relativity, Liv- +ing Reviews in Relativity 3, 10.12942/lrr-2000-5 (2000), +arXiv:gr-qc/0007085 [gr-qc]. +[71] W. Tichy, The initial value problem as it relates to nu- +merical relativity, Rept. Prog. Phys. 80, 026901 (2017), +arXiv:1610.03805 [gr-qc]. + +13 +[72] W. Tichy, Initial data for binary neutron stars with +arbitrary +spins, +Phys. +Rev. +D +84, +024041 +(2011), +arXiv:1107.1440 [gr-qc]. +[73] G. Sch¨afer and N. Wex, Phys. Lett. A 174, 196 (1993). +[74] W. Tichy, B. Br¨ugmann, M. Campanelli, and P. Diener, +Binary black hole initial data for numerical general rela- +tivity based on post-Newtonian data, Phys. Rev. D 67, +064008 (2003), gr-qc/0207011. +[75] W. Tichy and B. Br¨ugmann, Quasi-equilibrium binary +black hole sequences for puncture data derived from he- +lical killing vector conditions, Phys. Rev. D 69, 024006 +(2004), gr-qc/0307027. +[76] F. Swesty, Thermodynamically consistent interpolation +for equation of state tables, Journal of Computational +Physics 127, 118 (1996). +[77] A. Cumming, E. F. Brown, F. J. Fattoyev, C. J. +Horowitz, D. Page, and S. Reddy, Lower limit on the +heat capacity of the neutron star core, Phys. Rev. C 95, +025806 (2017), arXiv:1608.07532 [astro-ph.HE]. +[78] D. Page, U. Geppert, and F. Weber, The Cooling of com- +pact stars, Nucl. Phys. A 777, 497 (2006), arXiv:astro- +ph/0508056. +[79] S. W. Hawking and G. F. R. Ellis, The Large Scale Struc- +ture of Space-Time (Cambridge University Press, Cam- +bridge (UK), 1973). +[80] M. Shibata, A relativistic formalism for computation +of irrotational binary stars in quasi equilibrium states, +Phys. Rev. D 58, 024012 (1998), arXiv:gr-qc/9803085. +[81] J. P. Ostriker and J. E. Gunn, On the nature of pulsars. +i. theory., Astrophys. J., 157: +1395-1417(Sept. 1969). +10.1086/150160 (1969). +[82] M. Colpi, A. Possenti, S. Popov, and F. Pizzolato, Spin +and magnetism in old neutron stars, in Physics of Neu- +tron Star Interiors, edited by D. Blaschke, A. Sedrakian, +and N. K. Glendenning (Springer Berlin Heidelberg, +Berlin, Heidelberg, 2001) pp. 440–466, astro-ph/0012394. +[83] A. Rogers and S. Safi-Harb, On the diversity of com- +pact objects within supernova remnants – II. energy-loss +mechanisms, Monthly Notices of the Royal Astronomical +Society 465, 383 (2016), arXiv:1610.04685 [astro-ph.HE]. +[84] R. Arnowitt, S. Deser, and C. W. Misner, The dynamics +of general relativity, in Gravitation: An Introduction to +Current Research, edited by L. Witten (Wiley, New York, +1962) pp. 227–265. +[85] E. Gourgoulhon, 3+1 formalism and bases of numerical +relativity (2007), arXiv:gr-qc/0703035. +[86] E. Gourgoulhon, P. Grandcl´ement, K. Taniguchi, J.-A. +Marck, and S. Bonazzola, Quasiequilibrium sequences +of synchronized and irrotational binary neutron stars in +general relativity: Method and tests, Phys. Rev. D 63, +064029 (2001), gr-qc/0007028. +[87] G. L. Comer and R. Joynt, Relativistic mean field model +for entrainment in general relativistic superfluid neutron +stars, Phys. Rev. D 68, 023002 (2003), gr-qc/0212083. +[88] R. Prix, J. Novak, and G. L. Comer, Relativistic numeri- +cal models for stationary superfluid neutron stars, Phys. +Rev. D 71, 043005 (2005), gr-qc/0410023. +[89] N. Chamel, Two-fluid models of superfluid neutron star +cores, Monthly Notices of the Royal Astronomical Society +388, 737 (2008), arXiv:0805.1007 [astro-ph]. + diff --git a/EdE1T4oBgHgl3EQf-gYV/content/tmp_files/load_file.txt b/EdE1T4oBgHgl3EQf-gYV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..96ebbd6d87f974a896da949f9cf289e27d6224dc --- /dev/null +++ b/EdE1T4oBgHgl3EQf-gYV/content/tmp_files/load_file.txt @@ -0,0 +1,1303 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf,len=1302 +page_content='Quasi-equilibrium configurations of binary systems of dark matter admixed neutron stars Hannes R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' R¨uter ,1 Violetta Sagun ,1 Wolfgang Tichy ,2 and Tim Dietrich 3, 4 1CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal 2Department of Physics, Florida Atlantic University, Boca Raton, FL 33431, USA 3Institut f¨ur Physik und Astronomie, Universit¨at Potsdam, Haus 28, Karl-Liebknecht-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 24/25, Potsdam, Germany 4Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M¨uhlenberg 1, Potsdam 14476, Germany (Dated: January 10, 2023) Using an adapted version of the SGRID code, we construct for the first time consistent quasi- equilibrium configurations for a binary system consisting of two neutron stars in which each is admixed with dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The stars are modelled as a system of two non-interacting fluids min- imally coupled to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' For the fluid representing baryonic matter the SLy equation of state is used, whereas the second fluid, which corresponds to dark matter, is described using the equation of state of a degenerate Fermi gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We consider two different scenarios for the distribution of the dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In the first scenario the dark matter is confined to the core of the star, whereas in the second scenario the dark matter extends beyond the surface of the baryonic matter, forming a halo around the baryonic star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The presence of dark matter alters the star’s reaction to the companion’s tidal forces, which we investigate in terms of the coordinate deformation and mass shedding pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The constructed quasi-equilibrium configurations mark the first step towards consistent numerical-relativity simulations of dark matter admixed neutron star binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' INTRODUCTION In the present era of gravitational wave (GW) astron- omy, the internal properties of compact stars can be probed during their mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Using numerical-relativity (NR) simulations of the last stages of a binary coales- cence, it is possible to relate observational GW data to properties of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' While these simulations have undergone significant improvements in the past, the im- pact of dark matter (DM) on the binary neutron star (NS) dynamics has not yet been investigated in detail and is not taken into account in standard GW analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In fact, considering a coalescence of compact objects to occur in pure vacuum, could be an oversimplification that may lead to incorrect conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Due to their high compactness, NSs can trap and ac- cumulate DM in their interior throughout the star’s evo- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' DM alters the compact star’s properties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', its mass, its radius, its tidal deformability, its energy density and speed of sound profiles [1–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Its effect depends on the relative fraction of DM and on the exact equation of state (EoS) for the DM and baryonic matter (BM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' For an extended discussion of the impact of DM on compact star properties and its smoking gun signals, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' While the effect of DM on isolated NSs can be probed through electromagnetic observations, GW observations of binary systems of DM admixed compact stars open up a new observational window and the possibility to probe a density and temperature range larger that of isolated stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' To push forward our understanding of the imprint of DM, we construct quasi-equilibrium configurations of DM admixed NS binary system and study the impact of DM focusing on quantities pertaining to binary system, such as the orbital binding energy and the tidal deforma- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' It is worth noting that not only NSs, but also black holes could be embedded into DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' A step towards un- derstanding the impact of DM on black hole mergers was made in [19], where the behaviour of wave DM around equal mass black hole binaries was studied in numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Furthermore, GW signals from binary coa- lescences contain information of the binaries surrounding medium [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The effect of DM on the inspiral and post-merger phases of DM admixed NSs has been studied by a few groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' A first study by Ellis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [21] used a simple mechanical model, and showed that a DM core can lead to the appearance of additional peaks in the post-merger GW spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In [22] NR simulations of equal-mass bi- naries consisting of BM admixed with a bosonic Klein- Gordon field were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' For a DM mass fraction of 10%, a redistribution of fermionic matter by the bosonic cores was found, followed by the formation of a one-arm spiral instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Another approach approximating com- pact dark component as test particles was studied in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The simulations show the DM component to remain grav- itationally bound after the merger of BM and orbit the center of the remnant with an orbital separation of a few km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The DM core and a host star are likely to spin at different rotational frequencies just after the merger due to the absence of non-gravitational interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Further on, they may synchronise via the gravitational angular momentum transfer, including tidal effects [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Up to our knowledge, the first two-fluid NR simulations describing binaries of DM admixed NSs were performed by Emma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [25] for a mixture of BM and mirror DM only interacting via the gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The results demonstrate that these systems tend to have a longer in- spiral phase with increasing amount of DM, which could be associated to the lower deformability of DM admixed NSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' These simulations however, did not start from ini- tial data satisfying the Hamiltonian and momentum con- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='03568v1 [gr-qc] 9 Jan 2023 2 straints [26–28] and the fluids did not start in an equilib- rium configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Instead the initial data was approx- imated by superimposing TOV-like solutions of isolated DM admixed NSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In this work we construct consistent, constraint-solved, quasi-equilibrium conditions for a two- fluid system of BM and DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' One possible scenario for the formation of DM admixed NSs is the capture of DM particles during the lifetime of the star, from a progenitor to the equilibrated NS stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The core of a NS is very dense and hence the chance of a DM particle experiencing scattering is relatively high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In this scattering process the particle transfers its kinetic energy to the star, becoming gravitationally bound [29– 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' This process is more efficient towards the Galac- tic center, where the density of DM is many orders of magnitude greater than in the galaxy’s arms [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' A conservative estimate of DM capture in the most cen- tral part of the Galaxy shows that stars accumulate up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='01% of DM during the main sequence and equili- brated NS stages combined [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' However, also higher DM factions inside compact stars can be achieved through other scenarios, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', DM production during a supernova explosion, accretion of DM clumps formed at the early stage of the Universe, or initial star formation on a pre- existing DM seed or local DM rich environments [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' If DM is symmetric, it cannot reach a high fraction due to self-annihilation, producing an electromagnetic or neu- trino signal [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The latter scenario could lead to addi- tional heating of isolated NSs as well as post-merger rem- nants [37, 38], modification of kinematic properties [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Moreover, production of light DM particles, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', axions, in nucleon bremsstrahlung or in Cooper pair breaking and formation processes in the NS interior [40–43], could speed up the thermal evolution of a star by contributing an additional cooling channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We consider DM to be either concentrated in a core or extending beyond the surface of BM, forming a DM halo around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' As a first step, we consider non-interacting, fermonic DM with spin 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The single star properties of this DM candidate have been discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The baryonic component is modelled through a piecewiese- polytropic fit [44] of the SLy EoS [45] that reproduces nuclear matter ground state properties, fulfils heaviest pulsars measurements [46, 47], X-ray observations by NICER [48–52], and tidal deformability constraints from GW170817 [53] and GW190425 [54] binary NS mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The two components interact only through gravity, and therefore do not repel each other, but overlap due to the absence of non-gravitational interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' This assump- tion is in very good agreement with the observations of the Bullet Cluster [55, 56] and direct DM searches [57], which show that the DM-BM cross section to be many orders of magnitude lower than the typical nuclear one, σDM−BM ≈ 10−45 cm2 ≪ σBM ∼ 10−24 cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' By varying the particle mass and relative fraction of DM, we obtain either a core configuration with a ra- dius of the DM component less or equal to the baryonic one, RD ≤ RB, or a halo with RD > RB [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' For both scenarios, we construct initial configurations em- ploying SGRID [59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Many other codes exist for the construction of quasi-equilibrium NS binary systems, no- tably the spectral codes LORENE [61, 62], Spells [63], FUKA [64, 65], Elliptica [66], and the finite difference based code COCAL [67, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Up to our knowledge, these codes are only able to solve systems consisting of a sin- gle fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Here we construct for the first time quasi- equilibrium binary configurations with two fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The formalism and results are presented in geometric units in which the gravitational constant G = 1 and the speed of light c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In these units, lengths are given as multiples of the solar mass, M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' For the conversion to SI units a spatial length must be multiplied by L0 = 1476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='6250 m/M⊙ and a time by T0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='9254909 × 10−6 s/M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Where appropriate we also use MeV to specify en- ergy and mass of particles, as well as SI units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Through- out the paper, Greek letter indices denote four dimen- sional, spacetime indices, whereas Latin indices denote three-dimensional, spatial indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In Section II we summarize the two-fluid formalism and DM distribu- tion regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Its implementation to the SGRID code is described in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In Section IV we analyse the convergence properties of the constructed configurations, quantify the difference in the velocities of the two flu- ids and investigate some physical properties of the quasi- equilibrium configuration over a sequence of separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Section V summarizes the results and discusses future perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' FORMALISM We describe the matter as a system of two non- interacting perfect fluids only indirectly coupled through the gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' This model is well justified, be- cause the interaction between standard model BM and DM is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Utilisation of the perfect fluid model for DM is also justified, as the mean free path and the scattering time scale of DM particles can be small compared to the characteristic time scales of the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In the following, we estimate the mean free path and scattering time in a semi-classical approach for a degenerate Fermi gas of particles with the mass of 170 MeV (≈ 3 × 10−28 kg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The Fermi gas consists of non-interacting fermions, for which a self-scattering cross section σDM formally does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Instead, we use the value of the upper limit obtained from observations of merging galaxies, which yield σDM/m(DM) p < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='25 cm2/g, with m(DM) p the mass of the DM particles [56, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In this work we construct configurations with a particle density n(DM) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='7 fm−3 in the center of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Together with the upper limit for σDM this yields a mean free path λ = 1/(n(DM)σDM) of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='7 × 10−17 m, much smaller than the typical length scale of a NS, which is on the order of 104 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The scatter- ing time scale can be estimated using the Fermi velocity, which reaches values up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='8 c in the centre of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 3 Finally, using the value of the mean free path, this yields a scattering time of tc = λ/vDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='5 × 10−25 s, much smaller than for example the orbital period of the binary, which in our configurations is a small as 3 × 10−4 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' At the surface of the stars DM reaches the free streaming limit and the perfect fluid limit breaks down, but there the density is so small, that the impact on the gravita- tional field is low and hence the matter in this region can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' For non-interacting fluids, the energy-momentum ten- sor can be split into the two individual fluid components given by: T (s) µν = (e(s) + p(s))u(s) µ u(s) ν + p(s)gµν , (1) where e is the proper energy density, p is the pressure, uµ is the four velocity of the fluid and the label (s) denotes the particles species, which is either BM or DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The Einstein field equations are then given by Rµν + 1 2gµνR = 8π(T (BM) µν + T (DM) µν ) (2) and, because the two particle species do not interact, each fluid satisfies the equations of motion of a single fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Consequently, each fluid satisfies energy momen- tum conservation separately: ∇µT (s) µν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' For each fluid, we also define the rest mass density ρ(s) 0 , which is computed from the number density n(s) by ρ(s) 0 = m(s) p n(s) , (3) with m(s) p being the mass of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Furthermore, the specific enthalpy is then given by h(s) = e(s) + p(s) ρ(s) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (4) To make the equations tractable, the spacetime metric gµν is decomposed into a temporal and a spatial part by introducing the spatial metric γij, the lapse α, and the shift βi [27, 70, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The line element in this 3+1 split reads ds2 = −α dt2 + γij (βidt + dxi)(βjdt + dxj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (5) The extrinsic curvature Kij is related to the time deriva- tive of γij, by the formula Kij = − 1 2α(∂tγij − Diβj − Djβi) , (6) where Di denotes the covariant derivative compatible with the spatial metric γij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We construct the partial differential equations govern- ing quasi-equilibrium by following the derivation in [72], which is trivially applied to a system of non-interacting fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' To generate quasi-equilibrium configurations, we solve equations for velocity potentials φ(s), which are de- fined through the following split of the four-velocity γi µu(s)µ = 1 h(s) (Diφ(s) + w(s)i) , (7) where w(s)i is a divergence free vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', Diw(s)i = 0, describing the rotational part of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Following the derivation of [72], we fix the time derivatives of the fields by assuming the existence of an approximate Killing vec- tor ξ and a set of quasi-equilibrium conditions for the two fluids Lξe(s) ≈ 0 , (8) Lξp(s) ≈ 0 , (9) γi µLξ(∇µφ(s)) ≈ 0 , (10) γi µL ∇φ(s) h(s)u(s)0 w(s) µ ≈ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (11) We omit further details of the derivation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' since for non- interacting fluids everything can be directly carried over to the individual fluid components,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' and we state only the resulting partial differential equation for the velocity potentials φ(s): Di � ρ(s) 0 α h(s) (Diφ(s) + w(s)i) − ρ(s) 0 αu(s)0(βi + ξi) � = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (12) where the boost factor u(s)0 is given by u(s)0 = � h(s)2 + (Diφ(s) + w(s) i )(Diφ(s) + w(s)i) αh(s) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (13) and the specific enthalpy is given by the expression h(s) = � L(s)2 − (Diφ(s) + w(s) i )(Diφ(s) + w(s)i) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (14) with L(s)2 = b(s) + � b(s)2 − 4α4((Diφ(s) + w(s) i )w(s)i)2 2α2 (15) and b(s) = ((ξi+βi)Diφ(s)−C(s))2+2α2(Diφ(s)+w(s) i )w(s)i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (16) The variable C(s) is a constant, which can vary for each star and controls the mass of the fluid component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' For the approximate Killing vector ξi we make the fol- lowing ansatz: ξi = Ω(−y, x − xCM, 0) + vr D (ri − ri CM) , (17) where Ω is the instantaneous orbital frequency, D is the separation between the star centres, vr is the radial ve- locity, and xCM is the x-coordinate of the centre of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 4 At apsis the orbital frequency together with the sepa- ration of the stars control the orbital parameters like ec- centricity and length of the semi-major axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Away from apsis there is a non-vanishing radial component of the velocity to be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In cases like the “circu- lar” inspiral there is no apsis, but there is an always non- vanishing radially inward directed velocity component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The configurations presented in this work are constructed within the quasi-circular approximation for which the ra- dial component is neglected, vr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We set the value of Ω to its value at second Post-Newtonian order in Arnowitt-Deser-Misner (ADM) gauge [73–75] using the sum of the rest masses of the two fluids as the mass estimates of the stars, which are computed by m(s) 0i = � Vi ρ(s) i u(s)0α � det(γjk)d3x , (18) where Vi is the spatial volume over which the i-th star extends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The value of xCM is then given by xCM = (m(BM) 01 + m(DM) 01 )xc1 + (m(BM) 02 + m(DM) 02 )xc2 m(BM) 01 + m(DM) 01 + m(BM) 02 + m(DM) 02 , (19) where xc1/2 are the x-coordinates of the centres of the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In this work, we present results for equal-mass configurations only, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', xCM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Besides the continuity equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (12)) governing the fluid velocity potentials φ(s), the metric must be fixed in a way satisfying the ADM constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' To this end we choose a conformally flat ansatz for the spatial metric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', γij = ψ4¯γij, with γij = δij and ∂tγij = 0, and con- struct the data on maximally sliced hypersurfaces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', the trace of the extrinsic curvature vanishes: K = 0 and ∂tK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The free metric components are the lapse, shift, and conformal factor ψ and their governing equa- tions are formulated in terms of the extended conformal thin sandwich equations (XCTS) [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Together with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (12), the data is constrained by a set of seven coupled partial differential equations, which are solved iteratively one-by-one in a self-consistent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' SGRID We have adapted the pseudo-spectral SGRID code [59, 60] to generate quasi-equilibrium configurations for two fluid systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We use the same iteration scheme that is used in [60] for single-fluid NSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We sketch the iter- ation scheme in the following with an emphasis on the adaptions and changes made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' To ensure the convergence of the solver, it is nec- essary to provide an initial guess sufficiently close to the true solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' This initial guess is chosen as a superposition of two boosted TOV-like two fluid stars of a given mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' To generate solutions with particular rest masses for the fluid components, one has to find the central pressures for which the masses are realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Since we are dealing with two fluids, this is a two-dimensional root finding prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In our tests, we found that using the Newton- Raphson method is not always reliable, because the masses are not a monotonous function of the central pressures, hence, a Newton-Raphson solver easily gets caught in a local extremum of the mass func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Instead, we employ a series of bisections on the central pressure of one fluid component while keeping the central pressure of the other fluid fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The series of bisections iterates between the two fluid components in a self-consistent manner until the fluid masses are sufficiently close to the target parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' If the residuals of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (12) are larger than 10% of the combined residuals of the XCTS equations, we solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (12) and set the new φ(s) to be the average of the old solution φ(s) old and the just ob- tained solution φ(s) ell , using the following weights φ(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='8φ(s) old + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='2φ(s) ell .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We proceed by solving the XCTS equations and update α, β, and ψ in the same way, averaging the old and new solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We do not adjust the values of Ω and xCM as in [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The value of Ω would be fixed within an eccentric- ity reduction scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' xCM is left at its Newtonian value, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We adjust the constants C(s), such that the rest masses of each component and in each star match our desired target masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We then update the val- ues of h(s) keeping it fixed until the end of the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' If the sum of the residuals is below a certain toler- ance or a prescribed maximum number of iterations is reached, the iteration ends here and is concluded with a final solving of the XCTS equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The system of partial differential equations does not fix the position of the stars and, hence, they will slowly drift if not kept under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' To keep the stars in place, the center of the stars are driven back to the desired position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' For single fluids, the center is usually defined in an unambiguous way as the point of maximum density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' For two fluids the defi- nition is ambiguous, because the tidal deformations due to the companion star are different for each fluid component and, consequently, the maximum densities are at different points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In most cases, how- ever, the two maximum points will still be close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The results shown in this work are obtained by choosing the point with the maximum of the to- tal proper energy density, e(tot) = e(BM) + e(DM), as the center of the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We have chosen e(tot), 5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='3 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='2 25 20 15 10 5 0 5 10 15 20 25 X 10 8 6 4 2 0 2 4 6 8 10 Y 25 20 15 10 5 0 5 10 15 20 25 X 10 8 6 4 2 0 2 4 6 8 10 Y 25 20 15 10 5 0 5 10 15 20 25 X 10 8 6 4 2 0 2 4 6 8 10 Y 25 20 15 10 5 0 5 10 15 20 25 X 10 8 6 4 2 0 2 4 6 8 10 Y FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Specific enthalpy in the z = 0 plane for a config- uration with DM halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In the upper halves only the specific enthalpy of DM is shown, whereas in the lower halves the BM component lies on top of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The black lines indicate the boundaries of the spectral elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Each NS is comprised of a central cubical element and six cubed sphere elements (of which only four intersect the z = 0 plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The separation between the NS centres amounts to 32 M⊙ (47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='3 km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' in particular, because it is a covariant scalar and it is the major quantity determining the gravitational potential, hence giving an estimate for the center of mass of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' To drive the center of mass back, the values of h(s) are transformed by h(s),new = h(s) + ∆ri∂ih(s) , (20) where ∆ri = ri current − ri desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Continue with step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The SGRID code uses surface-fitted coordinates to re- duce the Runge phenomenon at the surface of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Each time we update the specific enthalpy h(s) (step 5 in the iteration), we adapt the grid such that the bound- aries of spectral elements coincide with the new surface of the outer fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' That means we only construct con- figurations in which the surfaces of the two fluids do not intersect, which would in principle be possible given the different deformabilities of the fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Furthermore, we do not construct domains that are adapted to the surface of the inner fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Therefore, at the surface of the inner fluid one can expect to observe the Runge phenomenon and a slight degradation of the convergence in the trun- cation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 1 shows a visualisation of the deformed spectral elements inside the NS and the distribution of matter in terms of the specific enthalpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' To close the system, the EoS is required to relate e(s), p(s), ρ(s) 0 , and h(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' For the EoS, SGRID reads in either parameters of piecewise polytropes or EoS tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' EoS tables are interpolated in a thermodynamically consis- tent manner [76] using a cubic Hermite interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' To find the thermodynamic quantities for a given specific enthalpy a Newton-Raphson root finder is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' At low densities we use a polytrope that is matched at the lowest density of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Parameters of Constructed Configurations We consider two different two-fluid configurations, one in which DM is confined to the core of the NS, the dark core configuration, and one in which DM extends beyond the surface of the BM, so that the NS has a DM halo, which we will refer to as the dark halo configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Furthermore we compare to configurations consisting of BM only: the single fluid configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We describe BM by a piecewise-polytropic fit [44] to the SLy EoS [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' As a model of DM, we investigate the degenerate, relativistic Fermi gas of spin- 1 2 particles at zero temperature, for which the EoS is read in as tabu- lated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' EoSs at zero temperature are sufficient for our calculations, because the Fermi energy of the sys- tem is much higher than its temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The typical temperature T0 of NS cores is of the order of 106 − 108 K [77, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We assume that DM has the same tempera- ture as the BM, because the captured DM particles keep scattering with baryons, rarely but often enough to ther- malise with the BM component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' A core temperature of approximately 108 K is much lower than the Fermi en- ergy of BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' This is also true for the Fermi gas EoS we consider, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' in the dark halo case the Fermi energy of DM reaches 403 MeV in the center of the star, an energy smaller than that of the BM, but still much larger than the temperature of the star, kBT0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='009 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' For the dark core configuration the DM particles have a mass of 1000 MeV and DM provides 5% of the NSs’ total rest mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Fermionic DM particles with mass of 1000 MeV present an interesting case, because they resemble nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In the dark halo case we model DM by particles with a mass of 170 MeV, for which the fluid is less dense and hence easily forms a halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Furthermore in the dark halo configuration DM only contributes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='5% of the total rest mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The choice of these values for the particle masses is motivated by the results of [8], where it was shown that for the DM particle masses below 174 MeV DM admixed NSs are in agreement with astrophysical observations of the heaviest NSs for arbitrary relative fraction of DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Moreover, the chosen mass of 170 MeV and the fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='5% leads to a relatively small halo of approximately twice the radius of the BM component, which is easy to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' When the size of the halos is big enough so that they touch each other, it is no longer possible to fit the element surfaces to the outer fluid of a star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Hence, we are discarding configurations with separations at which the two halos merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In all configurations the individual NSs have the same total rest mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', the combined rest mass of BM and DM is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='4M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In all setups, the NSs have equal masses and are irrotational, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', they have zero spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Convergence To validate the code, we check the convergence of the Hamiltonian constraint for a dark halo configuration of NSs with a separation of 44 M⊙ (65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='0 km) on a quasi- circular orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 2 shows the magnitude of the Hamiltonian con- straint H on the z = 0 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The constraint violations are largest in the interior of the star, where they reach values up to 4×10−5, whereas in the vacuum regions the error drops to values below 10−9, but with some spikes on the order of 10−7 at the element boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' A be- haviour commonly seen for spectral codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The Hamil- tonian constraint is largest in the region where the inner fluid is non-vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 2 one can observe a clear transition on the surface of the baryonic fluid to lower constraint violations in the DM halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 3 demonstrates the development of the volume- normalised L2-norm of the Hamiltonian constraint for the inner cube of one of the stars during the iterative solv- ing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The figure shows the behaviour for different number of points n in each dimension, which is the same for each spectral element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' All curves show a saturation in the norm of the Hamiltonian constraint towards the end of the iteration process, which for all configurations is stopped after 40 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Furthermore, it is visible that higher resolution leads to smaller violations of the Hamiltonian constraint in the final solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' For compar- ison Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 3 also shows the sequence for a corresponding single fluid configuration with the same mass and sepa- ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' After 40 iterations its Hamiltonian constraint is a factor 10 smaller than the dark halo configurations and it does not show any signs of saturation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' it would prob- ably reach even smaller constraint violations if iterated further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The convergence in the final solution is further inves- tigated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 4, which shows its L2-norm of the Hamil- tonian constraint with respect to the number of colloca- tion points in the spectral elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The figure shows the constraint violation for the inner cube element and for the cubed sphere facing towards the companion star, which is also representative for all other cubed sphere el- ements inside the NSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The curves are almost straight lines on the log-log-plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 4, which is compatible with a polynomial convergence of the constraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', |H|L2 ∼ n−p, with p the order of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' This is the expected convergence behaviour for non-smooth data, which we have due to the surface of the inner fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Using the highest and lowest resolution we can estimate the order of convergence in the inner cube element to be p ≈ log22/10(|H|L2,n=10/|H|L2,n=22) ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' To investigate the convergence of the actual solution variables we interpolate the data from different resolu- tions on a common set of points and compute norms of the estimated errors on these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We interpo- late the solution onto a 10 × 10 × 10-grid equidistant in each direction, with coordinate components given by ri ∈ {20m/9, m ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='.9]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' This grid includes some points 1e-13 1e-12 1e-11 1e-10 1e-9 1e-8 1e-7 1e-6 1e-5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='15 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Hamiltonian constraint in a dark halo configuration in the z = 0 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In the lower half the specific enthalpy of the two fluids is overlaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' with pure vacuum, points with only one fluid present and points with both fluids present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The error in the solution is estimated by taking the difference to the so- lution with the highest resolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', the solution that has 22 points in each dimension of the spectral elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 5 we show the convergence of the 1-norm and the maximum norm over the set of interpolated points for the gxx component of the metric and the lapse α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Both quantities do not show a monotonic decay of the er- ror, but there is an overall trend of decaying error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' This somewhat broken convergence behaviour can again be attributed to the presence of non-smooth fields on the surface of the inner fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 6 shows the convergence of the error in the specific enthalpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The DM in this configuration is fitted to the element boundaries and its specific enthalpy displays a relatively clear convergence behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The BM fluid on the other hand shows a very broken convergence and only very little improve- ment from the lowest to the highest number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The maximum norm of the error is actually growing for the two largest number of points, whereas the 1-norm of the error is also slightly broken, but with an overall behaviour similar to that of gxx and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' It should be noted, that it is not clear whether the formalism of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' II actually possesses a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The partial differential equation (12) is not strictly ellip- tic on the fluid surface and hence the standard theorems for the uniqueness of the solution can not be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In- stead our algorithm might find a solution of many possi- ble, which is another possible explanation for the slightly broken convergence behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Difference in the Fluid Velocities It is worth pointing out that even if the BM and DM fluid components are both irrotational, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', non- spinning, the exact velocity profiles are not the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The reason for this does not lie in the notion of an irro- tational fluid, but is caused by differences in the fluids’ equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' An irrotational fluid is defined by 7 0 5 10 15 20 25 30 35 40 iteration 10 5 10 4 10 3 10 2 |H| L 2 /V element n = 22, single fluid n = 12, dark halo n = 14, dark halo n = 18, dark halo n = 22, dark halo FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' L2-norm over the inner cube in one of the stars, normalised by the volume of the inner cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The different lines show configurations with different number of points n in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 10 12 14 16 18 20 22 numer of points in each dimension n 10 4 |H| L 2 /V element inner cube left cubed sphere FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Normalised L2-norm of the Hamiltonian constraint in a dark halo configuration for a different number of points per dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The norm is normalised by the volume of the spectral element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Note that the x-axis and y-axis are scaled logarithmically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 10 12 14 16 18 20 numer of points in each dimension n 10 4 10 3 10 2 10 1 10 0 10 1 error norm of variable Y, ||Y n − Y 22 || g xx , 1-norm g xx , maximum norm α, 1-norm α, maximum norm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Self-convergence of metric variables in dark halo configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Black: error norm of the gxx component of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Blue: error norm of the lapse, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We not that the 1-norm is not normalised by the number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 10 12 14 16 18 20 numer of points in each dimension n 10 4 10 3 10 2 10 1 10 0 ||h (s) n − h (s) 22 || h BM , 1-norm h BM , maximum norm h DM , 1-norm h DM , maximum norm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Self-convergence of the specific enthalpy in dark halo configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Black: error norm of the baryonic specific enthalpy h(BM), which is the inner fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Blue: error norm of the specific enthalpy of DM, h(DM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We not that the 1-norm is not normalised by the number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' the vanishing of its kinematic vorticity tensor [79] ωαβ := P µ α P ν β ∇[µuν] = 0 , (21) with P µ α = δµ α + uµuα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' This notion does not depend on the thermodynamic properties of the fluid and hence differences in the velocities can only be the result of the of the equations of motion, that are used in the derivation of the formlalism in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' II, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' the Euler equations [72, 80] u(s)µ∇µ(h(s)u(s) ν + ∇νh(s)) = 0 , (22) which follow from ∇µT (s) µν = 0, and the continuity equa- tion ∇µ(ρ(s) 0 u(s)µ) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (23) If for example the DM would have the same four-velocity as the BM, it would still be irrotational, but might be incompatible with the laws of energy-momentum or par- ticle number conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In nature the disparity in the fluid velocities is affected by two counter-acting effects, particle scattering between BM and DM on the one hand and physics determining spin-down on the other hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In our formulation the two fluids are modelled as non-interacting, but the BM- DM scattering cross-section might be non-zero in nature, which would drive the two fluids towards a common ve- locity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' This process is counter-acted by effects driving the fluid into an irrotational state, as for example magnetic braking for BM [81–83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' It is unclear whether a similar effect exists for DM and whether it is dominant over the effect of BM-DM scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' By assuming vanishing of the kinematic vorticity for the DM component, we as- sume that such an effect exists and it is also dominating over the scattering with BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We find that both fluids move with basically the same velocity, with coinciding velocities in the star center, 8 12 14 16 18 20 x −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='10 relative difference, V (s)x 1 − V (DM)x /V (BM)x , dark halo 1 − V (DM)x /V (BM)x , dark core V (BM)x , dark halo V (DM)x , dark core FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Relative difference in the velocities for configurations with a separation of 32 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The difference is shown along a diagonal with the parametrization ri(s) = s(1, 1, 0)+ri c, going through the center of the star located at ri c = (16M⊙, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' V (BM)x (black, dash-dotted line) and V (DM)x (grey, dotted line) show the x-component of the velocity of the respective inner fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' but increasing difference towards the surface of the in- ner fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We quantify this effect in terms of the residual three-velocity V (s)i, in which the orbital movement given by the Killing vector ξµ is split off, V (s)i = u(s)i/u(s)0 − ξi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (24) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 7 shows the x-component of V (s)i and the relative difference of the fluid velocities for the region in which both fluids are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We present results for configu- rations at a separation of 32 M⊙, a separation at which the DM halos in the dark halo configurations are already relatively close and deformed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We find that dif- ferences in the two fluids are smaller for larger separation, which is intuitively understandable, because for large sep- arations the system goes to the limit of isolated NSs in which the fluid velocities coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 7 is shown along a diagonal through the star parametrized in the following way: ri(s) = s(1, 1, 0)+ri c, where ri c is the center of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We choose to present the data along this diagonal because the differ- ence V (BM)i − V (DM)i has a quadrupolar structure with nodes going through ri c and being approximately parallel to the x and y axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Hence the difference is basically zero on the x and y-axis, but very prominent along the spec- ified diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The relative difference between the resid- ual velocities is below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='2% near the center of the star and reaches up to 10% on the surfaces of the inner fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The difference between the velocities of the dark halo and dark core configurations is relatively small, which can be seen from the fact the curves of the velocities of the inner fluids lie on top of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Binding Energy NSs with a DM component are more tightly bound, because the DM component adds gravitating mass, but provides no additional repulsion to balance the gravita- tional pressure [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The gravitational binding energy of the particles is the difference of the ADM mass [27, 84, 85] and the sum of the rest masses m(s) 0i of the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' If all fluid particles would fall in from infinity, the true ADM mass would equal the total rest mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' However, the configurations that we construct do not contain GWs and therefore they do not model the energy lost in gravita- tional radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The difference in our ADM mass esti- mate and the total rest mass is, therefore, a measure of the particle binding energy: Ebind,p = MADM − m(BM) 01 − m(DM) 01 − m(BM) 02 − m(DM) 02 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (25) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 8 shows the particle binding energy as a function of our estimate for the ADM angular momentum JADM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' It can be seen that dark core configurations are more tightly bound than single fluid configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The dark halo configurations seemingly coincide with the single fluid case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' This can be attributed to the relatively low DM fraction of only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='5% in these configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' All config- urations are more tightly bound for smaller JADM cor- responding to smaller stellar separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' This is due to the stronger orbital binding between the two stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Most of the binding energy is contained in the indi- vidual stars and the contribution of the orbital binding energy is universal in all configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The orbital bind- ing energy Ebind,orb is the energy necessary for the two NSs to escape to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' It can be computed using the gravitational mass m(s) i of the components, by Ebind,orb = MADM −m(BM) 1 −m(DM) 1 −m(BM) 2 −m(DM) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (26) The gravitational masses m(s) i are obtained by solving a TOV-like equation for isolated stars that have the same rest masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The gravitational mass m(s) i is smaller than the rest mass m(s) i0 , because it accounts for the binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Hence, Ebind,orb contains only contributions of the binding energy that are due to the mutual binding between the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 9 shows that the orbital binding energy is mostly independent of the DM configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The biggest effect is seen for the dark core configurations for which the magnitude of Ebind,orb is about 2% smaller than that of the other configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Deformation To quantify the deformation of the stars we compute the ratio of the diameters along the orbital radius and along the polar axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The diameter along the orbital ra- dius is taken as ∆x, largest difference in the x-coordinates of two points on the fluid surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The polar diameter 9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='50 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='00 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='75 J ADM /M 2 ⊙ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='300 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='295 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='290 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='285 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='280 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='275 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='270 particle binding energy E bind, p /M ⊙ single fluid dark halo dark core FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Particle binding energy Ebind,p as a function of the ADM angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='50 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='00 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='75 J ADM /M 2 ⊙ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='0300 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='0275 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='0250 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='0225 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='0200 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='0175 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='0150 orbital binding energy E bind, orb /M ⊙ single fluid dark halo dark core FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Orbital binding energy Ebind,orb as a function of the ADM angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' ∆z, is the largest difference in the z-coordinate of two points on the fluid surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The tidal force of the com- panion stretches the star in x-direction, whereas the poles are slightly flattened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' This measure of deformation is of course coordinate-dependent, but it still provides some physical insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 10 shows the deformation ∆x/∆z for each fluid surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' When the NSs are closer, the tidal forces on the companion are stronger and hence the de- formation is stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' It can be observed that NSs with a DM core are systematically less deformed than their one-fluid counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The strong deformation in the dark halo case can also be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 1, which shows a cut through the z = 0 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' For a separation of 32 M⊙ (47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='3 km) the de- formation is clearly visible by eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' At a separation of 28 M⊙ (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='3 km) the deformation becomes already so strong that the surfaces of the NSs touch and mass shed- ding occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The closeness to mass shedding can be quantified in terms of the mass-shedding parameter χ, which was first introduced in [86] and which we define as χ(s) = ∂xh(s)|eq ∂zh(s)|pole,avg , (27) 25 30 35 40 45 50 55 60 65 separation D [km] 20 25 30 35 40 45 separation D/M ⊙ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='050 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='075 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='125 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='150 deformation ratio ∆x/∆z BM, single fluid BM, dark halo BM, dark core DM, dark halo DM, dark core FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Deformation ∆x/∆z of the fluid surfaces as func- tion of the NS centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The deformation is computed as the ratio of the largest extents in x and z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Curves la- beled BM show the deformation of the surface of the baryonic fluid, whereas curves labeled DM show the deformation of the DM surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' where the label ”eq” denotes the point on the surface, which is facing towards the other companion and for which the x-coordinate is extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The label ”pole” denotes the surface points at which the z-coordinate is extremal and where in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (27) the label ”avg” indicates that we have averaged over the values at the ”north and south pole”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Note that for non-spinning stars the ”north” and ”south pole” values only differ slightly due to round- off error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In the mass shedding limit χ(s) will tend to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We evaluate the χ(s) for each fluid component indi- vidually on the respective fluid surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We show the resulting χ(s) as a function of the distance of the centres of the stars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The DM fluid in the dark halo scenario is easily deformable, which leads to a relatively small mass shedding parameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='9 already at a sep- aration of 44 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We find that a separation of 28 M⊙ leads to a configuration with touching star surfaces, from which we conclude that mass shedding occurs somewhere at a separation between 28 and 29 M⊙, which means the system will transition relatively slowly to the mass shed- ding regime over a time where the two NSs decrease their separation by 16 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' For the dark core configurations, on the other hand, the transition to mass shedding is rather sudden with χ reaching a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='9 at sepa- ration of approximately 23 M⊙ and the mass shedding occurring for the baryonic fluid at a separation of 16 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' CONCLUSION We have extended the SGRID code to construct constraint-solved, quasi-equlibrium configurations of bi- naries of NSs consisting of two non-interacting fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The second fluid represents DM that can comprise some part of the matter of NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In this study we have used the 10 25 30 35 40 45 50 55 60 65 separation D [km] 20 25 30 35 40 45 separation D/M ⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='0 mass shedding parameter χ BM, single fluid BM, dark halo BM, dark core DM, dark halo DM, dark core FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Mass shedding parameter χ as a function of the sep- aration of the NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Curves labeled BM show the deformation of the surface of the baryonic fluid, whereas curves labeled DM show the deformation of the DM surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' EoS of a degenerate, relativistic Fermi gas with different particle masses to model the DM fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' These quasi- equlibrium configurations can be used as initial data for NR inspiral simulations of DM admixed NS binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The BAM code can already evolve mirror DM [25] and could be easily extended to allow for general EoS for the DM fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Another possible application of the two fluid approach are superfluid NS cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' At sufficiently high density BM forms a state made of superfluid neutrons and supercon- ducting protons, which can be described in a two fluid approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' However, the two fluids still interact with each other due to the entrainment effect and the con- dition of beta-equilibrium [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Solutions of isolated NS with superfluid cores are constructed in [88, 89] taking into account the interaction of the fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' For a study of superfluid and superconducting cores in binary NS the formalism in this work could be extended using a similar model for the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In binary NS collisions the temperature will rise above the critical temperature for superfluidity and superconductivity, so that it becomes necessary to include even a third fluid representing the non-superfluid component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We have tested the convergence of the constructed con- figurations with respect to resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The Hamiltonian constraint converges polynomially with an order of ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The lack of exponential convergence can be attributed to the presence of the non-smooth transition of the density at the surface of the inner fluid, which is not fitted to the boundaries of the spectral elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Self-convergence tests for metric components and the specific enthalpies show that the solution improves with increasing reso- lution, but with a slightly broken convergence towards higher resolution, which we again attribute to the sur- face of the inner fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' For future improvements to the code it is a worthwhile consideration to implement a new grid layout that allows fitting to the surface of a second fluid We have shown that the two fluids do not have the ex- act same velocities, but that the difference in the resid- ual velocities reaches up to 10% on the surface of the inner fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The difference in the velocity profiles will be even stronger if one assumes independent rotational states for the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In this work we only inves- tigated only purely irrotational configurations, but our formalism, in principle, allows for to construct configu- rations with arbitrary spin for the individual stars and fluid components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' This is relevant in particular for the DM component, which might only have insufficient mech- anisms to lose angular momentum and hence could be in a state of rapid rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' The presence of DM affects the compactness and de- formability of NSs, which will change the merger dynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' We have shown that the presence of DM can delay the point of mass-shedding to a later stage of the inspi- ral, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', towards closer separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' This is in accordance with the findings in numerical evolutions of two-fluid bi- nary mergers [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' In the case of a DM halo, mass shed- ding could occur much earlier than for the baryonic com- ponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' However the matter contained in the DM halo is rather low and hence the impact of DM mass shedding on the dynamics of the BM is potentially small, never- theless, dynamical simulations are needed to verify this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by funding from the FCT – Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', within the Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' EXPL/FIS-AST/0735/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' also acknowledge the support from the project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' UIDB/04564/2020, and UIDP/04564/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' acknowledges funding from the National Science Foun- dation under grant PHY-2136036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Narain, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Schaffner-Bielich, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Mishustin, Compact stars made of fermionic dark matter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 74, 063003 (2006), arXiv:astro-ph/0605724 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ciarcelluti and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Sandin, Have neutron stars a dark matter core?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' B 695, 19 (2011), arXiv:1005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='0857 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [3] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Zurek, Asymmetric Dark Matter: Theories, Sig- natures, and Constraints, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 537, 91 (2014), arXiv:1308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='0338 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [4] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Goldman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Mohapatra, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Nussinov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rosen- baum, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Teplitz, Possible Implications of Asym- metric Fermionic Dark Matter for Neutron Stars, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 11 Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' B 725, 200 (2013), arXiv:1305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='6908 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Gresham, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lou, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Zurek, Nuclear Structure of Bound States of Asymmetric Dark Matter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 96, 096012 (2017), arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='02313 [hep- ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ellis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' H¨utsi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Kannike, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Marzola, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Raidal, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Vaskonen, Dark matter effects on neutron star prop- erties, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 97, 123007 (2018), arXiv:1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='01418 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Nelson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Reddy, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Zhou, Dark halos around neutron stars and gravitational waves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Cosm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Astrop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 2019, 012 (2019), arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='03266 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [8] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ivanytskyi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Sagun, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lopes, Neutron stars: New constraints on asymmetric dark matter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 102, 063028 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Das, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Kumar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Biswal, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Nakat- sukasa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Li, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Patra, Effects of dark matter on the nuclear and neutron star matter, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' As- tron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 495, 4893 (2020), arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='00594 [nucl-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Das, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Malik, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Nayak, Dark matter ad- mixed neutron star properties in the light of gravitational wave observations: a two fluid approach, arxiv (2020), arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='01318 [nucl-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [11] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Sagun, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Giangrandi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ivanytskyi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lopes, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Bugaev, Constraints on the fermionic dark matter from observations of neutron stars, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' PANIC2021, 313 (2022), arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='13289 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [12] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Leung, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='-c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Chu, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lin, Tidal deforma- bility of dark matter admixed neutron stars, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 105, 123010 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [13] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Karkevandi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Shakeri, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Sagun, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ivanyt- skyi, Bosonic dark matter in neutron stars and its ef- fect on gravitational wave signal, Physical Review D 105, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='1103/physrevd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='023001 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [14] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Sagun, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Giangrandi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ivanytskyi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Providˆencia, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Dietrich, How does dark matter affect com- pact star properties and high density constraints of strongly interacting matter, in 15th Conference on Quark Confinement and the Hadron Spectrum (2022) arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='10510 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [15] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Di Giovanni, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Sanchis-Gual, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Cerd´a-Dur´an, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Font, Can fermion-boson stars reconcile multimes- senger observations of compact stars?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 105, 063005 (2022), arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='11997 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Maselli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Pnigouras, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Nielsen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Kouvaris, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Kokkotas, Dark stars: gravitational and electro- magnetic observables, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 96, 023005 (2017), arXiv:1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='07286 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [17] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Giangrandi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Sagun, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ivanytskyi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Providˆencia, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Dietrich, The effects of self-interacting bosonic dark matter on neutron star properties (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='10905 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [18] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rutherford, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Raaijmakers, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Prescod-Weinstein, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Watts, Constraining bosonic asymmetric dark matter with neutron star mass-radius measurements, arXiv (2022), arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='03282 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Bamber, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Aurrekoetxea, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Clough, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Fer- reira, Black hole merger simulations in wave dark matter environments (2022), arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='09254 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [20] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Cole, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Bertone, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Coogan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Gaggero, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Kary- das, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Kavanagh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Spieksma, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Tomaselli, Disks, spikes, and clouds: distinguishing en- vironmental effects on BBH gravitational waveforms (2022), arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='01362 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ellis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Hektor, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' H¨utsi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Kannike, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Marzola, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Raidal, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Vaskonen, Search for Dark Matter Ef- fects on Gravitational Signals from Neutron Star Merg- ers, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' B 781, 607 (2018), arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='05540 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Bezares, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Vigan`o, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Palenzuela, Gravitational wave signatures of dark matter cores in binary neutron star mergers by using numerical simulations, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 100, 044049 (2019), arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='08551 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Bauswein, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Guo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lien, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lin, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='- R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Wu, Compact Dark Objects in Neutron Star Mergers (2020), arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='11908 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Hippert, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Dillingham, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Tan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Curtin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Noronha-Hostler, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Yunes, Dark Matter or Reg- ular Matter in Neutron Stars?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' How to tell the differ- ence from the coalescence of compact objects (2022), arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='08590 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Emma, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Schianchi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Pannarale, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Sagun, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Dietrich, Numerical Simulations of Dark Mat- ter Admixed Neutron Star Binaries, arXiv (2022), arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='10887 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [26] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Gourgoulhon, 3+1 Formalism in General Relativity (Springer, Berlin, 2012) gr-qc/0703035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [27] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Baumgarte and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Shapiro, Numerical Relativ- ity: Solving Einstein’s Equations on the Computer (Cam- bridge University Press, Cambridge, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [28] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Tichy, The initial value problem as it relates to nu- merical relativity, Reports on Progress in Physics 80, 026901 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [29] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Kouvaris, The Dark Side of Neutron Stars, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 2013, 856196 (2013), arXiv:1308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='3222 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [30] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Bell, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Busoni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Robles, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Virgato, Im- proved Treatment of Dark Matter Capture in Neutron Stars, JCAP 09, 028, arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='14888 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [31] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Bell, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Busoni, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Motta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Robles, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Thomas, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Virgato, Nucleon Structure and Strong Interactions in Dark Matter Capture in Neutron Stars, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 127, 111803 (2021), arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='08918 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [32] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Kouvaris and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Tinyakov, Can neutron stars con- strain dark matter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 82, 063531 (2010), arXiv:1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='0586 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='GA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Del Popolo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Le Delliou, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Deliyergiyev, Neu- tron Stars and Dark Matter, Universe 6, 222 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [34] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Bertone and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Fairbairn, Compact Stars as Dark Matter Probes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 77, 043515 (2008), arXiv:0709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='1485 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Baryakhtar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', Dark Matter In Extreme As- trophysical Environments, in 2022 Snowmass Summer Study (2022) arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='07984 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [36] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Kouvaris, Wimp annihilation and cooling of neutron stars, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 77, 023006 (2008), arXiv:0708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='2362 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [37] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' de Lavallaz and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Fairbairn, Neutron stars as dark matter probes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 81, 123521 (2010), arXiv:1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='0629 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='GA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [38] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Hamaguchi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Nagata, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Yanagi, Dark Matter Heating vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rotochemical Heating in Old Neutron Stars, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' B 795, 484 (2019), arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='02991 [hep- ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [39] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' P´erez-Garc´ıa and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Silk, Dark matter seeding and 12 the kinematics and rotation of neutron stars, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' B 711, 6 (2012), arXiv:1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='2275 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [40] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Sedrakian, Axion cooling of neutron stars, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 93, 065044 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [41] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Sedrakian, Axion cooling of neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Be- yond hadronic axions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 99, 043011 (2019), arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='00190 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [42] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Dietrich and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Clough, Cooling binary neutron star remnants via nucleon-nucleon-axion bremsstrahlung, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 100, 083005 (2019), arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='01278 [gr- qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [43] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Buschmann, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Dessert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Foster, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Long, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Safdi, Upper Limit on the QCD Axion Mass from Isolated Neutron Star Cooling, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 128, 091102 (2022), arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='09892 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [44] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Read, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lackey, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Owen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Fried- man, Constraints on a phenomenologically parameterized neutron-star equation of state, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 79, 124032 (2009), arXiv:0812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='2163 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [45] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Douchin and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Haensel, A unified equation of state of dense matter and neutron star structure, Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' As- trophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 380, 151 (2001), arXiv:astro-ph/0111092.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [46] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Antoniadis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Freire, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Wex, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Tauris, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lynch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' van Kerkwijk, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Kramer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Bassa, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Dhillon, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Driebe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Hessels, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Kaspi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Kondratiev, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Langer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Marsh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' McLaugh- lin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Pennucci, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ransom, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Stairs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' van Leeuwen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Verbiest, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Whelan, A Massive Pulsar in a Compact Relativistic Binary, Science 340, 448 (2013), arXiv:1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='6875 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [47] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Fonseca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', Refined Mass and Geometric Measure- ments of the High-mass PSR J0740+6620, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 915, L12 (2021), arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='00880 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [48] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Miller, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lamb, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Dittmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Bog- danov, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Arzoumanian, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Gendreau, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Guillot, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Harding, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lattimer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lud- lam, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Mahmoodifar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Morsink, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ray, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Strohmayer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Wood, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Enoto, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Foster, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Oka- jima, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Prigozhin, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Soong, Psr j0030+0451 mass and radius from nicer data and implications for the prop- erties of neutron star matter, The Astrophysical Journal 887, L24 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [49] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Riley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', A NICER View of PSR J0030+0451: Millisecond Pulsar Parameter Estimation, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 887, L21 (2019), arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='05702 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [50] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Raaijmakers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', Constraining the dense matter equation of state with joint analysis of NICER and LIGO/Virgo measurements, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 893, L21 (2020), arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='11031 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [51] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', The Radius of PSR J0740+6620 from NICER and XMM-Newton Data, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 918, L28 (2021), arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='06979 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [52] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Riley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', A NICER View of the Massive Pul- sar PSR J0740+6620 Informed by Radio Timing and XMM-Newton Spectroscopy, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 918, L27 (2021), arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='06980 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [53] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (LIGO Scientific, Virgo), Properties of the binary neutron star merger GW170817, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' X 9, 011001 (2019), arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='11579 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [54] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' (LIGO Scientific, Virgo), GW190425: Observation of a Compact Binary Coalescence with To- tal Mass ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='4M⊙, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 892, L3 (2020), arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='01761 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [55] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Clowe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Bradaˇc, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Gonzalez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Markevitch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Randall, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Jones, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Zaritsky, A direct em- pirical proof of the existence of dark matter, The Astro- physical Journal 648, L109 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [56] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Randall, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Markevitch, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Clowe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Gonza- lez, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Bradac, Constraints on the Self-Interaction Cross-Section of Dark Matter from Numerical Simula- tions of the Merging Galaxy Cluster 1E 0657-56, Astro- phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 679, 1173 (2008), arXiv:0704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='0261 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [57] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Billard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', Direct detection of dark mat- ter—APPEC committee report*, Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 85, 056201 (2022), arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='07634 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [58] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Sagun, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Giangrandi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ivanytskyi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lopes, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Bugaev, Constraints on the fermionic dark matter from observations of neutron stars, PoS PANIC2021, 313 (2022), arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='13289 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [59] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Tichy, Constructing quasi-equilibrium initial data for binary neutron stars with arbitrary spins, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D86, 064024 (2012), arXiv:1209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='5336 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [60] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Tichy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rashti, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Dietrich, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Dudi, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Br¨ugmann, Constructing binary neutron star ini- tial data with high spins, high compactnesses, and high mass ratios, Physical Review D 100, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='1103/phys- revd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='124046 (2019), arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='09690 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [61] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Gourgoulhon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Grandcl´ement, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Marck, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' No- vak, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Taniguchi, http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='lorene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='obspm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [62] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Gourgoulhon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Grandcl´ement, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Taniguchi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Marck, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Bonazzola, Quasiequilibrium sequences of synchronized and irrotational binary neutron stars in general relativity: Method and tests, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 63, 064029 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [63] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Tacik, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Foucart, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Pfeiffer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Haas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ossokine, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Kaplan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Muhlberger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Duez, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Kidder, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Scheel, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Szil´agyi, Binary Neutron Stars with Arbitrary Spins in Numerical Relativity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D92, 124012 (2015), arXiv:1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='06986 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [64] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Grandcl´ement, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Auliac, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Roussille, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Pa- penfort, https://kadath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='obspm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='fr/fuka/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [65] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Papenfort, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Tootle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Grandcl´ement, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Most, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rezzolla, New public code for initial data of unequal-mass, spinning compact-object binaries, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 104, 024057 (2021), arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='09911 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [66] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rashti, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Fabbri, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Br¨ugmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Chaurasia, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Dietrich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ujevic, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Tichy, New pseudospec- tral code for the construction of initial data, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 105, 104027 (2022), arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='14511 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [67] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ury¯u and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Tsokaros, New code for equilibriums and quasiequilibrium initial data of compact objects, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D85, 064014 (2012), arXiv:1108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='3065 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [68] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ury¯u, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Yoshida, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Gourgoulhon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Markakis, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Fujisawa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Tsokaros, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Taniguchi, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Eriguchi, New code for equilibriums and quasiequilibrium initial data of compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' rotating relativistic stars with mixed poloidal and toroidal magnetic fields, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 100, 123019 (2019), arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='10393 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [69] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Wittman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Golovich, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Dawson, The Mis- measure of Mergers: Revised Limits on Self-interacting Dark Matter in Merging Galaxy Clusters, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 869, 104 (2018), arXiv:1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='05877 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [70] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Cook, Initial data for numerical relativity, Liv- ing Reviews in Relativity 3, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='12942/lrr-2000-5 (2000), arXiv:gr-qc/0007085 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [71] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Tichy, The initial value problem as it relates to nu- merical relativity, Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 80, 026901 (2017), arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='03805 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 13 [72] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Tichy, Initial data for binary neutron stars with arbitrary spins, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 84, 024041 (2011), arXiv:1107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='1440 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [73] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Sch¨afer and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Wex, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' A 174, 196 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [74] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Tichy, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Br¨ugmann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Campanelli, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Diener, Binary black hole initial data for numerical general rela- tivity based on post-Newtonian data, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 67, 064008 (2003), gr-qc/0207011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [75] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Tichy and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Br¨ugmann, Quasi-equilibrium binary black hole sequences for puncture data derived from he- lical killing vector conditions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 69, 024006 (2004), gr-qc/0307027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [76] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Swesty, Thermodynamically consistent interpolation for equation of state tables, Journal of Computational Physics 127, 118 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [77] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Cumming, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Brown, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Fattoyev, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Horowitz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Page, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Reddy, Lower limit on the heat capacity of the neutron star core, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' C 95, 025806 (2017), arXiv:1608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='07532 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [78] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Page, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Geppert, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Weber, The Cooling of com- pact stars, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' A 777, 497 (2006), arXiv:astro- ph/0508056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [79] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Hawking and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ellis, The Large Scale Struc- ture of Space-Time (Cambridge University Press, Cam- bridge (UK), 1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [80] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Shibata, A relativistic formalism for computation of irrotational binary stars in quasi equilibrium states, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 58, 024012 (1998), arXiv:gr-qc/9803085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [81] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Ostriker and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Gunn, On the nature of pulsars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=', 157: 1395-1417(Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='1086/150160 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [82] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Colpi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Possenti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Popov, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Pizzolato, Spin and magnetism in old neutron stars, in Physics of Neu- tron Star Interiors, edited by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Blaschke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Sedrakian, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Glendenning (Springer Berlin Heidelberg, Berlin, Heidelberg, 2001) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 440–466, astro-ph/0012394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [83] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rogers and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Safi-Harb, On the diversity of com- pact objects within supernova remnants – II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' energy-loss mechanisms, Monthly Notices of the Royal Astronomical Society 465, 383 (2016), arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='04685 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [84] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Arnowitt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Deser, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Misner, The dynamics of general relativity, in Gravitation: An Introduction to Current Research, edited by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Witten (Wiley, New York, 1962) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' 227–265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [85] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Gourgoulhon, 3+1 formalism and bases of numerical relativity (2007), arXiv:gr-qc/0703035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [86] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Gourgoulhon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Grandcl´ement, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Taniguchi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Marck, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Bonazzola, Quasiequilibrium sequences of synchronized and irrotational binary neutron stars in general relativity: Method and tests, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 63, 064029 (2001), gr-qc/0007028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [87] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Comer and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Joynt, Relativistic mean field model for entrainment in general relativistic superfluid neutron stars, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 68, 023002 (2003), gr-qc/0212083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [88] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Prix, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Novak, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Comer, Relativistic numeri- cal models for stationary superfluid neutron stars, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' D 71, 043005 (2005), gr-qc/0410023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' [89] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content=' Chamel, Two-fluid models of superfluid neutron star cores, Monthly Notices of the Royal Astronomical Society 388, 737 (2008), arXiv:0805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} +page_content='1007 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf'} diff --git a/F9E0T4oBgHgl3EQfRAC9/vector_store/index.pkl b/F9E0T4oBgHgl3EQfRAC9/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..f3afd8d6a2a2d275620831e51f72f8021b6116dc --- /dev/null +++ b/F9E0T4oBgHgl3EQfRAC9/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 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mode 100644 index 0000000000000000000000000000000000000000..d39c72ff7120a29ab95ecce692506021ea5cb6ce --- /dev/null +++ b/INAzT4oBgHgl3EQfxv6_/content/tmp_files/2301.01744v1.pdf.txt @@ -0,0 +1,4497 @@ +Dynamic Maintenance of Monotone Dynamic +Programs and Applications +Monika Henzinger � � +Faculty of Computer Science, University of Vienna +Stefan Neumann � +KTH Royal Institute of Technology, Stockholm, Sweden +Harald Räcke � � +TU Munich, Munich, Germany +Stefan Schmid � � +TU Berlin, Germany and Fraunhofer SIT, Germany +Abstract +Dynamic programming (DP) is one of the fundamental paradigms in algorithm design. However, +many DP algorithms have to fill in large DP tables, represented by two-dimensional arrays, which +causes at least quadratic running times and space usages. This has led to the development of +improved algorithms for special cases when the DPs satisfy additional properties like, e.g., the Monge +property or total monotonicity. +In this paper, we consider a new condition which assumes (among some other technical as- +sumptions) that the rows of the DP table are monotone. Under this assumption, we introduce +a novel data structure for computing (1 + ϵ)-approximate DP solutions in near-linear time and +space in the static setting, and with polylogarithmic update times when the DP entries change +dynamically. To the best of our knowledge, our new condition is incomparable to previous conditions +and is the first which allows to derive dynamic algorithms based on existing DPs. Instead of using +two-dimensional arrays to store the DP tables, we store the rows of the DP tables using monotone +piecewise constant functions. This allows us to store length-n DP table rows with entries in [0, W] +using only polylog(n, W) bits, and to perform operations, such as (min, +)-convolution or rounding, +on these functions in polylogarithmic time. +We further present several applications of our data structure. For bicriteria versions of k-balanced +graph partitioning and simultaneous source location, we obtain the first dynamic algorithms with +subpolynomial update times, as well as the first static algorithms using only near-linear time and +space. Additionally, we obtain the currently fastest algorithm for fully dynamic knapsack. For +k-balanced partitioning, we show how to monotonize an existing non-monotone DP by Feldmann +and Foschini (Algorithmica’15); for simultaneous source location, we obtain an efficient algorithm +by considering the inverse DP function of the one used by Andreev, Garrod, Golovin, Maggs, and +Meyerson (TALG’09). Our result for fully dynamic knapsack improves upon a recent result by +Eberle, Megow, Nölke, Simon and Wiese (FSTTCS’21). +2012 ACM Subject Classification Theory of computation → Dynamic programming; Theory of +computation → Dynamic graph algorithms; Theory of computation → Packing and covering problems +Keywords and phrases Dynamic programming, dynamic algorithms, data structures +Funding Monika Henzinger: This project has received funding from the European Research Council +(ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant +agreement No. 101019564 “The Design of Modern Fully Dynamic Data Structures (MoDynStruct)” +and from the Austrian Science Fund (FWF) project “Fast Algorithms for a Reactive Network Layer +(ReactNet)”, P 33775-N, with additional funding from the netidee SCIENCE Stiftung, 2020–2024. +Stefan Neumann: This research is supported by the the ERC Advanced Grant REBOUND (834862) +and the EC H2020 RIA project SoBigData++ (871042). +Stefan Schmid: Research supported by Austrian Science Fund (FWF) project I 5025-N (DELTA), +2020-2024. +arXiv:2301.01744v1 [cs.DS] 4 Jan 2023 + +erc +EuropeanResearchCouncil +EstablishedbytheEuropeanCommission2 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +Contents +1 +Introduction +4 +2 +Maintaining Monotone Dynamic Programming Tables +8 +3 +Fully Dynamic Knapsack +11 +3.1 +Knapsack via Convolution of Monotone Functions +. . . . . . . . . . . . . . . +11 +3.2 +Dynamically Maintaining a Small Instance . . . . . . . . . . . . . . . . . . . . +14 +4 +Technical Overview +17 +4.1 +Monotonizing the DP of Feldmann and Foschini . . . . . . . . . . . . . . . . . +17 +4.1.1 +Computing the DP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +4.2 +Inverting the DP of Andreev et al. . . . . . . . . . . . . . . . . . . . . . . . . +22 +A Organization of the Appendix +26 +B Further Related Work +26 +C Preliminaries +28 +C.1 Räcke Tree +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +28 +C.2 Okay-Behaved DPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +D Balanced Graph Partitioning +30 +D.1 The Exact DP +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +31 +D.1.1 +DP Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +33 +D.1.2 +Computing the DP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +33 +D.2 The Approximate DP +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +D.3 Computing the Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +40 +D.4 Extension to General Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . +43 +D.5 Extension to the Dynamic Setting +. . . . . . . . . . . . . . . . . . . . . . . . +45 +E Simultaneous Source Location +48 +E.1 +The Exact DP +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +49 +E.1.1 +DP Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +49 +E.1.2 +Computing the DP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +E.2 +The Approximate DP +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +53 +E.3 +Extension to General Graphs (Proof of Theorem 4) . . . . . . . . . . . . . . . +54 +E.4 +Extension to the Dynamic Setting (Proof of Theorem 5) . . . . . . . . . . . . +54 +F Recourse Bounds +56 +F.1 +Proof of Theorem 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +57 +F.2 +Proof of Theorem 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +59 +G Non-Monotone Functions and ℓ∞-Necklace Alignment +60 +G.1 Piecewise Constant Functions With Non-Monotonicities . . . . . . . . . . . . +60 +G.1.1 +Proof of Lemma 40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +62 +G.2 ℓ∞-Necklace Alignment +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +64 +G.2.1 +The Static Algorithm +. . . . . . . . . . . . . . . . . . . . . . . . . . . +65 +G.2.2 +The Dynamic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . +68 + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +3 +H Omitted Proofs +69 +H.1 Proof of Lemma 6 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +69 +H.2 Proof of Lemma 7 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +70 +H.3 Proof of Theorem 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +71 +H.4 Proof of Theorem 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +72 +H.5 Property of the Räcke Tree +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +72 +H.6 Proof of Lemma 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +73 +H.7 Proof of Lemma 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +73 +H.8 Proof of Lemma 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +73 +H.9 Proof of Lemma 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +73 +H.10 Proof of Lemma 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +74 + +4 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +1 +Introduction +Dynamic programming (DP) is one of the fundamental paradigms in algorithm design. In the +DP paradigm, a complex problem is broken up into simpler subproblems and then the original +problem is solved by combining the solutions for the subproblems. One of the drawbacks +of DP algorithms is that in practice they are often slow and memory-intensive: for inputs +of size n their running time is typically Ω(n2), and when the DP table is stored using a +two-dimensional array they also need space Ω(n2). +This motivated researchers to develop more efficient DP algorithms with near-linear time +and space. Indeed, such improvements are possible under a wide range of conditions on the +DP tables [2,12,19,22,31,35,45,48,49,64], such as the Monge property, total monotonicity, +certain convexity and concavity properties, or the Knuth–Yao quadrangle-inequality; we +discuss these properties in more detail in Appendix B. When these properties hold, typically +one does not have to compute the entire DP table but instead only has to compute O(n) +DP entries which reveal the optimal solution. +However, we are not aware of any property for DPs that yields efficient dynamic algorithms, +i.e., algorithms that provide efficient update operations when the input changes. One might +find this somewhat surprising because, from a conceptual point of view, many dynamic +algorithms hierarchically partition the input and maintain solutions for subproblems; this +is quite similar to how many DP schemes are derived. Indeed, this conceptual similarity is +exploited by many “hand-crafted” algorithms (e.g., [26,38]) which start with a DP scheme and +then show how to maintain it dynamically under input changes. However, such algorithms +are often quite involved and their analysis often requires sophisticated charging schemes. +Hence, it is natural to ask whether there exists a general criterion which, if satisfied, +guarantees that a given DP can be updated efficiently under input changes. +Our Contributions. The main contribution of our paper is the introduction of a general +criterion which allows to approximate all entries of a DP table up to a factor of 1 + ϵ. We +show that if our criterion is satisfied by a DP (with suitable parameters) then: +In the dynamic setting, we can maintain a (1 + ϵ)-approximation of the entire DP table +using polylogarithmic update time (see Theorem 10). +In the static setting, we can compute a (1+ϵ)-approximation of the DP table in near-linear +time and space (see Theorem 9). +Our criterion essentially asserts that the rows of the DP tables should be monotone and +that the dependency graph of the DP should be a DAG, where the sets of reachable nodes +are small, among some other technical conditions (see Definition 8 for the formal definition). +Our criterion is incomparable to the Monge property, total monotonicity or other criteria +from the literature (see Appendix B for a more detailed discussion). +To obtain our results, we introduce a novel data structure for maintaining DPs which +satisfy our criterion. Our data structure is based on the idea of storing the DP rows using +monotone piecewise constant functions. The monotonicity of the DP rows will allow us to +ensure that our functions only contain very few pieces. Then we show that we can perform +operations on such functions very efficiently, with the running times only depending on the +number of pieces. This is crucial because it allows us to compute an entire (1+δ)-approximate +DP row in time just polylog(W), even when the DP has Ω(n) columns, assuming that the +DP entries are from [0, W]. Note that if W ≤ poly(n) then this decreases the running time +for computing an entire row from Ω(n) to just polylog(n). Additionally, this also allows us +to store each row using only polylog(W) space rather than storing it in an array of size Ω(n). +We present our criterion and the details of our data structure in Section 2. + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +5 +As applications of our data structure, we obtain new static and dynamic algorithms for +various problems. We present new algorithms for k-balanced partitioning, simultaneous source +location and for fully dynamic knapsack. Next, we describe these results in detail; we discuss +more related work in Appendix B. +Our Results for Fully Dynamic 0-1 Knapsack. First, we provide a novel algorithm +for fully dynamic 0-1 knapsack. In this problem, the input consists of a knapsack size B ∈ R+ +and a set of n items, where each item i ∈ [n] has a weight wi ∈ R+ and a price pi ∈ [1, ∞). +The goal is to find a set of items I that maximizes � +i∈I pi while satisfying the constraint +� +i∈I wi ≤ B. In the dynamic version of the problem, items are inserted and deleted. More +concretely, we consider the following update operations: insert(pi, wi), in which the price +and weight of item i are set to pi ∈ [1, ∞) and wi ∈ R+, respectively, and delete(i), where +item i is removed from the set of items. +Our main result is a dynamic (1 + ϵ)-approximation algorithm with worst-case update +time ϵ−2 · log2(nW) · polylog(1/ϵ, log(nW)), where W = � +i pi. Our algorithm improves +upon a recent result by Eberle, Megow, Nölke, Simon and Wiese [29] that also maintained a +(1 + ϵ)-approximate solution with update time O(ϵ−9 log4(nW)). +▶ Theorem 1. Let ϵ > 0. There exists an algorithm for fully dynamic knapsack that maintains +a (1 + ϵ)-approximate solution with worst-case update time +1 +ϵ2 log2(nW) polylog +� 1 +ϵ log(nW) +� +. +We will also show that we can return the maintained solution I in time O(|I|) and that +we can answer queries whether a given item i ∈ [n] is contained in I in time O(1). This +matches the query times of [29]. +To obtain this result, we first derive a slightly slower algorithm as a simple application of +our data structure for maintaining DPs with monotone rows. Then we use this algorithm +together with additional ideas to obtain the theorem (see Section 3). +Since our dynamic algorithm is based on a DP, it is possible that the solution changes +significantly after each update. However, in the appendix (Theorem 34) we prove a lower +bound, showing that every dynamic (1 + ϵ)-approximation algorithm for knapsack must +either make a lot of changes to the solution after each update or store many (potentially +substantially different) solutions between which it can switch after each update. This implies +that maintaining a single explicit solution with polylogarithmic update times is not possible +and the property of our algorithm cannot be avoided. +Our Results for k-Balanced Partitioning. Our most technically challenging result +is for k-balanced graph partitioning. In this problem, the input consists of an integer k and +an undirected weighted graph G = (V, E, cap) with n vertices, where cap : E → W∞ is a +weight function on the edges with weights in W∞ := [1, W] ∪ {0, ∞}. The goal is to find +a partition V1, . . . , Vk of the vertices such that |Vi| ≤ ⌈|V | /k⌉ for all i and the weight of +the edges which are cut by the partition is minimized. More formally, we want to minimize +cut(V1, . . . , Vk) := �k +i=1 +� +{u,v}∈E∩(Vi×(V \Vi)) cap(u, v). +We note that this problem is highly relevant in theory [5,32–34] and in practice [18,28, +44,55], where algorithms for balanced graph partitioning are often used as a preprocessing +step for large scale data analytics. Obtaining practical improvements for this problem is +of considerable interest in applied communities [18] and, for instance, the popular METIS +heuristic [44] has 1,400+ citations. +Since the above problem is NP-hard to approximate within a factor of n1−ϵ for any ϵ > 0 +even on trees [34], we consider bicriteria approximation algorithms. Given an undirected +weighted graph G = (V, E, cap), a partition V1, . . . , Vk of V is a bicriteria (α, β)-approximate +solution if |Vi| ≤ β⌈n/k⌉ for all i and cut(V1, . . . , Vk) ≤ α · cut(OPT), where OPT = + +6 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +(V ∗ +1 , . . . , V ∗ +k ) is the optimal solution with |V ∗ +i | ≤ ⌈n/k⌉ for all i. We note that the previously +mentioned hardness result implies that any algorithm that computes a bicriteria (α, 1 + ϵ)- +approximation for any α ≥ 1 and whose running time depends only polynomially on n, must +have a running time depending super-polynomially on 1/ϵ, unless P = NP.1 +Our main result for the static setting is presented in the following theorem. It gives the +first algorithm with polylogarithmic approximation ratio for this problem with near-linear +running time. More concretely, we compute a bicriteria (O(log4 n), 1 + ϵ)-approximation in +near-linear time for constant k. For comparison, the best approximation ratio achieved by +a polynomial-time algorithm [34] is a bicriteria (O(log1.5 n log log n), 1 + ϵ)-approximation +with running time Ω(n4). +▶ Theorem 2. Let ϵ > 0 and k ∈ N. Let G = (V, E, cap) be an undirected weighted graph +with n vertices and m edges and edge weights in W∞. Then for the k-balanced partition +problem we can compute: +An (O(log4 n), 1+ϵ)-approximation in time (k/ϵ)O(log(1/ϵ)/ϵ)·O′(m·log2(W))+(k/ϵ)O(1/ϵ2).2 +A (1 + ϵ, 1 + ϵ)-approximation in time (k/ϵ)O(log(1/ϵ)/ϵ) · O′(n · h2 · log2(W)) + (k/ϵ)O(1/ϵ2) +if G is a tree of height h. +A (1, 1 + ϵ)-approximation in time (k/ϵ)O(log(1/ϵ)/ϵ) · O′(n4 · log2(W)) + (k/ϵ)O(1/ϵ2) if G +is a tree. +Furthermore, we extend our results to the dynamic setting in which the graph G is under- +going edge insertions and deletions. In the following theorem, we present the first dynamic +algorithm with subpolynomial update time for this problem. We again consider bicriteria +approximation algorithms with update and query times depending super-polynomially on 1/ϵ; +this cannot be avoided since if we computed (α, 1)-approximations for any α ≥ 1 or if we +had a polynomial dependency on 1/ϵ, then the hardness result from above implies that our +update and query times must be super-polynomial in n (unless P = NP). +▶ Theorem 3. Let ϵ > 0 and k ∈ N. Let G = (V, E, cap) be an undirected weighted graph +with n vertices that is undergoing edge insertions and deletions. Then for the k-balanced +partition problem we can maintain: +An (no(1), 1 + ϵ)-approximate solution with amortized update time (k/ϵ)O(log(1/ϵ)/ϵ) · no(1) · +O′(log2(W)) and query time (k/ϵ)O(1/ϵ2) if G is unweighted. +A (1+ϵ, 1+ϵ)-approximate solution with worst-case update time (k/ϵ)O(log(1/ϵ)/ϵ) ·O′(h3 · +log2(W)) and query time (k/ϵ)O(1/ϵ2) if G is a tree of height h. +Our approach is inspired by the DP of Feldmann and Foschini [34]. However, the DP +rows in the algorithm of [34] are not monotone and, hence, their DP cannot directly be sped +up by our approach. Therefore, we first simplify and generalize the exact DP of Feldmann +and Foschini to make it monotone. The DP we obtain eventually is still slightly too complex +to fit into our black-box framework, but we show that the ideas from our framework can still +be used to obtain the result. In Section 4.1, we provide a technical overview. +Again, it is possible that the solution maintained by our algorithm changes substantially +after each update. Similar to above we show in the appendix (Theorem 35) that this cannot +be avoided when considering subpolynomial update times. +1 If we had an algorithm that computes a bicriteria (α, 1 + ϵ)-approximation in time poly(n, 1/ϵ) then we +could set ϵ = 1/(2n) which implies that all partitions have size ⌈n/k⌉. Thus we can compute a bicriteria +(α, 1)-approximate solution in time poly(n) which contradicts the hardness result, unless P = NP. +2 We use the notation O′(·) to suppress factors in poly(log n, k, log(1/ϵ), log log(W)). + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +7 +Our Results for Simultaneous Source Location. Next, we provide efficient algo- +rithms for the simultaneous source location problem by Andreev, Garrod, Golovin, Maggs and +Meyerson [4]. In this problem, the input consists of an undirected graph G = (V, E, cap, d) +with a capacity function cap: E → W∞ on the edges and a demand function d: V → W∞ on +the vertices. The goal is to select a minimum set S ⊆ V of sources that can simultaneously +supply all vertex demands. More concretely, a set of sources S is feasible if there exists a +flow from the vertices in S that supplies demand d(v) to all vertices v ∈ V and that does +not violate the capacity constraints on the edges. The objective is to find a feasible set of +sources of minimum size. +We will again consider bicriteria approximation algorithms. Let S∗ be the optimal +solution for the simultaneous source location problem. Then we say that S is a bicriteria +(α, β)-approximate solution if |S| ≤ α |S∗| and if S is a feasible set of sources when all edge +capacities are increased by a factor β. +The following theorem summarizes our main results. It presents the first near-linear time +algorithm for simultaneous source location that computes a (1+ϵ)-approximate solution while +only exceeding the edge capacities by a O(log4 n) factor. In comparison, the best algorithm +with arbitrary polynomial processing time computes a bicriteria (1, O(log2 n log log n))- +approximate solution in time Ω(n3) [4]. +▶ Theorem 4. Let ϵ > 0. Let G = (V, E, cap, d) be an undirected weighted graph with +n vertices and m edges. Then for the simultaneous source location problem we can compute: +A (1 + ϵ, O(log4(n)))-approximation in time3 ˜O( 1 +ϵ2 m). +A (1 + ϵ, 1)-approximation in time ˜O( 1 +ϵ2 h2 · n) if G is a tree of height h. +Next, we turn to dynamic versions of the problem. We consider the following update oper- +ations: SetDemand(v, d): updates the demand of vertex v to d(v) = d, SetCapacity((u, v), c): +updates the capacity of the edge (u, v) to cap(u, v) = c, Remove(u, v): removes the edge +(u, v), Insert((u, v), c): inserts the edge (u, v) with capacity cap(u, v) = c. +We obtain the first dynamic algorithms with subpolynomial update times for this problem, +which exceed the edge capacities only by a small subpolynomial factor. +▶ Theorem 5. Let ϵ > 0. Let G = (V, E, cap, d) be a graph with n vertices and m edges that +is undergoing the update operations given above. Then for the simultaneous source location +problem we can maintain: +A (1 + ϵ, no(1))-approximation with amortized update time no(1)/ϵ2 and preprocessing time +O(n2/ϵ2) if all edge capacities are 1. +A (1+ϵ, O(log4(n)))-approximation with worst-case update time ˜O(1/ϵ2) and preprocessing +time ˜O(m) if we only allow the update operation SetDemand(v, d). +A (1 + ϵ, O(log2(n) log log(n)))-approximation with worst-case update time ˜O(1/ϵ2) and +preprocessing time poly(n) if we only allow the update operation SetDemand(v, d). +A (1 + ϵ, 1)-approximate solution with worst-case update time ˜O(h3/ϵ2) and preprocessing +time O(n2/ϵ2) if G is a tree of height h. +To obtain these results, we use a similar DP approach as the one used by Andreev et +al. [4]. Interestingly, the DP function that we use essentially computes the inverse function +of the one used by Andreev et al. We sketch the details of this approach in Section 4.2. +After making these changes, the theorems become straightforward applications of our data +structure for maintaining DPs with monotone rows. +3 We write ˜O(f(n, ϵ, W)) to denote running times of the form f(n, ϵ, W) · polylog(n, ϵ, log W). + +8 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +Organization of Our Paper. In Section 2 we provide the details of our condition +for DPs with monotone rows. In Section 3 we present our results for 0-1 Knapsack which +nicely illustrate the applicability of our black-box framework from Section 2. We provide +a technical overview of our more involved results for k-Balanced Graph Partitioning and +for Simultaneous Source Location in Section 4. We give an overview of the appendix in +Appendix A. In the appendix we also present more related work and the full proofs of our +results. We present omitted proofs from the main text in Appendix H. +Open Problems and Future Work. In the future, it will be interesting to use our +framework to obtain more dynamic algorithms based on existing DPs. We believe that this +is interesting both in theory and in practice. Furthermore, it is intriguing to ask whether +our criterion from Definition 8 can be generalized. Indeed, our approach was built around +approximating monotone functions using piecewise constant functions, which can be viewed +as piecewiese degree-0 polynomials. An interesting question is whether we can obtain a more +general criterion if we approximate DP rows using pieces of higher-degree polynomials, such +as splines. Results in this direction might be possible; for example, in Appendix G we give a +side result for the case when the functions contain a small number of non-monotonicities and +derive a dynamic algorithm for the ℓ∞-necklace problem. +2 +Maintaining Monotone Dynamic Programming Tables +In this section, we introduce our notion of DP tables with monotone rows and the additional +technical assumptions that we are making. Then we present our data structure for efficiently +maintaining DP tables that satisfy our assumptions. In our data structure, we will store the +rows of the DP using piecewise constant functions, which we will introduce first. +List Representation of Piecewise Constant Functions. Let t ∈ R, W ∈ [1, ∞) and +set W∞ := {0}∪[1, W]∪{+∞}. A function f : [0, t] → W∞ is piecewise constant with p pieces +if there exist real numbers 0 = x0 < x1 < x2 < · · · < xp = t and numbers y1, . . . , yp ∈ W∞ +such that on each interval [xi−1, xi), f is constant and has value yi. More formally, for all +i ∈ {1, . . . , p} we have f(x) = yi for all real numbers x ∈ [xi−1, xi) and f(xp) = yp. Note +that we need the condition f(xp) = yp such that f is defined on the whole domain. +In the list representation of a piecewise constant function f, we use a doubly linked list +to store the pairs (x1, y1), . . . , (xp, yp). We also store the pairs (xi, yi) in a binary search tree +that is sorted by the xi-values, which allows us to compute a function value f(x) in time +O(log p) for all x ∈ [0, t]. In the following, we assume that all piecewise constant functions +we consider are stored in the list representation with an additional binary search tree. +One of the main observations we use is that many operations on piecewise constant +functions are efficient if there are only few pieces. The following lemma shows that several +operations can be computed in time almost linear in the number of pieces of the function, +rather than in time depending on the size of the domain of f.4 For δ > 0 and y ∈ W∞, we +write ⌈y⌉1+δ to denote the smallest power of 1+δ that is at least y, i.e., ⌈y⌉1+δ = min{(1+δ)i : +(1 + δ)i ≥ y, i ∈ N}; we follow the convention that ⌈0⌉1+δ = 0 and ⌈∞⌉1+δ = ∞. +▶ Lemma 6. Let t ∈ R and c ∈ R+. Let g, h : [0, t] → W∞ be monotone and piecewise +constant functions with pg and ph pieces, resp. Then we can compute the following functions: +fmin(x) := min{g(x), h(x)} with at most pg + ph pieces in time O((pg + ph) log(pg + ph)); +4 We note that computing the operations themselves can be done in linear time. However, since we also +store the pairs (xi, yi) of the list representations in a binary search tree, the running times in the lemma +include an additional logarithmic factor. + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +9 +fshift(x) := g(x − c) for x ≥ c, fshift(x) = g(0) for x < c with at most pg pieces in time +O(pg log(pg)); +fadd(x) := g(x) + h(x), with at most pg + ph pieces in time O((pg + ph) log(pg + ph)); +fround(x) := ⌈g(x)⌉1+δ for δ > 0 with at most 2+⌈log1+δ(W)⌉ pieces in time O(pg log(pg)). +Note that if we set ˜f = ⌈f⌉1+δ then ˜f is a (1 + δ)-approximation of f in the following sense. +For α > 1, we say that a function ˜f : [0, t] → W∞ α-approximates a function f : [0, t] → W∞ +if for all x ∈ [0, t], +f(x) ≤ ˜f(x) ≤ α · f(x). +(1) +Furthermore, if f is monotone then the rounded function ˜f contains at most O(log1+δ(W)) +pieces. This will be crucial later because this ensures that, if we perform a single rounding +operation for each row of our DP table, the resulting function will have few pieces and +operations on the function can be performed efficiently. +Next, consider functions f1, f2 : [0, t] → W∞. A function f : [0, t] → W∞ is the (min, +)- +convolution f1 ⊕ f2 if for all x ∈ [0, t], f(x) = (f1 ⊕ f2)(x) := min¯x∈[0,x] f1(¯x) + f2(x − ¯x). +Such convolutions are highly useful for the computation of many DPs. The following lemma +shows that we can efficiently compute the convolution of piecewise constant functions. +▶ Lemma 7. Let f1, f2 : [0, t] → W∞ be piecewise constant functions with at most p pieces +and assume that one of them is monotonically decreasing. Then we can compute the function +f : [0, t] → W∞ with f = f1 ⊕ f2 in time O(p2 log p) and f is a piecewise constant function +with O(p2) pieces. Furthermore, after computing f, for any x ∈ [0, t] we can return a value +¯x∗ ∈ [0, t] such that f(x) = f1(¯x∗) + f2(x − ¯x∗) in time O(log p). +Now observe that Lemma 7 has a drawback for our approach: The number of pieces (i.e., +the complexity of the functions) grows quadratically with every application. An important +property which can be used to mitigate this issue is that the result of the convolution is still +a monotone function, as we show in Lemma 22 in the appendix. Later, to keep the number +of pieces in our functions small, after each convolution that we perform via Lemma 7 (and +that might grow the number of pieces quadratically), we perform a rounding operation ⌈·⌉1+δ +(see Lemma 6). This loses a factor 1 + δ in approximation but guarantees that the resulting +function has O(log1+δ(W)) pieces. This will be crucial to ensure that our functions have +only few pieces. +Maintaining DPs With Monotone Rows. +Next, we introduce our DP scheme +formally. We consider DP tables with a finite set of rows I and a set of columns J , with +entries taking values in W∞. We will consider DP tables as functions DP: I × J → W∞.5 +Further, we will associate the i’th row of the DP with a function DP(i, ·): J → W∞, and we +store each such function DP(i, ·) using piecewise constant functions from above. +Next, we introduce the dependency graph for the rows of our DP. More concretely, the +dependency graph D = (I, ED) is a directed graph that has the rows I as vertices and a +directed edge (i′, i) between two rows if for some columns j, j′ ∈ J the entry DP(i′, j′) is +required to compute DP(i, j). We write In(i) = {i′ ∈ I : (i′, i) ∈ ED} to denote the set of +rows i′ that are required to compute row i. For the rest of the paper we will assume that the +dependency graph is a DAG, which is the case for all applications that we study. We will +also write Reach(i) to denote the set of vertices that are reachable from row i in D. +5 Even though our definition may suggest that we only consider two-dimensional DP tables, we do not +require an order on I and we allow I to be any finite set. For example, in Section D we will set I to +3-tuples corresponding to the parameters of a four-dimensional DP. + +10 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +Since we assume that the dependency graph is a DAG, we can compute the i’th DP row +as soon as we have computed the solutions for the DP rows in In(i). We assume that this +is done via a procedure Pi that takes as input the DP rows DP(i′, ·) for all i′ ∈ In(i) and +returns the row DP(i, ·) = Pi({DP(i′, ·): i′ ∈ In(i)}). +Next, we come to our condition which encodes when our scheme applies. In the definition +and for the rest of the paper, we write ADP to refer to an approximate DP table, which +approximates the exact DP table DP. Let β > 1. We say that ADP β-approximates DP if +DP(i, j) ≤ ADP(i, j) ≤ βDP(i, j) for all i ∈ I, j ∈ J . +▶ Definition 8. A DP table is (h, α, p)-well-behaved if it satisfies the following conditions: +1. (Monotonicity:) For all i ∈ I, the function DP(i, ·) is monotone. +2. (Dependency graph:) The dependency graph is a DAG and |Reach(i)| ≤ h for all i ∈ I. +3. (Sensitivity:) Suppose β > 1 and for all i′ ∈ In(i), we obtain a β-approximation ADP(i′, ·) +of DP(i′, ·). Then applying Pi on the ADP(i′, ·) yields a β-approximation of DP(i, ·), i.e., +DP(i, ·) ≤ Pi({ADP(i′, ·): i′ ∈ In(i)}) ≤ β · DP(i, ·). +4. (Pieces:) For each procedure Pi there exists an approximate procedure ˜Pi such that: +(a) ˜Pi({ADP(i′, ·): i′ ∈ In(i)}) is an α-approximation of Pi({ADP(i′, ·): i′ ∈ In(i)}), +(b) ˜Pi can be computed as the composition of a constant number of operations from +Lemma 6 and and at most one application of Lemma 7, and +(c) ˜Pi returns a monotone piecewise constant function with at most p pieces. +The definition is motivated in the following way: our operations on the piecewise constant +functions have efficient running times when the functions are monotone and have few pieces. +This is ensured by Properties (1), 4(b), and 4(c). Next, rounding errors cannot compound +too much if each row can only reach h other rows and the sensitivity condition is satisfied. +This is ensured by Properties (2), (3), and 4(a). +Even though the definition might look slightly technical at first glance, it applies in many +settings. In particular, Property (2) is satisfied when the dependency graph is a rooted tree +of height h in which all edges point towards the root; this is the case in all of our applications. +The other conditions are immediately satisfied by our DP for 0-1 Knapsack in Section 3 and +the DP for simultaneous source location in Section E. However, our DP for balanced graph +partitioning violates Property (4b) of Definition 8. Hence, we will also consider a weaker +assumption in Section C.2 which, however, will not allow for nice black-box results, such as +Theorems 9 and 10 below. +Next, we state our main results. They imply that we obtain static (1 + ϵ)-approximation +algorithms running in near-linear time and space for ( ˜O(1), ln(1+ϵ)/ ˜O(1), ˜O(1))-well-behaved +DPs. They also show that under this assumption, we can dynamically maintain (1 + ϵ)- +approximate DP solutions with polylogarithmic update times. +Our main theorem for static algorithms is as follows. +▶ Theorem 9. Consider an (h, α, p)-well-behaved DP. Then we can compute an approximate +DP table ADP which αh+1-approximates DP in time and space O(|I| · p2 log(p)). +Later, we will apply the theorem to DPs with dependency trees of logarithmic heights +h = O(log n), we will set the approximation ratio to α = ln(1 + ϵ)/(h + 1), and the number +of pieces to p = polylog(W). This will yield our desired algorithms with near-linear running +time ˜O(|I|) and space usage. Note that this is a big improvement upon the brute-force +running times and space usages of Ω(|I| · |J |). + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +11 +The proof of the theorem follows from observing that when moving from one vertex to +another in the dependency graph, we lose a multiplicative α-factor in the approximation +ratio; as each vertex can only reach h other vertices, this will compound to at most αh+1. +Combining the assumptions on the functions ˜Pi and the results from Lemmas 6 and 7, we get +that each row ADP(i, ·) can be computed in time O(p2 log(p)) which gives O(|I| · p2 log(p)) +total time. +We also give the following extension to the dynamic setting which shows that if one of +the DP rows changes, we can update the entire table efficiently. +▶ Theorem 10. Consider an (h, α, p)-well-behaved DP and suppose that row i is changed. +Then we can update our approximate DP table ADP such that after time O(h · p2 log(p)) it is +an αh+1-approximation of DP. +As before, we will typically use the theorem with h = O(log n), α = ln(1 + ϵ)/(h + 1) and +p = polylog(W). This will then result in our desired polylogarithmic update times. Note +that this is a significant speedup compared to storing the DP tables using two-dimensional +arrays: in that case even updating a single row would take time Ω(|J |), which in many +applications would already be linear in the size of the input. +The theorem follows from observing that after a row i changes, we only have to update +those rows which can be reached from i in the dependency graph. But these can be at most h +and each of them can be updated in time O(p2 log(p)) by Lemmas 6 and 7. +3 +Fully Dynamic Knapsack +In 0-1 knapsack, the input consists of a knapsack size B ∈ R+ and a set of n items, where +each item i ∈ [n] has a weight wi ∈ R+ and a price pi ∈ [1, ∞). The goal is to find a set of +items I that maximizes � +i∈I pi while satisfying the constraint � +i∈I wi ≤ B. For a set of +items I ⊆ [n], we refer to the sum � +i∈I wi as the weight of I. +For the rest of this section we set W = � +i pi and t = � +i∈[n] wi. +Next, we first derive a dynamic algorithm with update time ˜O(log3(n) log2(W)/ϵ2) which +is based on our framework for DPs with monotone rows. Then we will use this algorithm as +a subroutine to obtain a faster algorithm with update time ˜O(log2(nW)/ϵ2) in Section 3.2; +this will prove Theorem 1. +▶ Theorem 1. Let ϵ > 0. There exists an algorithm for fully dynamic knapsack that maintains +a (1 + ϵ)-approximate solution with worst-case update time +1 +ϵ2 log2(nW) polylog +� 1 +ϵ log(nW) +� +. +Below we will also show that we can return the maintained solution I in time O(|I|) and +that we answer queries whether a given item i ∈ [n] is contained in I in time O(1). This +matches the query times of [29]. +3.1 +Knapsack via Convolution of Monotone Functions +First, we give a brief recap of the knapsack approach by Chan [21]. We consider the more +general problem of approximating the function fJ : [0, t] → R+, where J ⊆ [n] is a set of +items and +fJ(x) = max +�� +i∈I +pi : +� +i∈I +wi ≤ x, I ⊆ J +� +. +(2) + +12 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +Intuitively, the value fJ(x) corresponds to the best possible knapsack solution if we can +only pick items which are contained in J and if the weight of the solution can be at most x. +Therefore, f[n](B) corresponds to the optimum solution of the global knapsack instance. +Note that for each J ⊆ [n], fJ(x) is a monotonically increasing piecewise constant function: +Indeed, consider x′ ≤ x. Any solution I ⊆ J that is feasible for x′ (i.e., the weight of I +is at most x′) is also a feasible solution for x. Thus, fJ(x′) ≤ fJ(x) and, therefore, fJ is +monotonically increasing. Furthermore, fJ is piecewise constant since each function value +fJ(x) corresponds to a solution I ⊆ J and the number of choices for I ⊆ J is finite. +Next, note that if we have two disjoint subsets J1, J2 ⊆ [n] then it holds that fJ1∪J2 is +the (max, +)-convolution of fJ1 and fJ2, i.e., for all x it holds that +fJ1∪J2(x) = max +¯x +fJ1(¯x) + fJ2(x − ¯x). +This can be seen by observing that for each x, the optimum solution I for the instance J1 ∪J2 +with weight at most x can be split into two disjoint solutions I1 ⊆ J1 and I2 ⊆ J2 such that +I1 has weight ¯x and I2 has knapsack weight at most x − ¯x (for suitable choice of ¯x ∈ [0, x]). +We conclude that if we have two knapsack instances over disjoint sets of items J1 and J2, +then we compute the solution for the knapsack instance with items J1 ∪ J2 by computing +the (max, +)-convolution of fJ1 and fJ2. +The Exact DP. The previous paragraphs imply a simple way of computing the exact +solution of a knapsack instance: For each item i ∈ [n], compute the function f{i} and +then recursively merge the solutions for sets of size 2j, j = 1, . . . , ⌈log n⌉, by computing +(max, +)-convolutions until we have computed the global solution f[n]. We perform the +recursive merging of the solutions using a balanced binary tree, resulting in a tree of height +O(log n). +More concretely, we build a rooted balanced binary tree T with n leaf nodes, where all +edges point towards the root. We have one leaf f{i} for each item i. Each internal node u +in T is associated with a function fJu as per Equation (2), where Ju is the set of all items in +the subtree rooted at u. To simplify notation, we will also refer to fJu as fu. +Now we consider the exact computation of the DP. This will reveal the procedures Pi +from Definition 8. As base case, for each i ∈ [n], the i’th leaf of T contains the function f{i}, +which is a piecewise constant function that has value 0 on the interval [0, wi) and value pi on +the interval [wi, t]. +Next, in each internal node u of T with children u1 and u2, we set fu to the (max, +)- +convolution of fu1 and fu2. By induction it can be seen that for every node u in T, it holds +that Ju = Ju1 ∪ Ju2 and thus Ju is the set of all items whose corresponding leaf is contained +in the subtree Tu. Hence, for the root r of T it holds that fr = f[n] and fr(B) is the optimal +solution for the global knapsack instance. +In the following, we check that our DP satisfies Properties (1–3) of Definition 8. +First, note that the tree T from above is also the dependency graph of our DP. Hence, +our DP has a row for every vertex of T and thus O(n) rows in total. Furthermore, since T +has height O(log n) and all edges point towards the root, every vertex can reach at most +h = O(log n) vertices. Hence, Property (2) of Definition 8 is satisfied. +Second, we observe that in both cases above, the function f{i} and fu which correspond to +the rows of our DP table are monotonically increasing (we argued this above for all functions +fJ). Thus, Property (1) is satisfied. +Third, observe that Property (3) is also satisfied since (max, +)-convolution satisfies our +sensitivity condition. + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +13 +We conclude that the first three properties of Definition 8 are satisfied. Unfortunately, +this does not yet imply that we can obtain efficient algorithms: Note that if we compute the +exact DP bottom-up then we compute one convolution per node and thus the total running +time of this approach is O(n·t(p)), where p is an upper bound on the number of pieces in our +functions and t(p) is the time it takes to compute a (max, +)-convolution of two functions +with p pieces. However, observe that computing the convolutions can potentially take a large +amount of time because the number of pieces of the functions might grow quadratically after +each convolution (see Lemma 7). We will resolve this issue below using rounding. +The Approximate DP. Next, we consider approximations which will reveal the functions +˜Pi from Definition 8. +First, note that we need to compute (max, +)-convolutions of monotonically increasing +functions efficiently. We observe that this can be done efficiently using our subroutine from +Lemma 7 for the (min, +)-convolution of monotonically decreasing functions: Indeed, suppose +that f is the (max, +)-convolution of two monotonically increasing functions g and h, then +for all x it holds that +f(x) = max +¯x {g(¯x) + h(x − ¯x)} = − min +¯x {−g(¯x) + (−h(x − ¯x))}. +Now observe that −g and −h are monotonically decreasing functions and, therefore, f = +−((−g) ⊕ (−h)), where ⊕ denotes the (min, +)-convolution. Thus, we can use the efficient +routine for (min, +)-convolution from Lemma 7 with the same running time.6 +Now we can define the subroutines ˜Pi. Let δ > 0 be a parameter that we set later. +Whenever we compute a function fu via a (max, +)-convolution, we use the efficient subroutine +from Lemma 7. After computing the convolution, we set fu = ⌈fu⌉1+δ via the subroutine +from Lemma 6. +Observe that this approach satisfies Property (4a) of Definition 8 with α = 1 + δ. +Furthermore, Property (4b) is satisfied since we only use a single convolution and a single +rounding step. Finally, Property (4c) is also satisfied because the resulting function is +monotone and has p = O(log1+δ(W)) after the rounding. +The above arguments show that our DP is (h, α, p)-well-behaved for h = ⌈log n⌉, α = 1+δ, +δ = ln(1+ϵ)/⌈log n⌉ and p = O(log1+δ(W)) = O(log(W)/δ). Hence, Theorem 10 immediately +implies the following lemma. +▶ Lemma 11. Let ϵ > 0. There exists an algorithm that computes a (1 + ϵ)-approximate +solution for 0-1 knapsack in time n · 1 +ϵ2 log2(n) log2(W) · polylog( 1 +ϵ log(nW)). +We note that we can return our solution I in time |I| log(n) · polylog( 1 +ϵ log(nW)) as +follows. +Recall that our global objective function value is achieved by fr(B) and that +fr(B) = fu1(¯x∗) + fu2(B − ¯x∗), where u1 and u2 are the nodes below the root node r of +the dependency tree. Now using the second part of Lemma 7 we can get the value of ¯x∗ in +time O(log p). If fu1(¯x∗) > 0 we recurse on fu1(¯x∗) and if fu2(B − ¯x∗) > 0 we also recurse +on fu2(B − ¯x∗). At some point we will reach a leaf node i and we include i in the solution +iff f{i}(x) > 0. Note that since we only recurse for function values which are strictly larger +than zero, for each item that we include into the solution we have to follow a single path +in the dependency tree of height O(log n) and our work in each internal node is bounded +6 We note that, formally, Lemma 7 can only be applied on functions with non-negative values. However, +this can be achieved by adding a number C to −g and −h, which is an upper bound on the values taken by +g and h, and at the end we subtract the constant function 2C, i.e., we set f = −((−g+C)⊕(−h+C))−2C. + +14 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +by O(log p). This gives the total time of O(|I| log(n) log(p)) and our claim follows from our +choice of p above. +Extension to the Dynamic Setting. Next, we extend our result to the dynamic +setting. For the sake of simplicity, we assume that n is an upper bound on the maximum +number of available items (items in S) and given to our algorithm in the beginning.7 We +consider update operations that insert and delete items from the set. More concretely, we +consider the following update operations: +insert(pi, wi), in which i is added to S by setting the price and weight of item i to +pi ∈ W∞ and wi ∈ R+, respectively, and +delete(i), where item i is removed from the set of items. +Our implementation is as follows. In the preprocessing phase, we build the same tree T +as above and use the subroutine from above to compute the function f{i}. For the operation +delete(i), we set pi = 0 and wi = 0, which changes exactly one row of our DP table. For +the operation insert(pi, wi), we set the price and weight of item i to pi and wi, resp., which +again changes a single row in our DP table. After changing such a row, we recompute the +global DP solution via Theorem 10. Since the DP is (h, α, p)-well-behaved with the same +parameters as above, the theorem implies the following proposition. +▶ Proposition 12. Let ϵ > 0. +There exists an algorithm for the fully dynamic knap- +sack problem that maintains a (1 + ϵ)-approximate solution with worst-case update time +1 +ϵ2 log3(n) log2(W) · polylog +� 1 +ϵ log(nW) +� +. +Observe that with the same procedure as for the static algorithm, we can return our +solution I in time |I| log(n) · polylog( 1 +ϵ log(nW)). Furthermore, given an item i ∈ [n], we +can return whether i ∈ I in time log(n) · polylog( 1 +ϵ log(nW)). This can be done by using the +same query procedure as in the static setting, where we only recurse on the unique subtree +in the depedency tree that contains the node for item i. +We note that the above proposition already improves upon the update time in the result +of Eberle et al. [29] in terms of the dependency on 1 +ϵ but it has a worse dependency on +log(nW). However, our query time is slower than the O(1)-time query operation in [29]. +We will resolve these issues in the next subsection, where we will use the algorithm from +Proposition 12 as a subroutine. +3.2 +Dynamically Maintaining a Small Instance +Next, we we obtain a faster dynamic algorithm with update time ˜O( 1 +ϵ2 log2(nW)) by combin- +ing the algorithm from Proposition 12 and with ideas from Eberle et al. [29]. Our high-level +approach is as follows. First, we partition the items into a small number of price classes. +Then we take a few items of small weight from each price class. This will give a very small +knapsack instance X for which we maintain an almost optimal solution using the subroutine +from Proposition 12; since this instance is very small (i.e., |X| ≪ n), the update time for +maintaining this instance essentially becomes O( 1 +ϵ2 log2(W)), i.e., we lose the O(log3 n) term +that made the update time in the proposition too costly. For the rest of the items which are +not contained in X, we show that we can compute a good solution using fractional knapsack, +7 It is possible to drop this assumption using an amortization argument. More concretely, every time the +number of items is less than n/2 or more than n, we rebuild the data structure with a new value of n. +Each rebuild can be done in time O(nt(n)), where t(n) is our update time. Since this only happens after +Ω(n) updates occured, we can amortize this cost over the updates that appeared since the last rebuild. + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +15 +which can be easily solved using a set of binary search trees. Then it remains to show that +the combination of the two solutions is a (1 + ϵ)-approximation. +The main differences of our algorithm and the one by Eberle et al. [29] are as follows. +Eberle et al. also partition the items into a small number of price classes. They also combine +solutions for a small set of heavy items X and solutions based on fractional knapsack for the +other items. However, they have to enumerate many different sets X and they also guess +the approximate price of the fractional knapsack solution; more concretely, they enumerate +Θ( 1 +ϵ2 log(W)) choices for X and the number of guesses they have to make for the fractional +knapsack solution is Θ( 1 +ϵ log(W)). Thus they have to consider Θ( 1 +ϵ3 log2(W)) guesses and +for each of them they have to compute approximate solutions, which takes time Θ( 1 +ϵ4 ) for +each X since they have to run a static algorithm from scratch. In our approach, we only +have to consider a single set X which we maintain in our data structure from Proposition 12, +which saves us a lot of time. Furthermore, the piecewise constant function, in which we store +the solution for X, essentially “guides” our Θ( 1 +ϵ log(W)) guesses for the weight of fractional +knapsack solution. In our analysis we have to be slightly more careful to ensure that our +guesses for the weight of the fractional knapsack solution guarantee the correct approximation +ratio. +Definitions. We assume that ϵ < 1 and that 1/ϵ is an integer. More concretely, we run +the algorithm with ϵ′ = max{ 1 +i : 1 +i ≤ ϵ, i ∈ N}. Set L = ⌈log1+ϵ(W)⌉ and recall that we set +W = � +i pi. +We define the price classes Vℓ = {i: (1 + ϵ)ℓ ≤ pi < (1 + ϵ)ℓ+1}. In the following, we +assume that all items from price class Vℓ have price exactly (1 + ϵ)ℓ+1. We only lose a factor +of 1 + ϵ by making this assumption. Furthermore, we set V 1/ϵ +ℓ +to the set of 1/ϵ items from +Vℓ with smallest weights wi (breaking ties arbitrarily). We also define V ′ +ℓ = Vℓ \ V 1/ϵ +ℓ +. +Next, we set X = � +ℓ≥0 V 1/ϵ +ℓ +and Y = � +ℓ≥0 V ′ +ℓ for all ℓ ≥ 0. Note that X and Y partition +the set of items and |X| = 1 +ϵ · L = O(ϵ−2 log(W)). +Now our strategy is to use our algorithm from Proposition 12 to maintain a solution for +the items in X. Then we show how we can combine the solution for X with a solution for Y +that is based on fractional knapsack and a charging argument. +Data Structures. For each ℓ ∈ [L], we maintain Vℓ sorted non-decreasingly by weight. +We also maintain the set X in a binary search tree, in which we sort the items by their +index, and we maintain our data structure from Proposition 12 on the items in X. +Furthermore, let Uℓ = � +ℓ′≤ℓ V ′ +ℓ′ denote the set of all items that are not contained in X +and of price class at most ℓ. For each ℓ, we maintain the set Uℓ in a binary search tree T in +which the items are stored as leaves and sorted by their density pi +wi . In each internal node u +of T, we store the total weight of the items in the subtree Tu rooted at u and the total profit +of the items in Tu. Observe that this allows us to answer queries of the type: “Given a +budget b, what is the value of the optimal fractional8 knapsack solution in Uℓ with weight at +most b?” in time O(log n). +Updates. Now consider an item insertion or deletion and suppose that the updated +item is of price class Vℓ. We first update the sets Vℓ, Uℓ′ for ℓ′ ≤ ℓ and the sets X and Y . +Note that for each of these sets at most one item can be removed and inserted. Thus, these +steps can be done in time O(ℓ · log(n)) = O(ϵ−1 log(W) log(n)). +Next, if X changed in the previous step, then we also perform the corresponding updates +8 In fractional knapsack, we may use items fractionally. An optimal solution is achieved by sorting the +items items by their density and greedily adding items to the solution until we have used up our budget b. +This approach uses at most one item fractionally (namely, the one at which we use up our budget). + +16 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +in the data structure from Proposition 12. Since |X| = O(ϵ−2 log(W)) holds by construction +of X, the update operations for the data structure maintaing the knapsack solution for X +take a total time of +O +� +ϵ−2 log3(|X|) log2(W) · polylog +�1 +ϵ log(|X| W) +�� += O +� +ϵ−2 log2(W) · polylog +�1 +ϵ log(nW) +�� +. +Furthermore, we can explicitly write down our solution IX for the items in X in time +ϵ−2 log(W) · polylog( 1 +ϵ log(nW)) since |X| = O(ϵ−2 log(W)). Also, for each i ∈ IX, we can +set a bit indicating that i ∈ IX. Note that the time for writing down IX and setting the bits +is subsumed by the update time above. +Queries. Returning the value of a solution: We return the value of a global knapsack +solution as follows. +Consider the data structure from Proposition 12 which maintains a solution for the items +in X. Note that this solution is stored as a piecewise constant function with p ≤ L pieces +and consider the list representation (x1, y1), . . . , (xp, yp) of this function. +Our strategy is as follows: For each i = 1, . . . , p, we consider a solution which spends +budget xi on items in X and budget B − xi on items in Y . Then we take the maximum over +all of the solutions we have considered. More concretely, for given i = 1, . . . , p, we obtain our +solution as follows. We pick ℓi such that (1 + ϵ)ℓi = ⌈ϵ · yi⌉1+ϵ (see Lemma 13 below for a +justification of this choice). Now we use the binary search tree for Uℓi to find the highest +profit that we can obtain from fractional knapsack on items in Uℓi ⊆ Y if we can spend +budget at most b = B − xi. Let y′ +i be the value of this query after removing any profit that +we gain from the (at most one) fractionally cut item. We also store the density of the final +item that is contained in the fractional knapsack solution. Now we return the maximum of +yi + y′ +i over all i = 1, . . . , p. +Note that since the solution for X has at most L = O(ϵ−1 log(W)) pieces and for each of +them we perform a single query in a binary search tree, the total time for return the solution +value is O(ϵ−1 log(W) log(n)). Note that this time is subsumed by the update time. +Returning the entire solution: Now we can return our global solution I in time O(|I|) as +follows. Observe that I is composed of the solution IX for the items in X and of the items in +the fractional knapsack solution. During our updates, we already stored the items in IX and +can write them down in time O(|IX|). Next, to return the items from the fractional knapsack +solution, recall that we stored the density of the final item in the fractional knapsack solution. +Thus, we only have to output the items ordered non-decreasingly by their density, while we +are above the desired density-threshold. This can be done in time linear in the size of the +fractional knapsack solution. This is essentially the same query procedure as in [29]. +Returning whether an item is in the solution: Furthermore, observe that the above implies +that we can answer whether an item i ∈ [n] is contained in our solution in time O(1): If +i ∈ X then we already stored a bit whether i ∈ IX. If i ̸∈ X then we can check whether i is +in the fractional knapsack solution by checking whether its density is above or below the +threshold given by the final item in the fractional knapsack solution. +Analysis. We start by making some simplifications to OPT. We let OPT′ denote the +version of OPT in which for each ℓ ∈ [L], we pick the |OPT ∩Vℓ| items of smallest weight from +Vℓ. This only loses a factor of 1+ϵ. Next, define OPT′ +X = OPT′ ∩X and OPT′ +Y = OPT′ ∩Y . +Observe that by how we picked OPT′, it holds that OPT′ +Y ∩Vℓ ̸= ∅ iff +��OPT′ ∩Vℓ +�� > 1/ϵ. +Let pX denote the total price of items in OPT′ +X and let wX denote the total weight of +the items in OPT′ +X. Let f denote the piecewise constant function that stores the solution + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +17 +for the items in X. Observe that by Proposition 12 we have that +pX ≤ f(wX) ≤ (1 + ϵ)pX. +Also, the function value f(wX) is part of a piece (xi∗, yi∗) with xi∗ ≤ wX and yi∗ = f(wX). +The next lemma justifies why we set ℓi such that (1 + ϵ)ℓi = ⌈ϵ · yi⌉1+ϵ in our algorithm. +To this end, let ℓi∗ be such that (1 + ϵ)ℓi∗ = ⌈ϵ · yi∗⌉1+ϵ and let ℓY be the price class of the +most valuable item in OPT′ +Y . In the lemma we show that ℓi∗ ≥ ℓY . We will use this to show +that our solution for X of profit yi∗ is valuable enough such that we can charge a fractionally +cut item from fractional knapsack onto the solution from X and only lose a factor of (1 + ϵ)2. +▶ Lemma 13. It holds that ℓi∗ ≥ ℓY . +Proof. Since OPT′ +Y ∩V ′ +ℓY ̸= ∅, +��OPT′ ∩VℓY +�� > 1/ϵ and thus OPT′ +X contains all 1/ϵ items +from V 1/ϵ +ℓY . Hence, pX ≥ 1 +ϵ · (1 + ϵ)ℓY . From above we get f(wX) = yi∗ and f(wX) ≥ pX. +By choice of ℓi∗, +(1 + ϵ)ℓi∗ = ⌈ϵ · yi∗⌉1+ϵ = ⌈ϵ · f(wX)⌉1+ϵ ≥ ⌈ϵ · pX⌉1+ϵ ≥ +� +ϵ · 1 +ϵ (1 + ϵ)ℓY +� +1+ϵ += (1 + ϵ)ℓY . +This implies ℓi∗ ≥ ℓY . +◀ +Next, consider the the fractional knapsack solution that we obtain from our query. Note +that this solution has a profit that is at least as large as the profit of OPT′ +Y (since fractional +knapsack is a relaxation of 0-1 knapsack). Furthermore, the fractional solution uses at +most one item fractionally and this item is from Uℓi∗ and has value at most (1 + ϵ)ℓi∗ = +⌈ϵ · yi∗⌉1+ϵ ≤ (1 + ϵ)ϵ · yi∗. Thus, we can charge this item on OPT′ +X and lose a factor of at +most (1 + ϵ)2. +We conclude that the solution yi∗ +y′ +i∗ is a (1+ϵ)O(1)-approximation of OPT. Combining +this with our previous running time analysis, we obtain Theorem 1. +4 +Technical Overview +We now present an overview of two techniques for making DPs fit our framework. We will +briefly discuss how we monotonized the DP for k-balanced partitioning and how we inverted +the DP for simultaneous source location. Due to space constraints, we only present excerpts +of our algorithms and we only consider special cases. More concretely, for both problems we +will consider the special case when the input graph is a binary tree. In the appendix we will +show that the results can be extended to general graphs. +4.1 +Monotonizing the DP of Feldmann and Foschini +We start by considering the k-balanced graph partitioning problem. Recall that in this +problem, the input is a graph G = (V, E, cap), where cap : E → W∞ is a weight function on +the edges, and an integer k. As discussed in the introduction, we assume that we can violate +the partition sizes by a (1 + ϵ)-factor and our goal is to find a partition V1, . . . , Vk of the +vertices such that |Vi| ≤ ⌈(1+ϵ) |V | /k⌉ for all i and such that we minimize cut(V1, . . . , Vk) := +�k +i=1 +� +{u,v}∈E∩(Vi×(V \Vi)) cap(u, v). +For the sake of better exposition, here we only consider the special case in which G is a +binary tree; in Appendix D.4 we show how to drop this assumption. + +18 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +In the following we present a DP in which the rows are monotone and we show how to +efficiently perform operations on these solution vectors using monotone piecewise constant +functions. Our DP is related to the DP by Feldmann and Foschini [34] which is non-monotone +and thus our DP can be viewed as the monotonization of the DP by Feldmann and Foschini. +We believe that our technique to monotonize the DP will have further applications in the +future. +High-Level Description of the DP. Our DP is computed bottom-up starting at the +leaves of the tree and then moving up in the tree. For each vertex v, we will compute a DP +solution of minimum cost that encodes whether the edge to the parent p of v is cut and +which edges shall be cut inside the subtree Tv that is rooted at v. Note that the removal of +the cut edges in our solution will decompose the tree into disjoint connected components +and exactly one of them contains v’s parent p. Additionally, we store information about the +number of vertices that are still connected to the parent p (and, therefore, to the outside of +Tv) after the cut edges are removed. We will assume that when we compute the DP cell for +a vertex v, we have access to the solutions for both of its children. +More concretely, when we have computed a solution for a subtree Tv, i.e., we know which +edges incident to nodes in this subtree we are going to remove (note that the edge leading to +the parent of v is incident to Tv and thus we consider it as part of this solution), we store +the following information in the DP table. First, we store its cost, i.e., the total capacity of +all edges that are incident to vertices in Tv and that are cut. As described above, we would +also like to store the number of vertices that are connected to the parent of v and the sizes +of connected components inside Tv. However, there are two difficulties: (1) We cannot store +the number of vertices that are connected to the root exactly because this would result in +a too large DP table. Instead, we store the cheapest solution in which vertices of at most +some given number are still connected to the parent of v. As we will see, this approach gives +rise to monotonically decreasing functions and allows for a very efficient computation of the +DP table. (2) We store implicitly the size of all connected components that are created +after the cut edges are removed and that lie completely inside Tv. As before, storing these +sizes exactly would result in a very large DP table and, therefore, we store them concisely +using the concept of a signature. The signatures will help us to characterize the sizes of the +components inside Tv very efficiently. +Signatures. We call a connected component in Tv large if it contains at least ϵ⌈|V | /k⌉ +vertices and otherwise we call it small. Let t = ⌈log1+ϵ(1/ϵ)⌉ + 1, and let M = ⌈k/ϵ⌉ + 1. A +signature is a vector g = (g0, . . . , gt−1) ∈ [M −1]t. Observe that each Pi is an integer between +0 and M − 1 and hence there are M t = (k/ϵ)O(ϵ−1 log(1/ϵ)) different signatures. Intuitively, +an entry Pi in g tells us roughly how many components of size (1 + ϵ)i · ϵ⌈|V | /k⌉ there are +in the DP solutions that we consider. Due to space constraints, we refer to the appendix for +the formal definition. +For x ∈ N, we let e(x) ∈ [M − 1]t denote the signature of a single component with x +vertices. More precisely, we set e(x) to the vector that has e(x)j = 1 for j = arg min{j ∈ +N: x ≤ (1 + ϵ)j · ϵ⌈|V | /k⌉} and e(x)j = 0, otherwise. If x < ϵ⌈|V | /k⌉, we define e(x) = ⃗0. +Formal DP Definition. Now we describe the DP formally. An entry DP(v, g, cut, x) ∈ +W∞ in the DP table for a vertex v is indexed by a signature g, a Boolean value cut and +x ∈ [n]. We will consider the tuples (v, g, cut) as the rows I of the DP table and x as the +columns; we associate each such row with a function DP(v, g, cut, ·): [n] → W∞. Note that +our DP has |V | · M t · 2 = (k/ϵ)O(ϵ−1 log(1/ϵ)) · n rows. Also, note that it has columns n; later, +even though x only takes discrete values, we will allow x to take values in [0, ∞). +An entry DP(v, g, cut, x) describes the optimum cost of cutting edges incident on the + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +19 +subtree Tv (including the cost of maybe cutting the edge to the parent of v). We will refer +to the set of vertices in Tv that are still connected to the parent of v after the cut edges are +removed as the root component. We impose the following conditions on DP(v, g, cut, x): +Once the cut edges are removed, the root component U ⊆ Tv has at most x vertices, i.e., +|U| ≤ x. +If cut is set to true then the edge between v and its parent is cut, otherwise it is kept. +The vertices inside Tv that (once the cut edges are removed) are not connected to the +parent of v form connected components that are consistent with the signature g. +Next, we observe that if we fix a vertex v, a signature g and a value for cut, then the +resulting function DP(v, g, cut, ·) is monotonically decreasing in x. +▶ Observation 14. Let v ∈ V , g ∈ [M − 1]t be a signature and cut ∈ {true, false}. Then the +function DP(v, g, cut, ·) : [0, ∞) → R+ is monotonically decreasing. +Proof. By definition, DP(v, g, cut, x) stores the cost of the optimum solution in which there +are at most x vertices in the root component. Since x ≤ x′, the solution DP(v, g, cut, x) is also +a feasible solution for DP(v, g, cut, x′). Hence, DP(v, g, cut, ·) is monotonically decreasing. +◀ +Comparison With the DP by Feldmann and Foschini. When comparing our DP +with the one by Feldmann and Foschini [34] then one of the crucial changes is that in our +DP, x encodes an upper bound on the number of vertices in the root component. Previously, +Feldmann of Foschini considered root components with exactly x vertices. This is why their +DP was non-monotone and why one can view our DP as the monotonization of the DP +in [34]. However, we also generalize the DP to the setting with vertex weights and, as we will +see below, parts of our algorithm for computing the DP approximately are rather involved. +4.1.1 +Computing the DP +We now give a flavor of what our algorithms for computing the DP look like. We start +by showing how to compute the exact DP solution DP(v, ·, ·, ·) for a vertex v of the tree, +where v has parent p and children vl, vr and it is connected to them via edges ep, el and er, +respectively. +Computing the DP is based on several case distinctions; here, we only consider the case +in which v is an internal vertex of the tree we do not cut the edges el and er. All other cases +are presented in the appendix. +When computing a DP row given by DP(v, ·, ·, ·), we will only require access to the DP +rows DP(vl, ·, ·, ·) and DP(vr, ·, ·, ·). This implies that the dependency tree of the DP is a +tree and has the same height as our input graph G (recall that here we assume that G is a +binary tree). Note that the height of the tree also implies an upper bound on the number of +reachable nodes. +Exact Computation. We start with the exact computation. Here, we can afford to +iterate over all values x ∈ [n] and g ∈ [M − 1]t to compute DP(v, ·, ·, ·). Therefore, we +consider the values for x and g as input to our algorithm. +Since we assume that we do not cut the edges el and er, we have to select subsolutions +for Tvl and Tvr, where each subsolution is characterized by the upper bound xl (resp. xr) +and its signature gl (resp. gr). +First, suppose that we cut the edge ep. If we let xl and xr denote the number of vertices of +the root components for the subsolutions, then the vertex v will be included in a component +of size xl + xr + 1 afterwards. Hence, we can combine the subsolutions to a solution for + +20 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +signature g as long as gl + gr + e(xl + xr + 1) = g. Consequently we set for every x ∈ [0, ∞), +DPB(v, g, true, x) = cap(v, p)+ +min +xl,xr,gl+gr=g−e(xl+xr+1) DP(vl, gl, false, xl)+DP(vr, gr, false, xr). +Second, suppose that we do not cut ep. Again we have to set DPB(v, g, false, x) = ∞ +for all signatures g and all x ∈ [0, 1), because v can reach p. For x ≥ 1 we have to select xl +and xr such that they sum to x − 1 as this guarantees that at most x vertices can reach the +parent p. Consequently, we set for all x ∈ [1, ∞) +DPB(v, g, false, x) = +min +gl+gr=g,xl+xr=x−1 DP(vl, gl, false, xl) + DP(vr, gr, false, xr). +Here, we can afford to exhaustively enumerate all O(M tn2) possibilities in the min- +operations above. +Approximate Computation. Now let us consider the approximate computation. We +denote the approximate DP solution by ADP. We assume that we have already computed the +children solutions ADP(vl, g, cut, ·) and ADP(vr, g, cut, ·) and that they are stored using our +data structure from Section 2. We will maintain as an invariant that each of these functions +has at most p = O(log1+δ(W)) pieces and we will ensure this by rounding our solution at the +end of every step, i.e., by setting ADP(v, g, cut, ·) = ⌈ADP(v, g, cut, ·)⌉1+δ using the rounding +procedure from Lemma 6. This will ensure the following two properties: (1) The functions +ADP(v, g, cut, ·) never have more than O(log1+δ(W)) pieces by Lemma 6. Thus, we can +perform all of our operations very efficiently. (2) For the function at the root of the tree, +the approximation error is at most (1 + δ)h, where h is the height of the tree. By picking +δ = O(ϵ/h), we will achieve that we obtain a (1 + ϵ)-approximate solution at the root. Now +we proceed to the explanation of our computation. +If we do not cut the edge to the parent of v, we proceed similar to the exact DP above. +We start by setting ADPB(v, g, false, x) = ∞ for all x ∈ [0, 1). Next, for x ∈ [1, ∞) we wish +to set +ADPB(v, g, false, x) = +min +gl+gr=g,xl+xr=x−1 ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr) +(3) += +min +gl+gr=g +min +xl+xr=x−1 ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr). (4) +Note that for fixed gl and gr, the inner min-operation in the second line describes a (min, +)- +convolution due to the constraint xl + xr = x − 1. Therefore, in the inner min-operation we +compute a convolution ADP(vl, gl, false, ·) ⊕ ADP(vr, gr, false, ·) and shift the result by 1 via +the shift operation from Lemma 6 (where for x ∈ [0, 1) we set ADPB(v, g, false, x) = ∞). We +need time O(p2 log p) for computing the convolution according to Lemma 7. To compute +the outer minimum in Equation (4), we iterate over all gl ∈ [M − 1]t using Lemma 6 and +thus perform O(M t) minimum computations over piecewise constant functions with at most +p2 pieces. Hence, we need time O(M tp2 log(M tp2)) according to Lemma 21. By Lemma 22, +ADPB(v, g, false, ·) is monotonically decreasing since it is the minimum over convolutions of +two monotonically decreasing functions. +If we cut the edge to the parent of v, then for all x ∈ [0, ∞) we would like to set +ADPB(v, g, true, x) = cap(v, p) + +min +xl,xr,gl+gr=g−e(xl+xr+1) ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr). +Note that here we need to be careful as the range of gl and gr depends on the choice of xl +xr. +Since there are Ω(n) possible values for xl +xr, we cannot afford to iterate over all values that + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +21 +xl + xr can take. Instead, we will show that we only need to consider O(log(k/ϵ)/ϵ) different +pairs (xl, xr) by exploiting the monotonicity of ADP(vl, gl, false, ·) and ADP(vr, gr, false, ·). +First, observe that we can assume xl ≤ |Tvl| and xr ≤ |Tvr|: increasing the upper bounds +on the number of vertices of the root component further would mean that the root component +contains than all vertices inside the sub-tree, which is impossible. Thus, xl + xr + 1 ∈ [1, n]. +Second, we partition the interval [1, n] into O(log(k/ϵ)/ϵ) intervals. We have intervals +Ij = (ξj−1, ξj] with ξj = (1 + ϵ)jϵ⌈n/k⌉ for all j = 1, . . . , log1+ϵ(k/ϵ). In addition, we add +an “interval” I0 := [ϵ⌈n/k⌉, ϵ⌈n/k⌉] and the interval I−1 := [1, ϵ⌈n/k⌉). We set ξ0 = ϵ⌈n/k⌉ +and we set ξ−1 to the largest integer that is less than ϵ⌈n/k⌉. Observe that for all j ≥ −1 +and x ∈ Ij, we have e(x) = e(ξj), i.e., the value of e(x) does not change inside the interval +Ij. Below, this property will allow us to separate the conditions on xl + xr and on gl + gr. +Now we can rewrite the above expression as +ADPB(v, g, true, x) = +cap(v, p) + min +j +min +xl+xr+1∈Ij +min +gl+gr=g−e(ξj) ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr). +Third, note that now the two min-operations only depend on the choice of j and, +importantly, the minimum over gl and gr does not depend on the choice of xl + xr any- +more. Therefore, we can swap the order of the two min-operations. Furthermore, since +ADPB(v, g, false, x) is monotonically decreasing with x, we can restrict the choice of xl +and xr such that xl + xr + 1 is the largest number in the corresponding interval Ij, i.e., +xl + xr + 1 = ξj. Thus, +ADPB(v, g, true, x) = +cap(v, p) + min +j +min +gl+gr=g−e(ξj) +min +xl+xr+1=ξj ADP(vl, gl, false, xl) + ADP(vr, gr, false, ξj − xl − 1). +Next, we explain how the above expression can be computed efficiently. Let us first argue +how we can efficiently compute the inner min-operation of the above expression. We start by +observing that this min-operation is not a convolution since in the constraint we sum up to +ξi which is a constant (rather than to the variable x). Now recall that ADP(vl, gl, false, ·) and +ADP(vr, gr, false, ·) are piecewise constant functions with O(p) pieces by our invariants. Since +xl, xr ≥ 0 this implies that there are only O(p2) choices for xl and xr such that xl, xr ∈ Ij +and either a new piece starts in ADP(vl, gl, false, xl) or in ADP(vr, gr, false, xr). Thus, we can +iterate over all these pairs (xl, xr) and evaluate ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr), +where xr = ξj − xl − 1. Thus, we can compute the inner min-operation in time O(p2 log p). +We note that since this min-operation is considering a super-constant number of terms, this +DP is not well-behaved (it violates Property (4b) of Definition 8). This is why in our analysis +we will use the more general notion from Section C.2. +Next, we can compute the outer two min-operations by simply iterating over j and all +choices for gl and setting gr = g − e(ξj) − gl as above in O(M t · log(k/ϵ)/ϵ) iterations. Hence, +we obtain a running time of O(M tp2 log p · log(k/ϵ)/ϵ). +Finally, we note that as ADPB(v, g, true, x) is independent of x, it is a constant. Thus, +ADPB(v, g, true, x) is a piecewise constant function with a single piece and it is monotonically +decreasing. +Rounding Step. As noted earlier, after computing the solutions ADPB(v, g, false, ·) and +ADPB(v, g, true, ·), we also round the solution by setting ADPB(v, g, cut, ·) = ⌈ADPB(v, g, cut, ·)⌉1+δ +for cut ∈ {true, false} to ensure that we only have p = O(log1+δ(W)) pieces in the result- +ing function. Note that this is the only approximate operation we perform and all other +operations above have been exact. + +22 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +4.2 +Inverting the DP of Andreev et al. +Now we briefly describe our DP for simultaneous source location. Recall that in this problem, +the input consists of an undirected graph G = (V, E, cap, d) with a capacity function +cap: E → W∞ and a demand function d: V → W∞. The goal is to select a minimum set +S ⊆ V of sources that can simultaneously supply all vertex demands. More concretely, a set +of sources S is feasible if there exists a flow from the vertices in S that supplies demand d(v) +to all vertices v ∈ V and that does not violate the capacity constraints on the edges. The +objective is to find a feasible set of sources of minimum size. +Here, we will again assume the special case in which G is a binary tree; we show in +Appendix E.3 how to drop this assumption. +DP Definition. Given a vertex v and a value x ∈ R, we let DP(v, x) denote the minimum +number of sources that we need to place in the subtree Tv such that when v receives flow at +most x from its parent then all demands in Tv can be satisfied. We note that x can take +positive and negative values: for x ≥ 0 this corresponds to the setting in which flow is sent +from the parent of v into Tv and for x < 0 this corresponds to the setting in which flow is sent +from Tv towards the parent of v. We further follow the convention that when the demands +in Tv cannot be satisfied when v receives flow x from its parent, then we set DP(v, x) = ∞. +Observe that this DP has rows I = V and columns J = R. Furthermore, DP(v, ·) is +monotonically decreasing since for x < x′, any solution in which Tv receives flow at most x +from the parent of v is also feasible when Tv receives flow at most x′ from the parent of v. +This satisfies Property (1) of Definition 8. +The Inverse DP. Interestingly, our DP is very related to the one by Andreev et al. [4]. +They defined a function f(v, i) which, given a vertex v and an integer i ∈ N, denotes the +minimum amount of flow that v needs to receive from its parent if all demands in Tv need to +be satisfied and if we can place i sources in the subtree Tv. Similar to above, f(v, i) takes +positive values if the demand in Tv can only be satisified by receiving flow from the parent +of v and it takes negative values if the demand in Tv is already satisfied by the sources in +the subtree Tv and v can send flow to its parent. +Now observe that our DP can essentially be viewed as the “inverse” of f(v, i). More +formally, observe that DP(v, x) = f −1(v, x) := min{i: f(v, i) ≤ x}. +The reason why we chose the inverse formulation for our DP is as follows. To ensure that +our algorithms are efficient, we have to make sure that our monotone piecewise constant +functions have only few pieces. One natural way to do is using rounding. However, since +the function values of f are positive and negative, it is not clear how we should perform the +rounding. For example, to only use a small number of pieces for representing f, we would +have to use different rounding mechanisms for those function values in [−1, 1] and those in +[−W, W]\[−1, 1], where W is the largest edge capacity: Indeed, if we rounded the values of f +to powers of (1 + δ)j then there are only O(log1+δ(W)) function values in [−W, W] \ [−1, 1] +but there are infinitely many function values in [−1, 1]. Similarly, if we rounded to multiples +of δ then there are only O(1/δ) function values in [−1, 1] but this would lead to O(W/δ) +function values in [−W, W] \ [−1, 1]. In both cases, our functions would have too many pieces +and we would have to pick a rounding function which provides a tradeoff between these two +cases. Furthermore, we would have to find an analysis that shows that this “more involved” +rounding function does not introduce much too error. +In our DP we bypass these issues because we move the negative numbers into the domain +of the function DP(v, ·): R → [n + 1]. Then in the codomain we only have non-negative +numbers to which we can apply the standard rounding function ⌈·⌉1+δ in a straightforward +way. This also has the positive side effects that instead of getting factors of polylog(W) in + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +23 +our running times, we only get factors of polylog(n) because our codomain became [n + 1] +rather than some potentially large interval [−W, W]. We believe this technique of considering +inverse DPs will be useful in the future to compute approximate solutions for DPs that can +take positive and negative values. +Due to lack of space, we present the details for computing the DP in Appendix E. +References +1 +Amir Abboud, Vincent Cohen-Addad, and Philip N Klein. New hardness results for planar +graph problems in p and an algorithm for sparsest cut. In STOC, pages 996–1009, 2020. +2 +Alok Aggarwal, Maria M. Klawe, Shlomo Moran, Peter W. Shor, and Robert E. Wilber. +Geometric applications of a matrix-searching algorithm. Algorithmica, 2:195–208, 1987. +3 +Greg Aloupis, Thomas Fevens, Stefan Langerman, Tomomi Matsui, Antonio Mesa, Yurai +Nuñez, David Rappaport, and Godfried Toussaint. Algorithms for computing geometric +measures of melodic similarity. Computer Music Journal, 30(3):67–76, 2006. +4 +Konstantin Andreev, Charles Garrod, Daniel Golovin, Bruce M. Maggs, and Adam Meyerson. +Simultaneous source location. ACM Trans. Algorithms, 6(1):16:1–16:17, 2009. +5 +Konstantin Andreev and Harald Räcke. Balanced graph partitioning. Theory of Computing +Systems, 39(6):929–939, 2006. +6 +Ali Aouad and Danny Segev. +An approximate dynamic programming approach to the +incremental knapsack problem. Operations Research, 2022. +7 +Kouji Arata, Satoru Iwata, Kazuhisa Makino, and Satoru Fujishige. Locating sources to meet +flow demands in undirected networks. Journal of Algorithms, 42(1):54–68, 2002. +8 +Yoan José Pinzón Ardila, Raphaël Clifford, and Manal Mohamed. Necklace swap problem +for rhythmic similarity measures. In International Symposium on String Processing and +Information Retrieval, pages 234–245. Springer, 2005. +9 +Kyriakos Axiotis and Christos Tzamos. Capacitated dynamic programming: Faster knapsack +and graph algorithms. In ICALP, pages 19:1–19:13, 2019. +10 +Arturs Backurs, Piotr Indyk, and Ludwig Schmidt. Better approximations for tree sparsity in +nearly-linear time. In SODA, pages 2215–2229, 2017. +11 +MohammadHossein Bateni, MohammadTaghi Hajiaghayi, Saeed Seddighin, and Cliff Stein. +Fast algorithms for knapsack via convolution and prediction. In STOC, pages 1269–1282, 2018. +12 +Wolfgang W. Bein, Mordecai J. Golin, Lawrence L. Larmore, and Yan Zhang. The knuth–yao +quadrangle-inequality speedup is a consequence of total monotonicity. ACM Trans. Algorithms, +6(1):17:1–17:22, 2009. +13 +Marcin Bienkowski, Miroslaw Korzeniowski, and Harald Räcke. A practical algorithm for +constructing oblivious routing schemes. In SPAA, pages 24–33, 2003. +14 +David Bremner, Timothy M. Chan, Erik D. Demaine, Jeff Erickson, Ferran Hurtado, John +Iacono, Stefan Langerman, Mihai Pătraşcu, and Perouz Taslakian. Necklaces, convolutions, +and X+Y. Algorithmica, 69(2):294–314, 2014. +15 +Karl Bringmann and Alejandro Cassis. Faster knapsack algorithms via bounded monotone +min-plus-convolution. In ICALP, volume 229, pages 31:1–31:21, 2022. +16 +Karl Bringmann, Fabrizio Grandoni, Barna Saha, and Virginia Vassilevska Williams. Truly +subcubic algorithms for language edit distance and RNA folding via fast bounded-difference +min-plus product. SIAM J. Comput., 48(2):481–512, 2019. +17 +Karl Bringmann, Marvin Künnemann, and Karol Wegrzycki. Approximating APSP without +scaling: equivalence of approximate min-plus and exact min-max. In STOC, pages 943–954. +ACM, 2019. +18 +Aydin Buluç, Henning Meyerhenke, Ilya Safro, Peter Sanders, and Christian Schulz. Recent +advances in graph partitioning. In Algorithm Engineering - Selected Results and Surveys, +volume 9220 of Lecture Notes in Computer Science, pages 117–158. 2016. + +24 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +19 +Rainer E. Burkard, Bettina Klinz, and Rüdiger Rudolf. Perspectives of monge properties in +optimization. Discret. Appl. Math., 70(2):95–161, 1996. +20 +Michael Bussieck, Hannes Hassler, Gerhard J Woeginger, and Uwe T Zimmermann. Fast +algorithms for the maximum convolution problem. Operations research letters, 15(3):133–141, +1994. +21 +Timothy M. Chan. Approximation schemes for 0-1 knapsack. In SOSA, volume 61, pages +5:1–5:12, 2018. +22 +Timothy M. Chan. Near-optimal randomized algorithms for selection in totally monotone +matrices. In SODA, pages 1483–1495, 2021. +23 +Timothy M. Chan and Moshe Lewenstein. Clustered integer 3sum via additive combinatorics. +In STOC, pages 31–40, 2015. +24 +Shucheng Chi, Ran Duan, Tianle Xie, and Tianyi Zhang. Faster min-plus product for monotone +instances. In STOC, pages 1529–1542, 2022. +25 +Justin Colannino, Mirela Damian, Ferran Hurtado, John Iacono, Henk Meijer, Suneeta +Ramaswami, and Godfried Toussaint. An o(n log n)-time algorithm for the restriction scaffold +assignment problem. Journal of Computational Biology, 13(4):979–989, 2006. +26 +Spencer Compton, Slobodan Mitrovic, and Ronitt Rubinfeld. New partitioning techniques and +faster algorithms for approximate interval scheduling. CoRR, abs/2012.15002, 2020. URL: +https://arxiv.org/abs/2012.15002. +27 +Marek Cygan, Marcin Mucha, Karol Wegrzycki, and Michal Wlodarczyk. +On problems +equivalent to (min, +)-convolution. ACM Trans. Algorithms, 15(1):14:1–14:25, 2019. +28 +Yihe Dong, Piotr Indyk, Ilya P. Razenshteyn, and Tal Wagner. Learning space partitions for +nearest neighbor search. In ICLR, 2020. +29 +Franziska Eberle, Nicole Megow, Lukas Nölke, Bertrand Simon, and Andreas Wiese. Fully +Dynamic Algorithms for Knapsack Problems with Polylogarithmic Update Time. In FSTTCS, +volume 213, pages 18:1–18:17, 2021. +30 +Eduard Eiben, Daniel Lokshtanov, and Amer E Mouawad. Bisection of bounded treewidth +graphs by convolutions. In ESA 2019. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, +2019. +31 +David Eppstein, Zvi Galil, and Raffaele Giancarlo. Speeding up dynamic programming. In +FOCS, pages 488–496. IEEE Computer Society, 1988. +32 +Guy Even, Joseph Naor, Satish Rao, and Baruch Schieber. Fast approximate graph partitioning +algorithms. SIAM J. Comput., 28(6):2187–2214, 1999. +33 +Uriel Feige and Robert Krauthgamer. A polylogarithmic approximation of the minimum +bisection. SIAM J. Comput., 31(4):1090–1118, 2002. +34 +Andreas Emil Feldmann and Luca Foschini. Balanced partitions of trees and applications. +Algorithmica, 71(2):354–376, 2015. +35 +Zvi Galil and Kunsoo Park. Dynamic programming with convexity, concavity, and sparsity. +Theor. Comput. Sci., 92(1):49–76, 1992. +36 +Gramoz Goranci, Harald Räcke, Thatchaphol Saranurak, and Zihan Tan. The expander +hierarchy and its applications to dynamic graph algorithms. In SODA, pages 2212–2228, 2021. +37 +Chris Harrelson, Kirsten Hildrum, and Satish Rao. A polynomial-time tree decomposition to +minimize congestion. In SPAA, pages 34–43, 2003. +38 +Monika Henzinger, Stefan Neumann, and Andreas Wiese. Dynamic approximate maximum +independent set of intervals, hypercubes and hyperrectangles. In SoCG, volume 164 of LIPIcs, +pages 51:1–51:14, 2020. +39 +Dorit S Hochbaum and David B Shmoys. Using dual approximation algorithms for scheduling +problems theoretical and practical results. Journal of the ACM, 34(1):144–162, 1987. +40 +Yiding Hua, Rasmus Kyng, Maximilian Probst Gutenberg, and Zihang Wu. Maintaining +expander decompositions via sparse cuts. CoRR, abs/2204.02519, 2022. +41 +Piotr Indyk, Fabian Kuhn, and Anca Muscholl. +On the fine-grained complexity of one- +dimensional dynamic programming. In ICALP 2017, volume 80, page 21, 2017. + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +25 +42 +Klaus Jansen and Lars Rohwedder. On integer programming, discrepancy, and convolution. +arXiv preprint arXiv:1803.04744, 2018. +43 +Ce Jin. An improved FPTAS for 0-1 knapsack. In ICALP, volume 132, pages 76:1–76:14, +2019. +44 +George Karypis and Vipin Kumar. Metis: A software package for partitioning unstructured +graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices. 1997. +45 +Donald E. Knuth. Optimum binary search trees. Acta Informatica, 1:14–25, 1971. +46 +Soh Kumabe and Yuichi Yoshida. Average sensitivity of dynamic programming. In SODA, +pages 1925–1961, 2022. +47 +Eduardo S Laber, Wilfredo Bardales, and Ferdinando Cicalese. On lower bounds for the +maximum consecutive subsums problem and the (min,+)-convolution. In ISIT, pages 1807– +1811. IEEE, 2014. +48 +Gary L. Miller, Richard Peng, Russell Schwartz, and Charalampos E. Tsourakakis. Approximate +dynamic programming using halfspace queries and multiscale monge decomposition. In SODA, +pages 1675–1682, 2011. +49 +Gaspard Monge. Mémoire sur la théorie des déblais et des remblais. Mem. Math. Phys. Acad. +Royale Sci., pages 666–704, 1781. +50 +Marcin Mucha, Karol Węgrzycki, and Michał Włodarczyk. A subquadratic approximation +scheme for partition. In SODA, pages 70–88. SIAM, 2019. +51 +Richard Peng. Approximate undirected maximum flows in O(mpolylog(n)) time. In SODA, +pages 1862–1867, 2016. +52 +Harald Räcke. Minimizing congestion in general networks. In FOCS, pages 43–52, 2002. +53 +Harald Räcke and Chintan Shah. Improved guarantees for tree cut sparsifiers. In ESA, pages +774–785, 2014. +54 +Harald Räcke, Chintan Shah, and Hanjo Täubig. Computing cut-based hierarchical decompo- +sitions in almost linear time. In SODA, pages 227–238, 2014. +55 +Peter Sanders and Christian Schulz. Think locally, act globally: Highly balanced graph +partitioning. In SEA, volume 7933, pages 164–175, 2013. +56 +Hiroshi Tamura, Masakazu Sengoku, Shoji Shinoda, and Takeo Abe. Location problems on +undirected flow networks. IEICE TRANSACTIONS (1976-1990), 73(12):1989–1993, 1990. +57 +Hiroshi Tamura, Masakazu Sengoku, Shoji Shinoda, and Takeo Abe. Some covering problems +in location theory on flow networks. IEICE Transactions on Fundamentals of Electronics, +Communications and Computer Sciences, 75(6):678–684, 1992. +58 +Godfried Toussaint. The geometry of musical rhythm. In Japanese Conference on Discrete +and Computational Geometry, pages 198–212. Springer, 2004. +59 +Godfried T Toussaint et al. A comparison of rhythmic similarity measures. In ISMIR, 2004. +60 +Nithin Varma and Yuichi Yoshida. Average sensitivity of graph algorithms. In SODA, pages +684–703, 2021. +61 +R. Ryan Williams. Faster all-pairs shortest paths via circuit complexity. SIAM J. Comput., +47(5):1965–1985, 2018. +62 +Virginia Vassilevska Williams. On some fine-grained questions in algorithms and complexity. +In Proceedings of the ICM, volume 3, pages 3431–3472. World Scientific, 2018. +63 +Andrew Chi-Chih Yao. Probabilistic Computations: Toward a Unified Measure of Complexity. +In FOCS, pages 222–227, 1977. +64 +F. Frances Yao. Efficient dynamic programming using quadrangle inequalities. In STOC, +pages 429–435, 1980. + +26 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +A +Organization of the Appendix +Our appendix is organized as follows: +In Appendix B we discuss more related work. +Appendix C introduces preliminaries. +Appendix D presents our results for k-balanced partitioning. +Appendix E presents our results for simultaneous source location. +Appendix F presents our recourse lower bounds for algorithms which only maintain few +solutions. +Appendix G presents our generalization to functions with non-monotonicities and our +results for ℓ∞-necklace. +Appendix H presents missing proofs. +B +Further Related Work +Speeding up DP algorithms is a well-studied topic, which has received attention for several +decades [2,12,19,22,31,35,45,48,49,64]. This line of work has led to several conditions, which, +if satisfied, imply that the underlying DP can be solved more efficiently. These conditions +include, for example, the Monge property, total monotonicity, certain convexity and concavity +properties, or the Knuth–Yao quadrangle-inequality, which are often related to each other. +For example, it is known that DP tables which satisfy the Monge property are also totally +monotone. One of the most popular methods in this area is the SMAWK algorithm [2] which +runs in near-linear time in the number of columns of the DP table if the DP table is totally +monotone. More concretely, a DP table is totally monotone if for each submatrix A of the +DP table and for every pair of consecutive rows i and i + 1 in A, the minimum entry for +row i + 1 appears in a column that is equal to or greater than the minimum entry for row i. +However, these conditions are quite different from our conditions in Definition 8 and +they are essentially incomparable. For the purpose of illustration, we will briefly argue this +for total monotonicity and Definition 8; similar arguments can also be made for the Monge +property and other criteria. On one hand, the totally monotone matrices do not imply that +the rows of the DP table are monotone. Indeed, when the rows are monotone then finding +the columns with the minimum entries is trivial (they are always in the first or last column, +depending on whether we consider monotonically increasing or decreasing rows, respectively). +Hence, total monotonicity does not imply our condition from Definition 8. On the other +hand, the ordering of the rows is highly important for the conditions above: just swapping +two rows of a totally monotone DP table can break total monotonicity. In our case, the +rows can be ordered arbitrarily in the DP table, as long as their dependency graph has good +properties. Hence, our property does not imply total monotonicity. This shows that these +definitions are incomparable. +Recently, Varma and Yoshida [60] and Kumabe and Yoshida [46] studied the sensitivity +of graph algorithms and of DP algorithms. They studied how much the solutions of such +algorithms change when a random element from the input is deleted. For several problems +including knapsack they showed that these algorithms have small sensitivity. However, we +show in Section F that when insertions are allowed, dynamic algorithms must have high +recourse or they have to maintain many different solutions. +The k-balanced graph partitioning problem has received a lot of attention in the theory +community [5,32–34]. The problem is also highly relevant in practice [18,28,44,55], where +algorithms for balanced graph partitioning are often used as a preprocessing step for large +scale data analytics. +For the special case of k = 2, this corresponds to the minimum + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +27 +bisection problem and Feige and Krauthgamer [33] presented polynomial-time algorithms +with polylogarithmic approximation ratios. For k ≥ 3, Andreev and Räcke [5] showed +that no polynomial-time algorithm can achieve a finite approximation ratio unless P = NP. +They also showed how to compute a bicriteria (O(log1.5(n)/ϵ2), 1 + ϵ)-approximate solution +in polynomial time. +Feldmann and Foschini [34] obtained a polynomial-time bicriteria +(O(log1.5(n) log log n), 1 + ϵ)-approximation algorithm which has the advantage that the +approximation ratio does not depend on the parameter ϵ of the partition sizes. Even et +al. [32] showed that one can compute a bicriteria (O(log n), 2)-approximation in polynomial +time. +The simultaneous source location problem that we study is closely related to the source +location problem introduced by Tamura et al. [56,57], in which a minimum number of sources +must be selected to be able to satisfy any single demand in an undirected edge-capacitated +graph. Arata et al. [7] showed that the problem is NP-hard and presented an exact algorithm +for the variant with uniform vertex costs. In the simultaneous source location problem that +was introduced by Andreev et al. [5] and that we study in this paper, all demands must be +satisfied simultaneously. Andreev et al. provide an O(log D)-approximation algorithm, where +D is the sum of demands, and a matching hardness result for this problem in general graphs. +They also present an exact polynomial-time algorithm when the input graph is a tree and +show that this result can be extended to general graphs when the edge capacities can be +violated by a O(log2 n log log n)-factor, where n is the number of vertices in the graph. +Chan [21] showed that one can consider the solutions for the 0-1 knapsack as monotone +piecewise constant functions and used this insight to obtain faster algorithms. Recently, these +results were improved by Jin [43] who showed how to compute a (1+ϵ)-approximation for 0-1 +knapsack with n items in time ˜O(n + ϵ−9/4). Bringmann and Cassis [15] derived faster exact +algorithms for 0-1 knapsack using bounded monotone min-plus-convolution. Aouad and +Segev [6] study the incremental knapsack problem, where the capacity constraint is increased +over time and the goal is to find nested subsets of items which maximize the average profit; +we note that this is different from our setting, where the goal is to obtain efficient update +times, while the solutions may change arbitrarily over time. +An ℓ1-necklace alignment problem was first considered by Toussaint [58], motivated by +computational music theory and rhythmic similarity [59]. Toussaint focused on a scenario +where the beads lie at integer coordinates. Ardila et al. [8] studied the problem for binary +strings. There also exist results for different distance measures between two sets of points on +the real line in which not every points needs to be matched [25], as well as for computing +the similarity of two melodies when they are represented as closed orthogonal chains on a +cylinder [3]. Bremner et al. [14] showed that ℓ2-necklace alignment can be solved in time +O(n log n), where n is the number of beads, using FFT. They also showed that ℓ∞-necklace +alignment can be solved using a constant number of (min, +)-operations and obtained +subquadratic-time algorithms for ℓ1- and ℓ∞-necklace alignment. +A common subroutine that is employed when solving DPs is (min, +)-convolution; note +that this subroutine is also of high importance in all of our algorithms. The complexity of +(min, +) convolution has received significant attention in the literature [9–11,14,17,20,23,24, +27,41,42,47,50]. It was shown that naive algorithm with running time O(n2) can be improved +to time n2/2Ω(√ +log n) [14,61] by a reduction to All Pairs Shortest Path [14] using Williams’ +algorithm for the latter [14]. However, so far, no O(n2−ϵ)-time algorithm was found, which +led to the MinConv hardness conjecture in fine-grained complexity theory [27, 41]. The +conjecture is particularly appealing because it implies other conjectures such as the 3-SUM +and the All-Pairs Shortest Paths conjectures, and dozens of lower bounds that follow from + +28 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +them (see [27,62]). There further exist many conditional lower bounds from the MinConv +conjecture and several MinConv-equivalent problems are known, e.g., related to the knapsack +problem or to subadditive sequences [27,41], among others [1,10,27,30,41,42,47,50]. There +have also been improvements for efficiently approximating the (min, +)-convolution in the +case of large weights [17] for the exact (min, +)-matrix product with bounded differences [16]. +C +Preliminaries +We introduce some preliminaries that we will use in the rest of the paper. For the sake of +better readability, we present some of the proofs in Appendix H. We write [m] to denote the +set {0, 1, . . . , m}. +Throughout the paper, we will consider input graphs G = (VG, EG, capG) with n vertices +and m edges, where capG : EG → W∞ ∪ {∞} is a weight function that for an edge e ∈ EG +describes the capacity of the edge. To simplify notation we extend capG to all vertex pairs +and define +capG(x, y) = +� capG({x, y}) +{x, y} ∈ EG +0 +otherwise. +. +Additionally, for disjoint sets A, B ⊆ VG, we set capG(A, B) := � +(a,b)∈A×B capG(a, b) and +capG(A) := capG(A, V \ A). We drop the subscript G of the capacity function cap whenever +the graph is clear from the context. +Let (VT , ET , r) be a rooted tree. For a vertex v ∈ VT we use Tv to denote the subtree +rooted at v and we say that the degree of v is its number of children. The height h of T is +the length of the longest path from the root to a leaf. +C.1 +Räcke Tree +A Räcke tree [52] (or tree cut sparsifier) T = (VT , ET ) for an undirected graph G = (VG, EG) +is a weighted, rooted tree in which the leaf nodes correspond to vertices of G. For a vertex +v ∈ VT , we write Vv ⊆ VG to denote the set of leaf vertices in Tv. Naturally, an edge e = (u, v) +of T corresponds to a cut in G, namely to the cut formed by the set Vu ∩ Vv in G. The +capacity capT of the tree edge (u, v) is set to the capacity of this cut, i.e., to capG(Vu ∩ Vv). +For a graph H = (VH, EH) and two disjoint subsets A, B ⊆ VH, we write +mincutH(A, B) := +min +S⊆VH:A⊆S,B⊆ ¯S capH(S) +to denote the minimum capacity of a cut that separates A and B. By definition of the +edge capacities in T we have mincutT (A, B) ≥ mincutG(A, B) for any two disjoint subsets +A, B ∈ VG. For the sake of completeness, we prove this property in Appendix H.5. +The goal of a Räcke tree T is to approximate the cut-structure of G, i.e., to guarantee +that for all disjoint sets of vertices A, B ⊆ VG, +mincutG(A, B) ≤ mincutT (A, B) ≤ q · mincutG(A, B) , +for a small value q ≥ 1. The parameter q is called the quality of the Räcke tree. +In the static setting, Räcke trees with polylogarithmic quality guarantees can be computed +in nearly linear time [51,54]. When larger running times are allowed, better qualities can be +achieved [13,37,53]. + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +29 +▶ Theorem 15 (Peng [51]). Let G be a connected undirected graph with n vertices and +m edges. Then there exist an algorithm that computes a Räcke tree of height O(log n) for G +with quality O(log4 n) in time ˜O(m). +Furthermore, there has recently been interest in maintaining Räcke trees dynamically [36, +40]. Here, we will use a result by Goranci, Räcke, Saranurak and Tan who showed that one +can maintain Räcke trees for unweighted graphs dynamically with subpolynomial update +time. +▶ Theorem 16 (Goranci, Räcke, Saranurak and Tan [36]). Let G be an undirected, unweighted +graph with n vertices that is undergoing edge insertions and deletions. +There exists a +deterministic algorithm with amortized update time no(1) that maintains a Räcke tree for G +with quality no(1) and height O(log1/6 n). +C.2 +Okay-Behaved DPs +We introduce a more general DP condition compared to the one in Definition 8 which, +however, will not allow us to obtain results like Theorems 9 or 10. We will consider the same +type of DP tables as in Section 2. +▶ Definition 17. A DP is okay-behaved if it fulfills the sensitivity condition of well-behaved +DPs: Suppose β > 1 and for all i′ ∈ In(i), we obtain a β-approximation ADP(i′, ·) of DP(i′, ·) +(as per Equation (1)). Then applying Pi on the ADP(i′, ·) yields a β-approximation of DP(i, ·), +i.e., +DP(i, ·) ≤ Pi({ADP(i′, ·): i′ ∈ In(i)}) ≤ β · DP(i, ·). +We also use routines ˜Pi to compute the DP rows ADP(i, ·). Again, if for all i it holds +that ˜Pi({ADP(i′, ·): i′ ∈ In(i)}) is an α-approximation of Pi({ADP(i′, ·): i′ ∈ In(i)}), we say +that ADP(1, ·), . . . , ADP(n, ·) is an α-approximate DP solution. +In the dependency graph, we call a vertex without any incoming edges a leaf. The level +of a vertex u is the length of the longest path from a leaf to u. Similar to the proof of +Theorem 9 we can show the following approximation guarantee for the approximate solutions +ADP(i, ·) and the exact solutions DP(i, ·). +▶ Lemma 18. Let i be a vertex of the dependency graph with level ℓ. Then the entry ADP(i, ·) +in the α-approximate ADP-solution for a okay-behaved DP problem fulfills +DP(i, ·) ≤ ADP(i, ·) ≤ αℓ+1 · DP(i, ·). +Next, suppose the dependency graph of the DP that we consider is derived from a tree as +follows. Let T = (VT , ET , r) be a rooted tree with root r and height h. We assume that the +children of a vertex are ordered from left to right. The dependency graph that we associate +with T is simply a directed copy of T in which we direct each edge towards the root. More +precisely, the dependency graph contains copies of all vertices in VT and for each vertex v +(except for r) an edge to its parent p. Clearly, this set of edges induces a DAG in which +the longest path has at most h edges. The following lemma summarizes the properties of +approximate DP solutions when using this approach. +▶ Lemma 19. Consider a rooted tree T = (VT , ET , r) with height h. Consider an okay- +behaved DP and the ADP-solution ADP(i, ·) corresponding to the dependency graph described +above. Assume that each ˜Pi is an α-approximation of Pi and can be computed in time at +most t. Then ADP(r, ·) is an αh+1-approximation of DP(r, ·) and can be computed in time +O(|VT | · t). + +30 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +The main difference of this lemma together with the definition of okay-behaved DPs and +Theorem 9 with well-behaved DPs is as follows. When applying Theorem 9, we only have to +consider how many pieces our functions have and we do not have to bother about deriving +running times bound for computing the operations on our functions (because the additional +conditions from the well-behaved DPs imply good running time bounds). Here, we have +to check less conditions for okay-behaved DPs (in particular, we do not have to bound the +number of pieces or operations) but we have to provide our own running time analysis. +Later, when we consider dynamic algorithms, we will have to consider the scenario when +the underlying tree T changes due to edge insertions and deletions (and therefore might +become a forest). In that case, the dependency graph and the DP solutions DP(i, ·) and +ADP(i, ·) change over time as well. The following lemma asserts that when a vertex i is +affected by an edge insertion or deletion, we only have to recompute the solutions DP(j, ·) +and ADP(j, ·) for vertices j that are reachable from i in the dependency graph and that there +are at most h such vertices. +▶ Lemma 20. Consider a rooted tree T = (VT , ET , r) with height h that is undergoing +edge insertions and deletions. Then after each insertion or deletion, we can recompute an +ADP-solution with the same guarantees as in Lemma 19 in time O(h · t), where t is the time +it takes to compute the functions ˜Pi. +Lemma 6 already provided a way to compute the minimum of two monotone piecewise +constant functions. When more than two functions are involved in the minimum computation, +the following version gives improved guarantees. +▶ Lemma 21. Let fi : [0, t] → W∞, i ∈ {1, . . . , k} be piecewise constant functions that +are either all monotonically increasing or all monotonically decreasing. Then fmin(x) := +mini{fi(x)} can be computed in time O(� +i pi · log(� +i pi)), where pi denotes the number of +pieces of function fi. +We also note the following well-known lemma for sake of completeness. +▶ Lemma 22. Let f1, f2 : [0, t] → W∞ and suppose that one of f1 and f2 is monotonically +decreasing. Then f = f1 ⊕ f2 is monotonically decreasing. +D +Balanced Graph Partitioning +In this section, we provide an algorithm for the k-balanced graph partitioning problem. In +this problem, the input consists of a graph G = (V, E, cap), where cap : E → W∞ is a weight +function on the edges, and an integer k. The goal is to find a partition V1, . . . , Vk of the +vertices such that |Vi| ≤ ⌈|V | /k⌉ for all i and the weight of the edges which are cut by the +partition is minimized. More formally, we want to minimize cut(V1, . . . , Vk) := � +i cap(Vi), +where cap(Vi) = � +{u,v}∈E∩(Vi,V \Vi) cap(u, v). +Since the above problem is NP-hard to approximate within any factor n1−ϵ for any ϵ +even on trees [34], we consider bicriteria approximation algorithms. Given a weighted graph +G = (V, E, cap), we say that a partition V1, . . . , Vk of V is an (α, β)-approximate solution +if |Vi| ≤ β⌈n/k⌉ for all i and cut(V1, . . . , Vk) ≤ α · cut(OPT), where OPT = (V ∗ +1 , . . . , V ∗ +k ) is +the optimal solution with |V ∗ +i | ≤ ⌈n/k⌉ for all i. +Our first main result in this section is summarized in the following theorem. We use the +notation O′(·) to suppress factors in poly(log n, k, log(1/ϵ), log log(W)). + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +31 +▶ Theorem 2. Let ϵ > 0 and k ∈ N. Let G = (V, E, cap) be an undirected weighted graph +with n vertices and m edges and edge weights in W∞. Then for the k-balanced partition +problem we can compute: +An (O(log4 n), 1+ϵ)-approximation in time (k/ϵ)O(log(1/ϵ)/ϵ)·O′(m·log2(W))+(k/ϵ)O(1/ϵ2).9 +A (1 + ϵ, 1 + ϵ)-approximation in time (k/ϵ)O(log(1/ϵ)/ϵ) · O′(n · h2 · log2(W)) + (k/ϵ)O(1/ϵ2) +if G is a tree of height h. +A (1, 1 + ϵ)-approximation in time (k/ϵ)O(log(1/ϵ)/ϵ) · O′(n4 · log2(W)) + (k/ϵ)O(1/ϵ2) if G +is a tree. +Furthermore, we can also extend our results to the dynamic setting in which the graph +G is undergoing edge insertions and deletions. Our second main result in this section is +summarized in the following theorem. +▶ Theorem 3. Let ϵ > 0 and k ∈ N. Let G = (V, E, cap) be an undirected weighted graph +with n vertices that is undergoing edge insertions and deletions. Then for the k-balanced +partition problem we can maintain: +An (no(1), 1 + ϵ)-approximate solution with amortized update time (k/ϵ)O(log(1/ϵ)/ϵ) · no(1) · +O′(log2(W)) and query time (k/ϵ)O(1/ϵ2) if G is unweighted. +A (1+ϵ, 1+ϵ)-approximate solution with worst-case update time (k/ϵ)O(log(1/ϵ)/ϵ) ·O′(h3 · +log2(W)) and query time (k/ϵ)O(1/ϵ2) if G is a tree of height h. +Our DP approach is inspired by the DP of Feldmann and Foschini [34]. However, the DP +cells in the algorithm of Feldmann and Foschini are not monotone and, therefore, their DP +cannot directly be sped up by the fast convolution of monotone functions approach. Hence, +we first simplify and generalize their DP to make it monotone such that we can apply the +fast convolution of monotone functions approach. +We note that in our static and dynamic algorithms, we can output the corresponding +solutions similarly to what we descriped after Proposition 12 for knapsack. +To obtain these results, we will first describe an exact DP in Section D.1 for the special +case of binary trees. Then we will show how to compute the DP more efficiently by introducing +approximation in Section D.2. In Section D.3 we show how to return a solution based on our +DP table. Sections D.4 and D.5 provide extensions from binary trees to more general graphs +and to the dynamic setting, respectively. +D.1 +The Exact DP +When describing the DP, we will make two assumptions. First, we assume that the input +graph T = (V, E) is a binary tree (we show in Section D.4 how to remove this assumption). +Second, we consider a slight generalization of the k-balanced partition problem on trees; +we note that we did not mention this generalization in Section 4. In this generalization, +we suppose that each vertex is assigned a weight by a weight function w: V → {0, 1}.10 +For convenience we set w(U) = � +u∈U w(u) for all U ⊆ V and refer to w(U) as the weight +of the vertices in U. Now our goal will be to find a partition V1, . . . , Vk of V such that +w(Vi) ≤ (1 + ϵ)⌈w(V )/k⌉ for all i and we will compare against OPT = (V ∗ +1 , . . . , V ∗ +k ), where +OPT is the optimal solution with w(V ∗ +i ) ≤ ⌈w(V )/k⌉ for all i. Note that by setting w(v) = 1 +9 We use the notation O′(·) to suppress factors in poly(log n, k, log(1/ϵ), log log(W)). +10 We note that our proofs and algorithms also work for more general weight functions w: V → R+. +However, in that case the functions DP(v, g, cut, ·) that we will introduce later will become more +complicated to compute and, therefore, we stick with the simpler case of vertex weights in {0, 1}. + +32 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +for all v ∈ V , we obtain the standard k-balanced partition problem and, therefore, our variant +is a strict generalization. +The reason for considering the above generalization is that later we want to use our +algorithm to find a balanced partitioning of general graphs G = (V ′, E′) using a Räcke +tree T = (V, E) (see Section C.1). However, the vertices V ′ of G are just a subset of the +vertices V of the Räcke tree T (since the vertices of G correspond to leaves in T and the +internal nodes of T do not correspond to any vertices in G). Thus, if we assigned weight +w(v) = 1 to all vertices in T and computed a balanced partitioning of T, this would not +necessarily correspond to a balanced partitioning of G. Instead, later we will consider the +weight function which assigns weight 1 to all leaves in T (corresponding to the vertices in G) +and weight 0 to all internal nodes of T (which can be ignored when deriving a partitioning of +G). Then each set Vi in T will correspond to a set V ′ +i in G with w(Vi) = |V ′ +i |. In particular, +if w(Vi) ≤ (1 + ϵ)⌈w(V )/k⌉ then we will obtain that |V ′ +i | ≤ (1 + ϵ)⌈|V ′| /k⌉ and, therefore, +the sets V1, . . . , Vk imply a balanced partition V ′ +1, . . . , V ′ +k of G. +High-Level Description of the DP. We start by giving a high-level description of the +DP. The DP is computed bottom-up starting at the leaves of the tree G and then moving up. +For each vertex v, we will compute a DP solution of minimum cost that encodes whether +the edge to the parent p of v is cut and which edges shall be cut inside the subtree Tv +that is rooted at v. Note that the removal of the cut edges in our solution will decompose +the tree into disjoint connected components and exactly one of them contains v’s parent p. +Additionally, we store information about the weight of the vertices that are still connected to +the parent p (and, therefore, to the outside of Tv) after the cut edges are removed. We will +assume that when we compute the DP cell for a vertex v, we have access to the solutions for +both of its children. +More concretely, when we have computed a solution for a subtree Tv, i.e., we know which +edges incident to nodes in this subtree we are going to remove (note that the edge leading to +the parent of v is incident to Tv and thus we consider it as part of this solution), we store +the following information in the DP table. First, we store its cost, i.e., the total capacity of +all edges that are incident to vertices in Tv and that are cut. As described above, we would +also like to store the weight of the vertices that are connected to the parent of v and the +sizes of connected components inside Tv. However, there are two difficulties: (1) We cannot +store the weight of the vertices that are connected to the root exactly because this would +result in a too large DP table. Instead, we store the cheapest solution in which vertices of at +most some given weight are still connected to the parent of v. As we will see, this approach +gives rise to monotonically decreasing functions and allows for a very efficient computation of +the DP table. (2) We store implicitly the size of all connected components that are created +after the cut edges are removed and that lie completely inside Tv. As before, storing these +sizes exactly would result in a very large DP table and, therefore, we store them concisely +using the concept of a signature. The signatures will help us to characterize the sizes of the +components inside Tv very efficiently. +Signatures. We call a connected component in Tv large if it contains vertices of total +weight at least ϵ⌈w(V )/k⌉ and otherwise we call it small. Let t = ⌈log1+ϵ(1/ϵ)⌉ + 1, and let +M = ⌈k/ϵ⌉ + 1. A signature is a vector g = (g0, . . . , gt−1) ∈ [M − 1]t. Observe that each gi +is an integer between 0 and M − 1 and hence there are M t = (k/ϵ)O(ϵ−1 log(1/ϵ)) different +signatures. Intuitively, an entry gi in g tells us roughly how many components of weight +(1 + ϵ)i · ϵ⌈w(V )/k⌉ there are in the DP solutions that we consider. The precise definition is +as follows. +Let S = {S1, . . . , Sr} be a set of connected components inside Tv (e.g., think of S as the + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +33 +components that are created after removing the cut edges in the DP solution for vertex v). +We say that a signature vector g = (g0, . . . , gt−1) ∈ [M − 1]t is consistent for S if we can +match the connected components in S to entries in g as follows. For each large component +Sj we let ℓ(Sj) = arg min{i ∈ [t]: w(Sj) ≤ (1 + ϵ)i · ϵ⌈w(V )/k⌉}, i.e., ℓ(Sj) is the smallest +number i such that Sj has weight at most (1 + ϵ)i · ϵ⌈w(V )/k⌉. Let si ∈ [M − 1] denote the +number of times the value i ∈ [t] has been chosen in this process, i.e., si = |{j : ℓ(Sj) = i}|, +and let s = (s0, . . . , st−1) denote the resulting vector. We say that g is consistent with the +set of components S if g = s. Thus, the above matching process can be viewed as rounding +up the component sizes and counting the number of components of each size. +For x ∈ N, we let e(x) ∈ [M − 1]t denote the signature of a single component with total +weight x. More precisely, we set e(x) to the vector that has e(x)j = 1 for j = arg min{j ∈ +N: x ≤ (1 + ϵ)j · ϵ⌈w(V )/k⌉} and e(x)j = 0, otherwise. If x < ϵ⌈w(V )/k⌉, we define e(x) = ⃗0. +D.1.1 +DP Definition +Now we describe the DP formally. An entry DP(v, g, cut, x) ∈ W∞ in the DP table for a vertex +v is indexed by a signature g, a Boolean value cut and x ∈ [n]. We will consider the tuples +(v, g, cut) as the rows I of the DP table and x as the columns; we associate each such row with +a function DP(v, g, cut, ·): [n] → W∞. Note that our DP has |V |·M t·2 = (k/ϵ)O(ϵ−1 log(1/ϵ))·n +rows. Also, note that it has columns n; later, even though x only takes discrete values, we +will allow x to take values in [0, ∞). +It describes the optimum cost of cutting edges incident on the subtree Tv (including the +cost of maybe cutting the edge to the parent of v). We will refer to the set of vertices in +Tv that are still connected to the parent of v after the cut edges are removed as the root +component. We impose the following conditions on DP(v, g, cut, x): +Once the cut edges are removed, the root component U ⊆ Tv has total weight at most x, +i.e., w(U) ≤ x. +If cut is set to true then the edge between v and its parent is cut, otherwise it is kept. +The vertices inside Tv that (once the cut edges are removed) are not connected to the +parent of v form connected components that are consistent with the signature g. +We observe that if we fix a vertex v, a signature g and a value for cut, then the resulting +function DP(v, g, cut, ·) is monotonically decreasing in x. This will be the crucial property +for the rest of the section. +▶ Observation 23. Let v ∈ V , g ∈ [M − 1]t be a signature and cut ∈ {true, false}. Then the +function DP(v, g, cut, ·) : [0, ∞) → R+ is monotonically decreasing. +Proof. By definition, DP(v, g, cut, x) stores the cost of the optimum solution in which +the vertices in the root component have weight at most x. Now observe that for x ≤ +x′, the solution DP(v, g, cut, x) is also a feasible solution for DP(v, g, cut, x′). Therefore, +DP(v, g, cut, ·) must be monotonically decreasing. +◀ +Since the DP cells are monotonically decreasing in x, we will use the shorthand notation +DP(v, g, cut, ∞) to denote the solution minx DP(v, g, cut, x). Note that this minimum is +obtained for the largest x-value at which DP(v, g, cut, ·) changes. +D.1.2 +Computing the DP +In the following, we describe how to compute DP(v, ·, ·, ·) exactly. For computing DP(v, ·, ·, ·) +we simply iterate over all possible choices of x, g and cut. Note that since each vertex has + +34 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +weight in {0, 1}, the function DP(v, g, cut, ·) only changes for x ∈ [n + 1] (i.e., when x is an +integer). Thus, we only need to consider n + 1 choices for x. We conclude that to compute +DP(v, ·, ·, ·) for a fixed vertex v, there are O(M t ·n) parameter choices that we need to iterate +over. +In our descriptions we use p to denote the parent of v, and vl and vr to denote v’s left +and right child, respectively, if these exist. +Case 1: v is a leaf. If we cut the edge to the parent of v, then the cost is cap(v, p), +there are no vertices in the root component and v forms its own connected component with +signature e(w(v)). Thus, we set DP(v, e(w(v)), true, x) = cap(v, p) for all x ∈ [0, ∞) and we +set DP(v, g, true, x) = ∞ for all x ∈ [0, ∞) and for all signatures g ̸= e(w(v)). +Now suppose we do not cut the edge (v, p) to the parent of v. Then we do not have to +pay any cost since we are not cutting any edge, the weight of vertices in the root component +is w(v) and the signature is g = 0 since there are no connected components in Tv that are +not connected to p. Therefore, for all x ∈ [0, w(v)) we set DP(v, 0, false, x) = ∞ and for all +x ∈ [w(v), ∞) we set DP(v, 0, false, x) = 0. For all signatures g ̸= 0 and all x ∈ [0, ∞), we +set DP(v, g, false, x) = ∞. +Case 2: v is not a leaf. If v is not a leaf then we assume that it has exactly two children +vl and vr (if it has only one child, we can add a second child v′ with w(v′) = 0, cap(v, v′) = 0 +and then v′ has no impact on the solution). We assume that for both vl and vr, we have +already computed the solutions DP(vl, g, cut, x) and DP(vr, g, cut, x) for all possible values +of x, g and cut. +Let el = (v, vl) and er = (v, vr) denote the edges to the respective child and let ep = (p, v) +denote the edge to the parent p of v. In the following we distinguish four cases (A, B, +C, D) depending on which of these edges we decide to cut. For each case, we compute +DPcase(v, g, cut, x)-values, case ∈ {A, B, C, D}, which are the optimum values under the +condition that we cut el and er according to the case. The final entry DP(v, g, cut, x) is then +obtained by minimizing over all cases, i.e., by setting +DP(v, g, cut, x) = +min +case∈{A,B,C,D} DPcase(v, g, cut, x) +for all x, g, cut. +Case A: cut el and er. Suppose we cut el and er. Then, given x and g, we have to select +subsolutions for the left and right sub-tree such that the weight of vertices that can reach p +is at most x and the connected components inside are consistent with g. +First, assume we cut the edge ep. +Then the cost for cutting this edge is cap(v, p). +Furthermore, the weight of vertices inside Tv that can reach p is zero and, hence, the value +of x is irrelevant by the monotonicity of DP(v, g, cut, ·). Next, if we have a solution with +signatures gl and gr in the left and right subtree, respectively, we can combine these solutions +as long as gl + gr + e(w(v)) = g (as the vertex v forms a single component of weight w(v) +since we cut both edges el and er). Note that in the subsolution for the child vl, the value +of x does not play a role for the feasibility of the solution DP(v, g, cut, x) since the size of +the root component in Tvl is already encoded in gl. Therefore, to obtain minimum cost we +consider DP(vl, gl, true, ∞); by symmetry, the same holds for vr. Therefore, we set for all +x ∈ [0, ∞), +DPA(v, g, true, x) = cap(v, p)+ +min +gl+gr=g−e(w(v)){DP(vl, gl, true, ∞)+DP(vr, gr, true, ∞)}. (5) +Second, assume we do not cut the edge to the parent p. Then there will be at least one +vertex (namely v) that can reach p. Hence, DPA(v, g, false, x) = ∞ for all signatures g and + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +35 +x ∈ [0, w(v)). For x ∈ [w(v), ∞), we can combine the solutions as above and we set +DPA(v, g, false, x) = +min +gl+gr=g{DP(vl, gl, true, ∞) + DP(vr, gr, true, ∞)}. +(6) +Case B: cut neither el nor er. Next, suppose we cut neither el nor er. In this case we +have to select subsolutions for Tvl and Tvr, where each subsolution is characterized by the +upper bound xl (resp. xr) and its signature gl (resp. gr). +First, suppose that we cut the edge ep. If we let xl and xr denote the exact weight of +the root components for the subsolutions, then the vertex v will be included in a component +of size xl + xr + w(v) afterwards. Hence, we can combine the subsolutions to a solution +for signature g as long as gl + gr + e(xl + xr + w(v)) = g. Consequently we set for every +x ∈ [0, ∞), +DPB(v, g, true, x) = +cap(v, p) + +min +xl,xr,gl+gr=g−e(xl+xr+w(v)) DP(vl, gl, false, xl) + DP(vr, gr, false, xr). +Second, suppose that we do not cut ep. Then again we have to set DPB(v, g, false, x) = ∞ +for all signatures g and all x ∈ [0, w(v)), because the vertex v of weight w(v) can reach p. For +x ≥ w(v) we have to select xl and xr such that they sum to x − w(v) as this guarantees that +vertices of weight at most x can reach the parent p. Consequently, we set for all x ∈ [w(v), ∞) +DPB(v, g, false, x) = +min +gl+gr=g,xl+xr=x−w(v) DP(vl, gl, false, xl) + DP(vr, gr, false, xr). +Case C: cut el but not er. Now suppose we cut the edge to the left child vl but we do not +cut the edge to the right child vr. In this case, v stays connected to the root component of +vr and we need to choose a subsolution with parameters xr and gr for Tvr and a subsolution +with parameter gl for Tvl. Note that since we cut el, the upper bound on the weight of the +root component of vl is irrelevant as this is implicitly encoded in gl. +First, suppose we cut ep. If we let xr denote the exact weight of the root component for the +subsolution in Tvr then v will be included in a component of size xr +w(v) afterwards. Hence, +we can combine the subsolutions to a solution for signature g as long as gl+gr+e(xr+w(v)) = +g. Consequently, for every x ∈ [0, ∞) we set +DPC(v, g, true, x) = cap(v, p)+ +min +xr,gl+gr=g−e(xr+w(v)) DP(vl, gl, true, ∞)+DP(vr, gr, false, xr). +(7) +Second, suppose we do not cut ep. Then we have to set DPC(v, g, false, x) = ∞ for all +signatures g and all x ∈ [0, w(v)), because vertex v with weight w(v) can reach p. For +x ∈ [w(v), ∞), we have to select xr ≤ x−w(v) as this guarantees that vertices of total weight +at most x can reach the parent p. Due to the monotonicity of DP(vr, gr, false, ·) we can just +choose xr = x − w(v). Consequently, for all x ∈ [w(v), ∞) we set +DPC(v, g, false, x) = +min +gl+gr=g,xr=x−w(v) DP(vl, gl, true, ∞) + DP(vr, gr, false, xr). +(8) +Case D: cut er but not el. Symmetric to Case C. +Next, we argue that this DP is okay-behaved, i.e., it satisfies Definition 17. In particular, +we note that this DP is not well-behaved because it does not satisfy Property (4b) of + +36 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +Definition 8 since in Case 2, Step B below we will have to perform too many min-operations +(see Equation (11)). We will also show that the DP’s dependency graph is exactly the input +tree and hence the conditions of Lemma 19 are satisfied. Furthermore, all entries for a DP +cell DP(v, ·, ·, ·) can be computed in time O(M 2tn3) by simply enumerating all choices in the +different min-operations above. +▶ Lemma 24. The DP is okay-behaved and the dependency tree and the input tree T are +identical. Furthermore, given a vertex v, we can compute all entries in DP(v, ·, ·, ·) in time +O(M 2tn3). +Proof. First, note that in the DP each cell DP(v, ·, ·, ·) only depends on the solutions of its +two children. Note that these are exactly the edges which are present in the dependency +graph and also in T. Therefore, the dependency graph and T are identical. Furthermore, +when the input for a child solution is a β-approximation, the output of the DP will also +be an β-approximation because we perform all computations exactly. Thus, the DP is also +okay-behaved. +Second, let us consider the running time. Recall that for fixed x, g and cut, we set +DP(v, g, cut, x) = mincase∈{A,B,C,D} DPcase(v, g, cut, x) and this quantity can be computed +in time O(1) by a simple table lookup. Thus, we only have to consider the time it takes to +compute DPcase(v, g, cut, x) for each case ∈ {A, B, C, D} and for fixed x, g and cut. +For Case A, observe the min-operations can be computed by iterating over all M t choices +of gl and setting gr = g − e(w(v)) − gl as long as gr is a non-negative vector. Then the +expressions inside the min-term can be computed by table lookup in constant time. Thus, +the time is O(M t). For Case B, in case cut = true note that we can iterate over all choices +of xl, xr and iterate over gl as described above. This takes time O(M tn2). In the case +cut = false we can again iterate over the gl as above and we can iterate over all xl ∈ [n + 1] +and set xr = x − w(v) − xl as long as xr ≥ 0; thus, the case can be solved in time O(M tn). +For Cases C and D, we can iterate over all choices of xr and then iterate over the gl as above. +This gives a total running time of O(M tn). +We conclude that for fixed x, g and cut, the time to compute DPcase(v, g, cut, x) for +all case ∈ {A, B, C, D} is O(M tn2). Since there are O(n) choices of x, M t choices for g +and two choices for cut, we conclude that the total running time to compute DP(v, ·, ·, ·) is +O(M 2tn3). +◀ +D.2 +The Approximate DP +In this section we show how to construct the approximate DP table in an efficient manner. +For this we essentially perform the same computations as above, but instead of computing +the exact solution DP(v, ·, ·, ·) by computing exact solutions to the cases DPcase(v, ·, ·, ·), we +compute an approximate solution ADP(v, ·, ·, ·) which will be the minimum of approximate +solutions ADPcase(v, ·, ·, ·), where case ∈ {A, B, C, D}. +However, there are a few crucial differences. +First, for fixed v, g, cut and case ∈ +{A, B, C, D}, we interpret ADPcase(v, g, cut, ·) as a piecewise constant function which is +stored in an efficient list representation (as per Section 2). After we computed the solutions +ADPcase(v, g, cut, ·), we compute the function +ADP(v, g, cut, ·) := +⌈min{ADPA(v, g, cut, ·), ADPB(v, g, cut, ·), ADPC(v, g, cut, ·), ADPD(v, g, cut, ·), }⌉1+δ, +(9) + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +37 +i.e., instead of just taking the minimum over the different cases, we also perform a rounding +step to multiples of 1 + δ. This rounding step introduces an approximation error of α = 1 + δ +but reduces the number of pieces within the piecewise constant function DP(v, g, cut, ·) to +p := O(log1+δ(W)) according to Lemma 6 (for this to work we need to guarantee that the +function to be rounded is monotone and therefore we will show that ADPcase(v, g, cut, ·) is +monotone for each case ∈ {A, B, C, D}). The second crucial difference is, of course, that +we perform the above computations with values that already have been rounded, i.e., with +entries from ADP instead of entries from DP. We note that Equation (9) is the only place in +the approximate DP which is not exact; all other computations are done precisely (without +any rounding) and, therefore, the approximate DP only loses a factor 1 + δ. +In order to guarantee a highly efficient implementation we rely on the following invariants +for entries in the approximate DP: +1. For all v, g, and cut, the function ADP(v, g, cut, ·) is monotonically decreasing. +2. For all v, g, and cut, the function ADP(v, g, cut, ·) is piecewise constant with at most +p := O(log1+δ(W)) pieces. +Note that the first property resembles the fact that for the exact DP, DP(v, g, cut, ·) is +monotonically decreasing as per Observation 23. However, here we state this property +as an invariant because there could exist approximations of DP(v, g, cut, ·) which are non- +monotone and, therefore, we need to prove that each of our functions ADP(v, g, cut, ·) is +indeed monotone. Note the second property follows immediately from the monotonicity and +the rounding step in Equation (9) and thus we will not need to prove it in the following. +Similar to the description of the exact DP, we will now go through each of the cases and, +given v, describe how to compute ADP(v, g, cut, ·) in time ˜O(1) for all g and cut. The cases +are exactly the same as for the exact DP and thus for the sake of brevity we do not repeat +the correctness argument. +Case 1: v is a leaf. Then, we do the same in the exact case. We set ADP(v, e(w(v)), true, x) = +cap(v, p) for all x ∈ [0, ∞) and we set ADP(v, g, true, x) = ∞ for all x ∈ [0, ∞) and all sig- +natures g ̸= e(w(v)). Furthermore, we set ADP(v, 0, false, x) = ∞ for all x ∈ [0, w(v)) +and ADP(v, 0, false, x) = 0 for all x ∈ [w(v), ∞). +For all signatures g ̸= 0 and all +x ∈ [0, w(v)), we set ADP(v, g, false, x) = ∞. Note that in all cases, the corresponding +functions ADP(v, g, cut, ·) are monotonically decreasing and have O(1) pieces. +Case 2: v is not a leaf. We distinguish the same four cases as for the exact DP. Again, +we will assume that v has exactly two children vl and vr and we let el = (v, vl), er = (v, vr) +and ep = (p, v), where p is the parent of v. +Case A: cut el and er. First, suppose we cut el and er. Then, as in the exact DP, if we +cut the edge to the parent of v, we wish to set +ADPA(v, g, true, x) = +cap(v, p) + +min +gl+gr=g−e(w(v)){ADP(vl, gl, true, ∞) + ADP(vr, gr, true, ∞)}. +for all x ∈ [0, ∞). Note that in the equation above, the quantities cap(v, p), ADP(vl, gl, true, ∞) +and ADP(vr, gr, true, ∞) are simply numbers and can be viewed as a piecewise constant +function with a single piece. Thus, ADPA(v, g, true, ·) is a piecewise constant function with +a single piece and, therefore, it is also monotonically decreasing. Hence, the invariants +are satisfied for ADPA(v, g, true, ·). Furthermore, ADPA(v, g, true, ·) can be computed via a +sum and a minimum over monotonically decreasing piecewise functions via Lemma 6. Note +that the minimum takes O(M t) different values because it is computed by iterating over all + +38 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +gl ∈ [M − 1]t and setting gr = g − e(w(v)) − gl as long as all entries in gr are non-negative. +Since each function ADP(vl, gl, true, ·) has O(p) pieces according to our invariants, we can +compute the value ADP(vl, gl, true, ∞) in time O(1); the same holds for ADP(vr, gl, true, ∞). +Thus, computing ADPA(v, g, true, ·) takes time O(M t). +Next, suppose we do not cut the edge to the parent of v. Then, as in the exact DP, we +wish to set: +ADPA(v, g, false, x) = +min +gl+gr=g{ADP(vl, gl, true, ∞) + ADP(vr, gr, true, ∞)} +for all x ∈ [0, ∞). Then by the same arguments as above, ADPA(v, g, false, ·) is a piecewise +constant monotonically decreasing function with a single piece. It can be computed in time +O(M t) as described above. +Case B: cut neither el nor er. Now suppose we do not cut any edge to the children. +If we do not cut the edge to the parent of v, we proceed similar to the exact DP. We start +by setting ADPB(v, g, false, x) = ∞ for all x ∈ [0, w(v)). Next, for x ∈ [w(v), ∞) we wish to +set +ADPB(v, g, false, x) = +min +gl+gr=g,xl+xr=x−w(v) ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr) += +min +gl+gr=g +min +xl+xr=x−w(v) ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr). +(10) +Note that for fixed gl and gr, the inner min-operation in the second line describes a (min, +)- +convolution due to the constraint xl+xr = x−w(v). Therefore, in the inner min-operation we +compute a convolution ADP(vl, gl, false, ·)⊕ADP(vr, gr, false, ·) and shift the result by w(v) via +the shift operation from Lemma 6 (where for x ∈ [0, w(v)) we set ADPB(v, g, false, x) = ∞). +We need time O(p2 log p) for computing the convolution according to Lemma 7. To compute +the outer minimum in Equation (10), we iterate over all gl ∈ [M −1]t and thus perform O(M t) +minimum computations over piecewise constant functions with at most p2 pieces. Hence, we +need time O(M tp2 log(M tp2)) according to Lemma 21. By Lemma 22, ADPB(v, g, false, ·) is +monotonically decreasing since it is the minimum over convolutions of two monotonically +decreasing functions. +If we cut the edge to the parent of v, then for all x ∈ [0, ∞) we would like to set +ADPB(v, g, true, x) = +cap(v, p) + +min +xl,xr,gl+gr=g−e(xl+xr+w(v)) ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr). +Note that here we need to be careful as the range of gl and gr depends on the choice of xl +xr. +Since there are Ω(n) possible values for xl +xr, we cannot afford to iterate over all values that +xl + xr can take. Instead, we will show that we only need to consider O(log(k/ϵ)/ϵ) different +pairs (xl, xr) by exploiting the monotonicity of ADP(vl, gl, false, ·) and ADP(vr, gr, false, ·). +First, observe that we can assume xl ≤ w(Tvl) and xr ≤ w(Tvr): increasing the upper +bounds on the weight of the root component further would mean that the root component +contains more weight than all vertices inside the sub-tree, which is impossible. +Thus, +xl + xr + w(v) ∈ [1, w(V )]. +Second, we partition the interval [1, w(V )] into O(log(k/ϵ)/ϵ) intervals. We have intervals +Ij = (ξj−1, ξj] with ξj = (1+ϵ)jϵ⌈w(V )/k⌉ for all j = 1, . . . , log1+ϵ(k/ϵ). In addition, we add +an “interval” I0 := [ϵ⌈w(V )/k⌉, ϵ⌈w(V )/k⌉] and the interval I−1 := [1, ϵ⌈w(V )/k⌉). We set +ξ0 = ϵ⌈w(V )/k⌉ and we set ξ−1 to the largest integer that is less than ϵ⌈w(V )/k⌉. Observe + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +39 +that for all j ≥ −1 and x ∈ Ij, we have e(x) = e(ξj), i.e., the value of e(x) does not change +on in the interval Ij. Below, this property will allow us to separate the conditions on xl + xr +and on gl + gr. +Now we can rewrite the above expression as +ADPB(v, g, true, x) = +cap(v, p) + min +j +min +xl+xr+w(v)∈Ij +min +gl+gr=g−e(ξj) ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr). +Third, note that now the two min-operations only depend on the choice of j and, +importantly, the minimum over gl and gr does not depend on the choice of xl + xr any- +more. Therefore, we can swap the order of the two min-operations. Furthermore, since +ADPB(v, g, false, x) is monotonically decreasing with x, we can restrict the choice of xl and +xr such that xl + xr + w(v) is the largest number in the corresponding interval Ij, i.e., +xl + xr + w(v) = ξj. Thus, +ADPB(v, g, true, x) = cap(v, p)+ +min +j +min +gl+gr=g−e(ξj) +min +xl+xr+w(v)=ξjADP(vl, gl, false, xl) + ADP(vr, gr, false, ξj − xl − w(v)). +(11) +Next, we explain how the above expression can be computed efficiently. Let us first argue +how we can efficiently compute the inner min-operation of the above expression. We start by +observing that this min-operation is not a convolution since in the constraint we sum up to +ξi which is a constant (rather than to the variable x). Now recall that ADP(vl, gl, false, ·) and +ADP(vr, gr, false, ·) are piecewise constant functions with O(p) pieces by our invariants. Since +xl, xr ≥ 0 this implies that there are only O(p2) choices for xl and xr such that xl, xr ∈ Ij +and either a new piece starts in ADP(vl, gl, false, xl) or in ADP(vr, gr, false, xr). Thus, we can +iterate over all these pairs (xl, xr) and evaluate ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr), +where xr = ξj −xl −w(v). Thus, we can compute the inner min-operation in time O(p2 log p). +Next, we can compute the outer two min-operations by simply iterating over j and +all choices for gl and setting gr = g − e(ξj) − gl as above in O(M t · log(k/ϵ)/ϵ) iterations. +Hence, we obtain a running time of O(M tp2 log p · log(k/ϵ)/ϵ). We note that this is the step +which makes the okay-behaved rather than well-behaved (since it violates Property (4b) of +Definition 8). +Finally, we note that as ADPB(v, g, true, x) is independent of x, it is a constant. Thus, +ADPB(v, g, true, x) is a piecewise constant function with a single piece and it is monotonically +decreasing. +Case C: cut el but not er. Now suppose we cut the edge to the left child but not to the +right child. +First assume that we cut the edge to the parent of v. As in the exact DP, for all x ∈ [0, ∞) +we want to set +ADPC(v, g, true, x) = +cap(v, p) + +min +xr,gl+gr=g−e(xr+w(v)) ADP(vl, gl, true, ∞) + ADP(vr, gr, false, xr). +As in the previous case, observe that in the minimum the constraint gl +gr = g −e(xr +w(v)) +depends on the choice of xr. Thus, we rewrite the above equation analogously to the previous + +40 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +case: +ADPC(v, g, true, x) += cap(v, p)+ min +j +min +xr+w(v)∈Ij +min +gl+gr=g−e(ξj) ADP(vl, gl, true, ∞) + ADP(vr, gr, false, xr) += cap(v, p)+ min +j +min +gl+gr=g−e(ξj) +min +xr+w(v)∈Ij ADP(vl, gl, true, ∞) + ADP(vr, gr, false, xr) += cap(v, p)+ min +j +min +gl+gr=g−e(ξj) ADP(vl, gl, true, ∞) + ADP(vr, gr, false, ξj − w(v)), +where in the last step we used that ADP(vr, gr, false, ·) is monotonically decreasing. The +evaluation of the function values of the two piecewise constant functions with O(p) pieces +can be done in time O(log(p)). Furthermore, by exhaustively enumerating all choices for j +and proceeding for gl and gr as above, we obtain O(M t log(k/ϵ)/ϵ) iterations giving a total +running time of O(M t log p log(k/ϵ)/ϵ). As before, ADPC(v, g, true, ·) is a constant (since +the computation does not depend on x) and therefore it has only a single piece and it is +monotonically decreasing. +Next, suppose we do not cut the edge to the parent of v. Then we set DPC(v, g, false, 0) = +∞ for all signatures g and all x ∈ [0, w(v)). For all x ∈ [w(v), ∞), we set +ADPC(v, g, false, x) = +min +gl+gr=g ADP(vl, gl, true, ∞) + ADP(vr, gr, false, x − w(v)). +Note that inside the min-operation, the first term is a constant and the second term is a +piecewise constant function that is shifted by w(v). Furthermore, the minimum is taken +over O(M t) piecewise constant functions (one for each choice of gl by the same argument as +above). We can perform the addition and shift operation via Lemma 6 (time O(p log p) per +application). Then we perform a minimum operation over M t functions where each function +has just p pieces. This can be done in time O(M tp log(M tp)) by Lemma 21. In total we get +a running time of O(M tp log(M tp)). +Case D: cut er but not el. Symmetric to Case C. +We conlucde this subsection with the following lemma which summarizes the properties of +the approximate DP computation The lemma follows immediately from the above discussion. +▶ Lemma 25. The approximate DP computes a (1 + δ)-approximate DP solution and the +dependency tree and the input tree T are identical. Given a vertex v, a signature g and value +cut ∈ {true, false}, we can compute the corresponding approximate DP entry ADP(v, g, cut, ·) +in time O(M tp2 log(M tp) log(k/ϵ)/ϵ)). +Proof. The approximation ratio of the approximate DP is (1 + δ)-approximate because, +as we pointed out earlier, we only use exact computations except in the rounding step in +Equation (9). Thus, we only use (1 + δ)-factor in the computation. +The claim about the running time follows immediately from the discussion above the +lemma, where we already analyzed the running times for all steps. +◀ +D.3 +Computing the Result +In this section we describe how the previously described DPs can be used to extract the +result for the k-balanced partition problem. Recall that we consider the generalized version +of the k-balanced partition problem, where each vertex v has a weight w(v) ∈ {0, 1} (see +Section D.1 for the definition). + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +41 +We focus on the value version of the problem in which we only need to output an +approximation of the value of the optimal cut OPT but we do not have to return the actual +partition V1, . . . , Vk that obtains this cut value. We note, however, that by analyzing the DP +solution from top to bottom, we could also construct a concrete partition V1, . . . , Vk in time +˜O(n) that achieves the cut value which is returned by the value version. +Feasible Signatures. Before we describe our algorithm, we first need to introduce the +notion of feasible signatures. More concretely, recall that in Section D.1 we introduced +signatures as a succinct way of storing the sizes of connected components in a solution. +Now, feasible signatures will refer to signatures in which the connected components can be +partitioned such that we obtain a nearly k-balanced partitioning of the vertices. We make +this intuition more formal below. +For every signature g = (g0, . . . , gt−1) ∈ [M − 1]t, we say that its associated machine +scheduling instance11 I(g) is the instance which contains exactly gi jobs of size (1 + ϵ)i · +ϵ⌈w(V )/k⌉ of all i. We say that g is a feasible signature if the jobs in I(g) can be scheduled on +k machines with makespan at most (1 + ϵ)⌈w(V )/k⌉. Later, we will identify the machines of +the scheduling problems with partitions in the k-balanced partitioning solution and the jobs +with connected components. In this way, we will be able to ensure the balance constraints of +the k-balanced partitioning solution. +Algorithm. We now describe our two static algorithms for binary trees. The only +difference between the algorithms is whether to use the exact DP from Section D.1 or the +approximate DP from Section D.2; we will refer to these algorithms as the exact and the +approximation algorithm, respectively. We assume that the input is an error parameter ϵ > 0 +and a rooted, weighted tree T = (V, E, cap) with root r and vertex weights w(v) ∈ {0, 1} for +which we wish to solve the k-balanced partitioning problem. +First, our algorithm augments T by adding a fake root r′. We make r′ the parent of r +and set w(r′) = 0 and cap(r, r′) = 0. Then we compute the DP bottom-up as described in +Section C.2, where we interpret T as its own dependency graph. In the exact algorithm, +we use the DP from Section D.1, and in the approximation algorithm, we use the DP from +Section D.2. +Second, we compute the set of all nearly feasible signatures. To obtain this set, we +enumerate all M t signatures and for each of them, we check whether it is nearly feasible +or not. We do this as follows. For each signature g, we construct the machine scheduling +instance I(g) and run the PTAS by Hochbaum and Shmoys [39] for this problem with +approximation ratio 1 + ¯ϵ and running time (N/¯ϵ)O(1/¯ϵ2), where N denotes the total number +of jobs in I(g) and we will see later that N is a constant if k, ϵ and ¯ϵ are constants. We add +a signature g to the set of nearly feasible signatures if the returned makespan for I(g) is at +most (1 + ¯ϵ)(1 + ϵ)⌈w(V )/k⌉. We note that by using the PTAS, the set that we compute +can potentially contain some signatures which are infeasible but they still do not violate the +balance constraint too much. +Third, we consider the entries in the DP table at the (true) root r of the tree for the case +that the edge to its (artificial) parent is not cut (recall that we added an edge of weight 0 +from the true root r to the fake root r′ and so cutting it does not incur any cost), i.e., we +consider the DP entries DP(r, ·, true, w(V )) or ADP(r, ·, true, w(V )) depending on whether +we are in the approximate or in the exact case. We iterate over all feasible signature vectors +11 Recall that in the makespan minimization problem with identical machines, the input consists of a set of +N jobs of sizes s1, . . . , sN and an integer k. The goal is to find an assignment of the jobs to k machines +such that the makespan is minimized. Here, the makespan refers to maximum load of all k machines. + +42 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +g and then take the minimum value that we have seen. +We conclude the algorithms’ guarantees in the following proposition. We note that for +constant k, ϵ, ¯ϵ and W, the running time of the exact algorithm is ˜O(n4) and the running +time of the approximation algorithm simplifies to ˜O(n · h2), where h is the height of the +input tree. Thus, for trees of height ˜O(1), the approximation algorithm is very efficient and +runs in time ˜O(n). +▶ Proposition 26. Let ϵ, ¯ϵ > 0 and k ∈ N. Let T = (V, E, cap) be a rooted binary tree that +has edge weights cap(e) and vertex weights w(v) ∈ {0, 1}. Then: +The exact algorithm obtains a bicriteria (1, (1+¯ϵ)(1+ϵ))-approximation for the k-balanced +partitioning problem on T in time O(M 2tn4). +The approximation algorithm obtains a bicriteria (1+ϵ, (1+¯ϵ)(1+ϵ))-approximation for the +k-balanced partitioning problem on T in time O +� +nh2·M 2t log2(W) log(k/ϵ) log(M th log(W)/ϵ)/ϵ3� ++ +M t(k/(ϵ¯ϵ))O(1/¯ϵ2), where h denotes the height of T. +Proof. To prove the proposition, we need to argue about the approximation ratios of the +algorithms and we also need to prove that the partitioning does not violate the balance +constraints. We will also need to analyze the running times. +We start by analyzing the balance constraints. We show that in the solution returned by +the algorithm, the connected components V1, . . . , Vk can be partitioned such that w(Vi) ≤ +(1 + ¯ϵ)(1 + ϵ)⌈w(V )/k⌉ for all i = 1, . . . , k. +Consider the DP entry DP(r, g, true, w(V )) for the (true) root r, where the edge to the +parent is cut and any signature vector g that is in the set of nearly feasible signatures that +we computed. Then this corresponds to the cost of some partition of T = Tr where, after +removing the cut edges, the large connected components S in Tr can be matched to entries +in g such that: +a component S ∈ S is matched to entry gi with |S| ≤ (1 + ϵ)iϵ⌈w(V )/k⌉ and +exactly gi components are matched to gi. +Hence, we can obtain a partitioning V1, . . . , Vk as follows. First, we compute the (1 + ¯ϵ)- +approximate solution of I(g) in which (by assumption on g) the makespan is at most +(1 + ¯ϵ)(1 + ϵ)⌈w(V )/k⌉. This gives us an assignment of jobs to machines. Now we identify +components with jobs and the sets Vi with machines and obtain an assignment of the +large components to the Vi. In particular, each Vi receives large components for which the +(rounded) weights sum to at most (1 + ¯ϵ)(1 + ϵ)⌈w(V )/k⌉. Now we need to assign the small +components in the algorithm’s solution. These can be assigned greedily by always assigning +a small component (of weight less than ϵ⌈n/k⌉) to set Vi of (currently) smallest weight. +In the end, all Vi will have weight at most (1 + ¯ϵ)(1 + ϵ)⌈w(V )/k⌉ (this follows from the +standard argument that, when considering exact component weights, on average each server +has makespan at most w(V )/k and thus there will always be a server of makespan at most +w(V )/k to which the current small component can be assigned without violating the capacity +constraint). This means if the algorithm returns an objective function value then there is a +partition V1, . . . , Vk with the same objective function value that is nearly feasible, i.e., that +satisfies w(Vi) ≤ (1 + ¯ϵ)(1 + ϵ)⌈w(V )/k⌉ for all i = 1, . . . , k. +Next, let us consider the approximation ratios of the algorithms. Consider the optimum +partition OPT = (V ∗ +1 , . . . , V ∗ +k ) that minimizes cut(V ∗ +1 , . . . , V ∗ +k ) such that w(V ∗ +i ) ≤ ⌈w(V )/k⌉ +for all i. +We first argue that OPT gives rise to a DP entry with a feasible signature +and cost OPT in the exact DP. To see this, take the optimum partition V ∗ +1 , . . . , V ∗ +k and +round up the weight of every large connected component to the next value of the form +(1 + ϵ)i · ϵ⌈n/k⌉. Let g = (g0, . . . , gt−1) ∈ [M − 1]t be the signature where gi denotes the + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +43 +number of large components in OPT whose rounded weight is (1 + ϵ)i · ϵ⌈n/k⌉. Note that +since w(V ∗ +i ) ≤ ⌈w(V )/k⌉ for all i, the total rounded weight of components in V ∗ +i is at most +(1 + ϵ) · ⌈w(V )/k⌉ as component weights are increased at most by a (1 + ϵ)-factor. Hence, +the constructed signature vector g is feasible because the partition V ∗ +1 , . . . , V ∗ +k gives rise +to a feasible solution for I(g). Furthermore, the rounding did not have any effect on the +objective function value and, thus, OPT gives rise to a DP entry with a feasible signature +and cost OPT in the exact DP. This implies that the optimum value for the exact DP +is at most cut(V ∗ +1 , . . . , V ∗ +k ). Together with the above claim that the DPs approximately +satisfy the balance constraint, we obtain that the exact algorithm computes a bicriteria +(1, (1 + ¯ϵ)(1 + ϵ))-approximation. +Now let us turn to the approximation ratio of the approximation algorithm from Sec- +tion D.2. Recall that by Lemma 24 the exact DP is okay-behaved and in Lemma 25 we show +that in each step the approximation algorithm loses a factor of at most 1 + δ at every level +of the tree T. Now, we can apply Lemma 19 to obtain that the approximation in the root is +(1 + δ)h+1, where h is the height of the tree T. Thus, the approximation ratio of the approxi- +mate DP is 1 + ϵ if we set δ = ln(1 + ϵ)/(h + 1) since then (1 + δ)h+1 ≤ exp(δ(h + 1)) = 1 + ϵ. +Since the notion of approximation from Lemma 19 holds for all functions of the form +ADP(r, g, true, ·) and all possible values of x, we obtain that the approximation algorithm +computes a bicriteria (1 + ϵ, (1 + ¯ϵ)(1 + ϵ))-approximation. +We conclude the proof of the proposition by considering the running times of the algorithms. +Note that w.r.t. running time, both algorithms only differ by how long it takes to fill the DP +cells and the time for computing the solution is the same. +Let us first consider the time for computing the solution as per Section D.3. First, let +us consider the time for solving the PTAS which is (N/¯ϵ)O(1/¯ϵ2), where N denotes the total +number of jobs. Note that in our case there are at most N ≤ k(1 + 1/ϵ) jobs: each job has +size at least ϵ⌈n/k⌉ and therefore a machine can take at most 1 + 1/ϵ jobs in an optimum +solution. Hence, if we have more than k(1+1/ϵ) jobs, a PTAS can directly reject the instance +and declare it infeasible. Thus, the time for running the PTAS a single time is (k/(ϵ¯ϵ))O(1/¯ϵ2). +Since we have to run the PTAS for each of the M t signatures, the total time for finding the +nearly feasible configurations is M t(k/(ϵ¯ϵ))O(1/¯ϵ2). +Finally, let us consider the time for filling the DP cells. For the exact DP, Lemma 24 +states that filling a cell DP(v, ·, ·, ·) takes time O(M 2tn3). Then, by applying Lemma 19, +the total time to compute all DP cells is O(M 2tn4). For the approximate DP, it takes +time O(M tp2 log(M tp) log(k/ϵ)/ϵ)) to fill a single DP cell ADP(v, g, cut, ·) by Lemma 25. +Since there are M t choices for g and by again applying Lemma 19, we obtain that the total +running time for filling the approximate DP table is O(nM 2tp2 log(M tp) log(k/ϵ)/ϵ)). Since +in Section D.2 we picked the number of pieces to be p = O(log1+δ(W)) and above we picked +δ = O(ϵ/h), the running time is upper bounded by O +� +nM 2t · +� +1/ϵ · h log W +�2 · log(k/ϵ)/ϵ · +log(M th log(W)/ϵ) +� += O +� +nh2 · M 2t log2(W) log(k/ϵ) log(M th log(W)/ϵ)/ϵ3� +. +◀ +D.4 +Extension to General Graphs +Now we generalize the results of Proposition 26 from binary trees to general graphs. +We start with the generalization to general graphs in which we will make use of Räcke +trees (see Section C.1). Since Räcke trees might be non-binary, we now introduce the notion +of binarized Räcke trees which essentially describe a way of turning a non-binary Räcke tree +into a binary tree that is very similar to a Räcke tree. Later, the binarized Räcke trees will +allow us to apply Proposition 26 on them. + +44 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +▶ Definition 27 (Binarized Räcke Tree). Let G = (VG, EG, capG) be a weighted graph. We +say that a weighted, rooted tree T = (VT , ET , capT ) is a binarized Räcke tree for G if the +following properties hold: +T is a rooted binary tree. +VG ⊆ VT . +All edges in T have weights in W∞. +Let T ′ be the tree that is obtained by contracting all edges with weight ∞ in T. Then T ′ +is a Räcke tree for G. +We call the tree T ′ from the last bullet point the corresponding (non-binarized) Räcke tree of +T. We say that T has quality q if the corresponding Räcke tree T ′ has quality q. +Next, we observe that each cut in T of finite cost corresponds to a cut in the corresponding +Räcke tree T ′ and vice versa. Therefore, cuts of finite cost in T ′ approximate the cut structure +of the initial graph G. We make this more formal in following observation. +▶ Observation 28. Let G = (VG, EG, capG) be a weighted graph and let T = (VT , ET , capT ) +be a binarized Räcke tree for G with quality q. Then for all disjoint subsets A, B ⊆ VG it +holds that mincutG(A, B) ≤ mincutT (A, B) ≤ q · mincutG(A, B). +Proof. Let T ′ be the corresponding (non-binarized) Räcke tree of T and consider two disjoint +subsets of vertices A, B ⊆ VG. We show that minT (A, B) = mincutT ′(A, B). Then the +observation follows immediately since T has quality q (by assumption) and, therefore, T ′ is a +Räcke tree for G with quality q which satisfies the property from the observation. +Since T ′ can be obtained from T only by contracting edges, we have mincutT ′(A, B) ≥ +mincutT (A, B). Next, let us argue that mincutT (A, B) ≥ mincutT ′(A, B). First, note that +mincutT ′(A, B) ≤ q · mincut(A, B) < ∞. Since we contract only edges with weight ∞ to +go from T to T ′, T does not contain any cut with finite cost that is not contained in T ′. +Therefore, mincutT (A, B) ≥ mincutT ′(A, B). +◀ +Additionally, we show that we can compute a binarized Räcke tree of good quality in +nearly-linear time. +▶ Lemma 29. Let G = (VG, EG, capG) be a weighted graph with n vertices and m edges. We +can compute a binarized Räcke tree T = (VT , ET , capT ) with O(n) vertices, height O(log2 n) +and quality O(log4 n) in time ˜O(m). +Proof. Let T ′ = (VT ′, ET ′, capT ′) be the Räcke tree for G from Theorem 15 that can be +computed in time ˜O(m). First, note that T ′ has nT ′ := O(n) vertices and height O(log n). +Second, note that T ′ can have unbounded degree. Therefore, we will show how to compute +a binarized Räcke tree T that has T ′ as its corresponding (non-binarized) Räcke tree. We +do so replacing in T ′ each vertex u by a balanced binary tree τu with deg(u) leaves, where +deg(u) denotes the number of children of u. The internal edges of τu will have weight ∞ and +the edges connecting subtrees τu and τv, u ̸= v, in T will correspond to the edges in T ′ and +will have the same (finite) weight as in T ′. We will see that by contracting all edges with +weight ∞ in T, we will obtain T ′. We now elaborate on this process. +We construct T as follows. First, we compute T ′ as per the algorithm from Theorem 15. +Now we construct T as follows. For each vertex u ∈ VT ′, we add a balanced rooted binary +tree τu with deg(u) leaves. We refer to the root of τu as ru. We identify each leaf of τu with a +child of u and denote the leaf of τu that corresponds to the child v by cu,v. We set the weight +of edges inside τu to ∞. Note that for each vertex u, the tree τu has O(deg(u)) vertices, and, +therefore, T has O(nT ′) = O(n) vertices. Next, for each edge (u, v) ∈ ET ′ (where we assume + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +45 +that v is a child of u), we insert the edge (cu,v, rv) in T and set capT (cu,v, ru) = capT ′(u, v). +Finally, if u is the root of T ′ then we set ru to the root of T. +It is left to show that T is a binarized Räcke tree of height O(log2 n) and quality O(log4 n). +Clearly, T is a binary tree since all vertices inside each subtree τu have at most two child +nodes and, additionally, each vertex cu,v has at most one child node (namely rv). Next, T +has height O(log2 n) since T ′ has height O(log n) and the subtrees τu have height O(log n). +Finally, let T ′′ be the tree obtained from T by contracting all edges with weight ∞. We +argue that T ′ = T ′′. Indeed, consider any vertex u ∈ VT ′ and its subtree τu in T. Then after +contracting the edges in τu, we are left with a subtree that only contains ru. Furthermore, all +edges between vertices of different subtrees τu and τv, u ̸= v, have finite weight. Therefore, +T ′ = T ′′. This implies that T is binarized Räcke tree for G. Since T ′ has quality O(log4 n), +the quality of T is also O(log4 n). +◀ +We conclude the subsection by proving Theorem 2. +Proof of Theorem 2. We can obtain the proof for the claim about general graphs as follows. +Let G = (VG, EG, capG) be a weighted graph with n vertices. We compute a binarized Räcke +tree T = (VT , ET , capT ) with O(n) vertices as per Lemma 29 in time ˜O(n). In T, we assign +weight w(v) = 1 to all vertices v ∈ VG ∩VT (i.e., to the leaves in T that correspond to vertices +in G) and weight w(v) = 0 to all vertices v ∈ VT \ VG (i.e., to the internal nodes of T that +do not correspond to any vertex in G). Now observe that w(V ) = n and thus a balanced +partitioning V1, . . . , Vk of T with w(Vi) ≤ (1 +ϵ)⌈w(V )/k⌉ for all i corresponds to a balanced +partitioning V ′ +1, . . . , V ′ +k of G with |V ′ +i | ≤ (1+ϵ)⌈n/k⌉ for all i, where V ′ +i = {v ∈ Vi : w(v) = 1}. +Now by combining Observation 28, Proposition 26 and the fact that T has quality O(log4 n), +we obtain the claim. +To obtain the result about general trees T ′ (with unbounded degrees), we proceed similarly. +We construct a binarized tree T exactly as in the proof of Lemma 29. Now, in T we set +w(rv) = 1 for all root vertices of the subtrees τv and we set w(v) = 0 for all other vertices of +the subtrees τv. Similar to before, observe that w(VT ) = n and thus a balanced partitioning +V1, . . . , Vk of T with w(Vi) ≤ (1 +ϵ)⌈w(V )/k⌉ for all i corresponds to a balanced partitioning +V ′ +1, . . . , V ′ +k of T ′ with |V ′ +i | ≤ (1 + ϵ)⌈n/k⌉ for all i, where V ′ +i = {v ∈ VT ′ : rv ∈ Vi}. Then by +Proposition 26, this implies the proof for trees with unbounded degrees. +◀ +D.5 +Extension to the Dynamic Setting +Next, we provide new dynamic algorithms in which edges are inserted and deleted from the +graph. We give new algorithms for trees and for general graphs. +Extension to Dynamic Trees. Let us start with the case when T is a binary tree that +is undergoing edge insertions and deletions. We will use Lemma 20 to make the result from +Proposition 26 dynamic. However, there is a slight technical difficulty: due to edge deletions, +T will become a forest and fall apart into several connected components. This becomes an +issue, when an edge (u, v) is inserted for which both u and v already have parents in their +respective components. In that case, we cannot immediately make u the root of v (or vice +versa). Therefore, we need to find an efficient way of re-rooting the tree containing v, i.e., +we need to make v the root of its component and we need to ensure that we do not have +to recompute the DP solution for all vertices in the component of v. We now describe our +dynamic algorithm in more detail. +First, suppose that an edge (u, v) is removed from T and assume that (before the edge +deletion) u is closer to the root of T than v. Then T becomes a forest with multiple connected +components. In that case, we make v the root of its component and recompute the DP + +46 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +solution for v (since v does not have a parent, we only have to recompute the DP cell for v). +Furthermore, for u and all of its ancestors we recompute the DP solution as per Lemma 20. +Next, suppose an edge (u, v) is inserted, where u and v are in different connected +components. Further suppose that after the edge insertion, u is the parent of v. Then we +distinguish two cases whether v is the root of its component or not. +First, suppose that v is the root of its component. Then we simply insert the edge +(u, v) into T and recompute the solution for v and all of its ancestors (including u) as per +Lemma 20. +Second, suppose that neither u nor v is the root of its component. Now, we first have +to re-root the component containing v such that it has v as its root and such that all DP +solution are valid. We do this as follows. Let v = v1, . . . , vℓ denote the vertices on the path +from v to the root vℓ of its component (before the edge insertion). Then we first remove all +edges (vℓ, vℓ−1), . . . , (v2, v1) from T (in this order) as per the edge deletion routine described +above. Note that after the deletions, none of the vi has a parent and, therefore, each vi +is the root of its own component. Furthermore, by how we picked the order of the edge +deletions, after the i’th deletion we only have to recompute the DP cells for the vertices vℓ−i +and vℓ−i−1. Now we insert the edges again but with flipped direction, i.e., we insert the +edges (vℓ−1, vℓ), . . . , (v1, v2) (in this order). Thus, v = v1 becomes the root of the component. +To insert the edges, we use the subroutine from the paragraph above, where we exploit that +each vi is the parent of its own component, which implies that the DP solutions can be +updated efficiently: by how we picked the order of the edge insertions, after the i’th edge +insertion we only need to recompute the DP cells for vertices vℓ−i−1 and vℓ−i. After the +rebalancing of the component containing v is done, v has become the parent of its component +and, therefore, we can use the routine from above to insert the edge (u, v). This concludes +the edge insertion procedure. +Next, when we want to output the value of the DP solution, we simply use the subroutine +described in Section D.3. +We summarize the guarantees of our dynamic algorithm in the following proposition. +Note that when the parameters ϵ, ¯ϵ, k and W are constants, the update time becomes ˜O(h3) +and the query time is just O(1). Therefore, the algorithm is very efficient for trees that have +polylogarithmic or subpolynomial height in the number of vertices. +▶ Proposition 30. Let ϵ, ¯ϵ > 0 and k ∈ N. +Let T = (V, E, cap) be a rooted binary +tree with edge weights cap(e) ∈ W∞ and vertex weights w(v) ∈ {0, 1}, that is under- +going edge insertions and deletions. Let h be an upper bound on the height of the tree +T at all times. Then there exists a fully dynamic algorithm that maintains a bicriteria +(1 + ϵ, (1 + ¯ϵ)(1 + ϵ))-approximation for the k-balanced partition problem on T with update +time O +� +h3 · M 2t log2(W) log(k/ϵ) log(M th log(W)/ϵ)/ϵ3� +and query time M t(k/(ϵ¯ϵ))O(1/¯ϵ2). +Proof. The fact that the algorithm maintains a bicriteria (1+ϵ, (1+¯ϵ)(1+ϵ))-approximation +follows immediately from Lemma 20 and the same arguments as in the proof of Proposition 26, +where we argued that the approximate DP satisfies the conditions of Lemma 19. +It is left to analyze the update and query times. For the query times, note that all we do +is run the subroutine from Section D.3. This subroutine runs in time M t(k/(ϵ¯ϵ))O(1/¯ϵ2) as +we argued in the proof of Proposition 26. This proves the claim about the query time. +For the update times, let us first consider edge deletions (u, v). In this case, we need +to update the DP cell for v and the DP solutions for u and all of its ancestors. +By +Lemma 20, Lemma 25 and by our choice of p = O(h log W/ϵ), this can be done in time +O +� +h3 · M 2t log2(W) log(k/ϵ) log(M th log(W)/ϵ)/ϵ3� +. + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +47 +Next, consider the case in which (u, v) is inserted and v is the root of its component. +Then we need to recompute the DP solutions for v and all of its ancestors (including u) +which, by Lemma 20, Lemma 25 and by our choice of p = O(h log W/ϵ), can be done in +the time claimed in the lemma. In the case that we need to re-root the component of v, +note that we have to recompute the solutions for all ancestors of v. Since the height of T is +bounded by h, there are at most h such ancestors. Furthermore, we have picked the order +of edge deletions such that whenever we delete or insert an edge in the re-rooting process +then we only need to recompute two DP cells. Hence, in total we only need to recompute +the solutions for O(h) DP cells in the re-rooting process and thus by Lemma 25, the total +time for this process is O +� +h3 · M 2t log2(W) log(k/ϵ) log(M th log(W)/ϵ)/ϵ3� +. +◀ +Extension to Dynamic General Graphs and Non-Binary Trees. Now suppose +that our input is a dynamic (general) graph G that is undergoing edge insertions and +deletions. Essentially we will solve this problem by maintaining a dynamic Räcke tree and +running the algorithm from Proposition 30 on top of it. However, the dynamic Räcke tree +from Theorem 16 is non-binary and, therefore, we start by arguing that we can maintain a +binarized Räcke tree dynamically in the following lemma. +▶ Lemma 31. Let G = (VG, EG) be a dynamic unweighted graph with n vertices that is +undergoing edge insertions and deletions. We can maintain a binarized Räcke tree T = +(VT , ET , capT ) with O(n2) vertices, height O(log7/6 n) and quality no(1) in amortized update +time no(1). The preprocessing time is O(n2). +Proof. Let T ′ = (VT ′, ET ′, capT ′) be the fully dynamic Räcke tree for G from Theorem 16. +First, note that T ′ has nT ′ := O(n) vertices and height O(log1/6 n). Second, note that T ′ +can have unbounded degree. Therefore, similar to the proof of Lemma 29, we will show how +to maintain a binarized Räcke tree T that has T ′ as its corresponding (non-binarized) Räcke +tree. We do so by taking T ′ and replacing each vertex u in T ′ by a balanced binary tree +τu with nT ′ leaves; the internal edges of τu will have weight ∞ and the edges connecting +subtrees τu and τv, u ̸= v, in T will correspond to the edges in T ′ and will have the same +(finite) weight as in T ′. We will see that by contracting all edges with weight ∞ in T, we +will obtain again T ′. We now elaborate on this process. +During the preprocessing, we first build T ′. Note that this takes time O(n2). Now we +construct T as follows. For each vertex u ∈ VT ′, we add a balanced rooted binary tree +τu with nT ′ leaves. We refer to the root of τu as ru. We identify each leaf of τu with a +vertex v ∈ VT ′ and denote the leaf of τu that corresponds to v by cu,v. We set the weight of +the edges inside τu to ∞. Note that T has O(n2 +T ′) = O(n2) vertices. Next, for each edge +(u, v) ∈ ET ′ (where we assume that v is a child of u), we insert the edge (cu,v, rv) in T and +set capT (cu,v, ru) = capT ′(u, v). Finally, if u is the root of T ′ then we set ru to the root of T. +Next, suppose that G is changed due to an edge insertion or deletion. Then we first +update the tree T ′ via the algorithm from Theorem 16. Now, whenever an edge (u, v) is +inserted (deleted) in T ′, we insert (delete) the edge (cu,v, rv) into (from) T. Each of these +insertions and deletions in T can be done in time O(1). Since it takes amortized time no(1) +to update T ′ (via Theorem 16), the total update time is no(1). +It is left to show that T is a binarized Räcke tree of height O(log7/6 n) and quality no(1). +Clearly, T is a binary tree since all vertices inside each subtree τu have at most two child +nodes and, additionally, each vertex cu,v has at most one child node (namely rv). Next, +T has height O(log7/6 n) since T ′ has height O(log1/6 n) and the subtrees τu have height +O(log n). Finally, let T ′′ be the tree obtained from T by contracting all edges with weight +∞. We argue that T ′ = T ′′. Indeed, consider any vertex u ∈ VT ′ and its subtree τu in T. + +48 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +Then after contracting the edges in τu, we are left with a subtree that only contains ru. +Furthermore, all edges between vertices of different subtrees τu and τv, u ̸= v, have finite +weight. Therefore, T ′ = T ′′. This implies that T is binarized Räcke tree for G. Since T ′ has +quality no(1), the quality of T is also no(1). +◀ +Given the lemma above, our dynamic algorithm for dynamic general graphs G works as +follows. We maintain the dynamic binarized Räcke tree T as per Lemma 31 on our input +graph G, i.e., whenever an edge is inserted or deleted in G, we update the data structure +from the lemma as well. Note that this causes edge insertions and deletions in T as well. As +before, we set the vertex weights in T such that w(v) = 1 if v corresponds to a vertex in +G and w(v) = 0 if v is an internal node of T that does not correspond to any vertex in G. +Furthermore, we run our dynamic algorithm from Proposition 30 for binary trees on T. In +particular, whenever T gets updated, we also update the DP solution as per Proposition 30. +We also use the same query procedure as in the proposition. +We conclude the subsection by proving Theorem 3. +Proof of Theorem 3. We prove the result for general graphs first. +Since the dynamic +binarized Räcke tree T that we maintain has quality no(1), the same argumentation as in the +proof of Theorem 2 implies that we maintain a bicriteria (no(1), (1 + ¯ϵ)(1 + ϵ))-approximation +for G. +Since by Lemma 31 we can maintain T with amortized update time no(1), the +amortized number of edge insertions and deletions into T is no(1) per update operation. Since +T has height O(log7/6 n) and by Proposition 30, the total amortized update time no(1). This +implies the claim about dynamic general graphs. +To obtain our result for non-binary trees, we can proceed similar to above. Consider a +non-binary T ′ that is undergoing edge insertions and deletions. We can maintain the same +data structure as in the proof of Lemma 31 to obtain a binary tree T with O(n2) vertices +with worst-case update time O(1). Now we assign weight w(rv) = 1 to all vertices rv that +are roots of the subtrees τv in T and weight w(v) = 0 to all other vertices of the subtrees τv. +By the same arguments as in the proof of Theorem 2, we obtain the claim. +◀ +E +Simultaneous Source Location +In this section, we provide efficient algorithms for the simultaneous source location problem +as studied by Andreev et al. [4]. Recall that in this problem, the input consists of a graph +G = (V, E, cap, d) with a capacity function cap: E → W∞ on the edges and a demand +function d: V → W∞ on the vertices of the graph. The goal is to select a minimum set +S ⊆ V of sources that can simultaneously supply all vertex demands. More concretely, a set +of sources S is feasible if there exists a flow from the vertices in S that supplies demand d(v) +to all vertices v ∈ V and that does not violate the capacity constraints on the edges. Here, +we assume that each source vertex can potentially send an infinite amount of flow that is +only constrained by the edge capacities. The objective is to find a feasible set of sources of +minimum size. +Next, we summarize our main results for the simultaneous source location problem. First, +we introduce our notion of bicriteria approximation. Let S∗ be the optimal solution for +the simultaneous source location problem. Then we say that a solution S is a bicriteria +(α, β)-approximate solution if |S| ≤ α |S∗| and if S is a feasible set of sources after all edge +capacities are increased by a factor β. +The following theorem summarizes our main result for static algorithms. + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +49 +▶ Theorem 4. Let ϵ > 0. Let G = (V, E, cap, d) be an undirected weighted graph with +n vertices and m edges. Then for the simultaneous source location problem we can compute: +A (1 + ϵ, O(log4(n)))-approximation in time12 ˜O( 1 +ϵ2 m). +A (1 + ϵ, 1)-approximation in time ˜O( 1 +ϵ2 h2 · n) if G is a tree of height h. +Next, we turn to our dynamic algorithms which support the following update operations: +SetDemand(v, d): updates the demand of vertex v to d(v) = d, +SetCapacity((u, v), c): updates the capacity of the edge (u, v) to cap(u, v) = c, +Remove(u, v): removes the edge (u, v) from the graph, +Insert((u, v), c): inserts the edge (u, v) into the graph with capacity cap(u, v) = c. +The next theorem summarizes our main results for dynamic algorithms. +▶ Theorem 5. Let ϵ > 0. Let G = (V, E, cap, d) be a graph with n vertices and m edges that +is undergoing the update operations given above. Then for the simultaneous source location +problem we can maintain: +A (1 + ϵ, no(1))-approximation with amortized update time no(1)/ϵ2 and preprocessing time +O(n2/ϵ2) if all edge capacities are 1. +A (1+ϵ, O(log4(n)))-approximation with worst-case update time ˜O(1/ϵ2) and preprocessing +time ˜O(m) if we only allow the update operation SetDemand(v, d). +A (1 + ϵ, O(log2(n) log log(n)))-approximation with worst-case update time ˜O(1/ϵ2) and +preprocessing time poly(n) if we only allow the update operation SetDemand(v, d). +A (1 + ϵ, 1)-approximate solution with worst-case update time ˜O(h3/ϵ2) and preprocessing +time O(n2/ϵ2) if G is a tree of height h. +We note that in our static and dynamic algorithms, we can output the corresponding +solutions similarly to what we descriped after Proposition 12 for knapsack. +We start by presenting an exact DP for the special case of binary trees in Section E.1 +and then present an approximate DP in Section E.2. After that, we generalize the result +from binary trees to general graphs in Section E.3 and then also to the fully dynamic setting +in Section E.4. +E.1 +The Exact DP +We consider the special case of the simultaneous source location problem on binary trees and +provide a DP that solves this problem exactly. We let T = (V, E, cap, d) denote the rooted +binary tree with root r that we obtain as input. Additionally, we assume that for each vertex +v ∈ V we obtain as input whether we are allowed to make v a source or not; note that this +only generalizes the problem (as in the original problem all vertices can be made sources). +Later in Section E.3, this generalization will be helpful when we apply Räcke trees because +then we only want to allow leaves to act as sources. +E.1.1 +DP Definition +We now define our exact DP. We will also discuss its relationship with the DP by Andreev +el al. [4] and why we did not use the DP of Andreev et al. Given a vertex v and a value +x ∈ R, we denote by DP(v, x) the minimum number of sources to place in Tv such that when +v receives flow at most x from its parent then all demands in Tv can be satisfied. We note +that x can take positive and negative values: for x ≥ 0 this corresponds to the setting in +12 We write ˜O(f(n, ϵ, W)) to denote running times of the form f(n, ϵ, W) · polylog(n, ϵ, log W). + +50 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +which flow is sent from the parent of v into Tv and for x < 0 this corresponds to the setting +in which flow is sent from Tv towards the parent of v. We further follow the convention that +when the demands in Tv cannot be satisfied when v receives flow x from its parent, then we +set DP(v, x) = ∞. +Observe that this DP has rows I = V and columns J = R. We will store the rows +DP(v, ·) using our data structure from Section 2 using monotone piecewise constant functions. +Next, we observe that each DP(v, ·) is monotonically decreasing. Hence, the DP satisfies +Property (1) of Definition 8. +▶ Observation 32. The function DP(v, ·): R → [n + 1] ∪ {∞} is monotonically decreasing. +Proof. This follows immediately from the definition of DP(v, x): Consider x, x′ ∈ R with +x ≤ x′. Then any solution in which Tv receives flow at most x from the parent of v is also +feasible when Tv receives flow at most x′ from the parent of v. Therefore, DP(v, x) ≥ DP(v, x′), +which finishes the proof. +◀ +Observe that the global solution for the simultaneous source location problem on T can +be obtained by evaluating DP(r, 0), where r is the root of T: First, r has no parent and, +therefore, it must be a source itself or have its demand satisfied by its children; this explains +the choice of x = 0. Furthermore, (by definition) DP(r, 0) is the minimum number of sources +that we need to satisfy all demands in Tr = T and, thus, the flow that we obtain is feasible. +We conclude that DP(r, 0) gives the global optimum solution. +Relationship to the approach by Andreev et al. [4]. Next, let us elaborate on the +relationship of our DP and the function f used by Andreev et al. [4]. In [4], the function f +computed by a dynamic program is defined as follows. Given a vertex v and an integer i ∈ N, +Andreev et al. define a function f(v, i) that denotes the minimum amount of flow that v +needs to receive from its parent if all demands in Tv need to be satisfied and if we can place +i sources in the subtree Tv. Similar to above, f(v, i) can take positive and negative values: +if the demand in Tv can only be satisfied by receiving flow from the parent, then f(v, i) is +positive; if the demand in Tv is already satisfied by the sources in the subtree Tv, then it is +possible that v can send flow to its parent and f(v, i) is negative. It is not hard to see that +the function f(v, i) is monotonically decreasing in i.13 +Now consider f(v, ·): N → R as a function and consider its “inverse”14 function f −1(v, ·): R → +N, where f −1 is defined on the whole set of real numbers (including negative numbers). That +is, f −1(v, x) denotes the minimum number of sources that we need to place in Tv such that +the demand that v requires from its parent is at most x. But this was exactly the definition +of DP(v, x). Thus, DP(v, x) = f −1(v, x) for all v and x. +Why Did We Not Use f? +In [4] it is shown how the function f can be computed +in polynomial time by a bottom-up dynamic program using just a few case distinctions +and a (min, +)-convolution in each DP cell. +Thus, one might wonder why we picked +DP(v, ·) = f −1(v, ·) and not f for our DP? Indeed, it seems quite natural to interpret the +function f as a monotone piecewise constant function and to use it for our dynamic program. +13 This follows immediately from the definition of f and the fact that by adding more sources to a +subtree Tv, the amount of flow that Tv needs to receive from the parent of v only decreases. +14 We note that, formally, f(v, ·) has no inverse since it is possible that multiple values map to the same +number, i.e., f(v, i) = f(v, i′) for i ̸= i′. Thus, formally, we set f −1(v, x) = min{i: f(v, i) ≤ x}, where +we follow the convention min{∅} = ∞. Then we interpret f −1(v, ·) as a piecewise constant function +from R to [n + 1]. + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +51 +While for the case of exact computations this is possible, we now sketch why this appears +unhandy for the approximate case later. +Suppose that we used the function f in our approximate computations. To obtain efficient +approximation algorithms, we will have to ensure that f has only few pieces and our main +way to achieve this is by rounding f as per Lemma 6. However, this becomes tricky because +the function values of f are positive and negative. In the following, it will be illustrative to +think of positive function values for f as vertex demands that need to be satisfied and of +negative values for f as available edge capacities. The main issue is that since the function +values of f are positive and negative, it is not clear how we should perform the rounding: +if we rounded positive and negative values up (towards +∞) then this would correspond +to increasing the vertex demands while at the same time decreasing the edge capacities; +however, this could render some feasible solutions (in the exact computation) infeasible (in +the rounded computation). On the other hand, it is conceivable that by always rounding f +down (towards −∞), we would essentially decrease the vertex demands while increasing the +edge capacities. Potentially, this approach could work when we are allowed to violate the +edge capacities by a (1 + ϵ)-factor. However, even if we did that, we would have another +issue: to only use a small number of pieces for representing f, we would have to use different +rounding mechanisms for those function values in [−1, 1] and those in [−W, W] \ [−1, 1], +where W is the largest edge capacity. Indeed, if we rounded the values of f to powers of +(1 + δ)j then there are only O(log1+δ(W)) function values in [−W, W] \ [−1, 1] but there are +infinitely many function values in [−1, 1]. Similarly, if we rounded to multiples of δ then +there are only O(1/δ) function values in [−1, 1] but this would lead to O(W/δ) function +values in [−W, W] \ [−1, 1]. In both cases, our functions would have too many pieces and, +thus, one would have to pick a rounding function which provides a tradeoff between these two +cases. Furthermore, we would have to find an analysis that shows that this “more involved” +rounding function does not introduce too much error. +Note that in the above discussion, all of the issues come from the fact that f(v, ·) can also +take negative values. On the other hand, our DP (which is f −1(v, ·)) only takes non-negative +function values and, therefore, we avoid all of the above complications because we can use +the standard rounding function ⌈·⌉1+δ that rounds to powers of 1 + δ. Thus, we bypass all of +the issues above. +Our approach also has the positive side effects that instead of getting factors of polylog(W) +in our running times, we only get factors of polylog(n) because the codomain of our monotone +piecewise constant functions became [n + 1] rather than some potentially large interval +[−W, W]. +E.1.2 +Computing the DP +Now we describe the exact computation of our DP. This will reveal the procedures Pi from +Definition 8. +Let v ∈ V be any vertex in T. We describe how to compute DP(v, ·) efficiently assuming +that we have already computed the solutions for the children of v (if they exist). Recall that +for each vertex v we also obtain as input, whether v can be used as a source or not. In our +following case distinctions, whenever we consider the case that v is used as a source, we will +implicitly condition on the fact that it is also possible to use v as source; if v cannot be used +as a source, we simply skip this case. +In the construction for each DP cell DP(v, ·) for a vertex v with parent p, we will +additionally ensure that we do not violate the capacity of the edge (p, v) when x is very small +or very large. More concretely, we will ensure that DP(v, ·) satisfies the additional property + +52 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +that DP(v, x) = ∞ for x < − cap(p, v) and DP(v, x) = DP(v, cap(p, v)) for all x > cap(p, v). +We will denote this property as the feasible capacity property. +Case 1: v is a leaf. Suppose that v is a leaf. We initialize DP(v, ·) as the function +which takes value ∞ on all of R. In the following, we add at most two pieces to DP(v, ·) +depending on whether v can be used as a source or not. +First, suppose v can be used as a source. Then we can send flow up to cap(p, v) to the +parent p of v. Furthermore, since v is a leaf, there is exactly one source in Tv. Thus, we +update DP(v, ·) and set DP(v, x) = 1 for all x ≥ − cap(p, v). This adds one piece to DP(v, ·). +Second, suppose v is not a source. Then if x ≥ d(v) and cap(p, v) ≥ d(v), v can receive +all of its demand d(v) from its parent and the flow is feasible because we do not exceed the +capacity of the edge (p, v). Therefore, if cap(p, v) ≥ d(v), then we update DP(v, ·) again and +add the piece with DP(v, x) = 0 for all x ≥ d(v). If x ≥ d(v) but d(v) > cap(p, v) then we +do nothing because the parent of v cannot satisfy the demand of v. +Observe that DP(v, ·) is a monotonically decreasing function with at most three pieces. +Furthermore, it clearly satisfies the feasible capacity property and Property (3) of Definition 8. +Case 2: v is not a leaf. Suppose that v is not a leaf and that v has children v1 and +v2, as well as a parent p. Recall that we assume that we have already computed the DP +entries DP(v1, ·) and DP(v2, ·) for both children of v. We now show how to compute two DP +solutions DPA(v, ·) and DPB(v, ·) depending on whether v is a source (in Case A) or not (in +Case B). Then, if v can be used as a source, we set +DP(v, ·) = min{DPA(v, ·), DPB(v, ·)}, +where we compute the min-operation via Lemma 6. If v cannot be used as a source, we set +DP(v, ·) = DPB(v, ·). +Case A: Suppose v is used as a source. We initialize DPA(v, ·) as the function which takes +value ∞ on all of R. Now, since v can be used as a source, v can send flow cap(p, v) to its +parent and flow cap(v, v1) and cap(v, v2) to its children. Therefore, for x ≥ − cap(p, v), the +number of sources in DPA(v, x) is 1 (since v is a source) plus the number of sources that we +require in Tv1 when v1 can receive flow cap(v, v1) from its parent v plus the same quantity +for Tv2. Thus, it suffices to set +DPA(v, x) = 1 + DP(v1, cap(v, v1)) + DP(v2, cap(v, v2)) +for all x ≥ − cap(p, v). Note that here we exploited that the functions DP(v1, ·) and DP(v2, ·) +are monotonically decreasing and that both of them satisfy the feasible capacity property. +We conclude that in this case DPA(v, ·) is a monotonically decreasing function with two +pieces. +Case B: Suppose that v is not used as a source. We initialize DPB(v, ·) as the function +which takes value ∞ on all of R. To compute the value of DPB(v, x), we need to obtain +the minimum number of sources such that v receives flow at most x from its parent and +such that all demands in Tv are satisfied. Since v is not a source, its demand d(v) must +be satisfied either by its parent p or by its children (or a combination of them). Therefore, +to obtain that we have to pick the children solutions DP(v1, x1) and DP(v2, x2) such that +d(v) ≤ x − x1 − x2. +Since we did not make v a source, the number of sources in DPB(v, x) is the number of +sources that we need to place in the subtrees Tv1 and Tv2. Thus, we get +DPB(v, x) = min +x1∈R{DP(v1, x1) + DP(v2, x − x1 − d(v))}, + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +53 +where we used that x2 ≤ x − x1 − d(v) and by monotonicity of DP(v2, ·) we minimize the +number of sources in Tv2 if we consider x2 = x − x1 − d(v). Here, the flows that we computed +for DPB(v, x) are set feasible because the solutions DP(vi, ·) satisfy the feasible capacity +property and therefore we do not violate the edge constraints to the children. +Since the above equality holds for all values of x, DPB(v, ·) corresponds to a shifted +(min, +)-convolution of two monotonically decreasing functions. More concretely, via Lemma 6 +we can first compute the shifted function DP(v2, · − d(v)) and then we can set +DPB(v, ·) = DP(v1, ·) ⊕ DP(v2, · − d(v)), +which we compute via Lemma 7. +Finally, as a postprocessing step, we set DP(v, ·) = min{DPA(v, ·), DPB(v, ·)} if v can be +used as a source and DP(v, ·) = DPB(v, ·) otherwise, as we already mentioned above. But we +also need to ensure that DP(v, ·) satisfies the feasible capacity property. Therefore, we set +DP(v, x) = ∞ for x < − cap(p, v) and we set DP(v, ·) = DP(v, cap(p, v)) for x > cap(p, v). +Observe that these changes to DP(v, ·) can be done in time linear in the number of pieces of +DP(v, ·). +Properties of the DP. Observe that in the DP above for each vertex v we only required +the DP solutions for its children v1 and v2. Hence, our dependency graph is given by +our input tree T where all edges are directed towards the root. This implies that every +node in the dependency graph can only reach those nodes on a path to the root and thus +Property (2) of Definition 8 is satisfied with h being the height of T. Additionally, one can +verify that above all operations also satisfy Property (3) of Definition 8. Finally, observe +that in each step we only used a constant number of operations from Lemma 6 and at most +one (min, +)-convolution from Lemma 7. +E.2 +The Approximate DP +Now we explain how we solve the above DP more efficiently by computing approximate +solutions ADP(v, ·). This will reveal the procedures ˜Pi from Definition 8. +In our approximation algorithm, we do everything exactly as above except that we +replace each exact solution DP(v, ·) with the approximate solution ADP(v, ·). Then we add a +postprocessing step in which we round ADP(v, ·), i.e., we set +ADP(v, ·) = ⌈ADP(v, ·)⌉1+δ +(12) +for a parameter δ > 0 that we will set later. +Note that all of our operations are exact except the rounding step which loses a factor of +α = 1 + δ. Thus, Property (4a) of Definition 8 is satisfied. Additionally, observe that in each +step we only used a constant number of operations from Lemma 6 and at most one (min, +)- +convolution from Lemma 7. This implies that Property (4b) is satisfied. Furthermore, all +functions we consider are monotone and our rounding step ensures that each row ADP(v, ·) +has at most p = O(log1+δ n) pieces. Hence, Property (4c) is also satisfied. +This implies that the DP is (h, 1+δ, O(log1+δ(n)))-well-behaved. By applying Theorem 9 +with δ = ln(1 + ϵ)/(h + 1), we obtain the following proposition which shows that on binary +trees, the approximation algorithm computes a bicriteria (1 + ϵ, 1)-approximate solution. +We note that for constant ϵ, the running time essentially becomes ˜O(n · h2), where h is the +height of the tree. Thus, for trees of height ˜O(1), we obtain a near-linear running time. +▶ Proposition 33. Let ϵ > 0. The approximation algorithm computes a bicriteria (1 + ϵ, 1)- +approximate solution for the simultaneous source location problem on binary trees in time +O(n · (h log(n)/ϵ)2 log(h log(n)/ϵ)), where h is the height of the tree. + +54 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +E.3 +Extension to General Graphs (Proof of Theorem 4) +We prove Theorem 4 by giving reductions to the binary setting. +First, suppose that G is a tree with potentially unbounded degree. Then we turn G +into a binary tree T using the same construction as in the proof of Lemma 29. That is, we +replace each vertex u in G by a balanced binary tree τu with deg(u) leaves cu,v1, . . . , cu,vdeg(u), +where the vi are the children of u in G; the internal edges of τu have capacity ∞ and we +denote the root of each τu by ru. Furthermore, for each edge (u, v) in G, we insert the edge +(cu,v, rv) into T with capacity cap(cu,v, rv) = cap(u, v). By the same arguments as in the +proof of Lemma 29, T has O(n) vertices and height O(h log n), where h is the height of G. +It is straight-forward to see that with this construction, there exists a flow from u to v in +G if and only if there exists a flow from ru to rv in T. Now, in T we have already set the +edge capacities and it remains to set the vertex demands. For each vertex ru in T, we set +d(ru) = d(u), and for all other vertices v in T, we set d(v) = 0. Furthermore, in our instance +of the simultaneous source location problem we set that each vertex ru can be picked as +a source and none of the other vertices in T can be picked as a source. Note that there +exists a one-to-one correspondence between sources in G and sources in T. Together with +our observation for flows above, this means that solving the simultaneous source location +problem on T gives a solution for G. +To obtain the bicriteria (1 + ϵ, 1)-approximation result for trees, we apply the approxima- +tion algorithm from Proposition 33 on T. +Finally, to obtain the (1 + ϵ, O(log4 n))-approximate solution for a general graph G, we +proceed as follows. We build the binarized Räcke tree T for G as per Lemma 29 and recall that +T has quality q = O(log4 n) and height ˜O(1). In T, we set the bits to indicate that all leaves +can be used as sources but none of the other vertices might be used as a source. We apply +the approximation algorithm from Proposition 33 on T to obtain a (1 + ϵ, 1)-approximate +solution on T in time ˜O(n). Now let us point out that the Räcke tree from Theorem 15 +(and, therefore, also the binarized Räcke tree from Lemma 29) is also a tree flow sparsifier. +That is, if there exists a feasible flow F in G, then there exists a flow of the same value +between the corresponding leaves in T. Additionally, for any feasible flow F with value v +between leaves in T, there exists a feasible flow with value 1 +qv between the corresponding +vertices in G. Therefore, if we are allowed to exceed the edge capacities in G by a factor +of q = O(log4 n), the flow that we compute in T is feasible in G. This gives that we can +compute a (1 + ϵ, O(log4 n))-approximate solution in time ˜O(n). +E.4 +Extension to the Dynamic Setting (Proof of Theorem 5) +To prove Theorem 5, we first consider the special case of dynamic binary trees (which is +not mentioned in the theorem). We show that for binary trees we can maintain a bicriteria +(1 + ϵ, 1)-approximate solution with worst-case update time ˜O(h3/ϵ2), where h is an upper +bound on the height of the tree. Then we show that the results of the theorem can be derived +from this result. +Consider a dynamic binary tree on which we maintain the approximate DP from Sec- +tion E.2. We will exploit that T and the dependency tree of our DP coincide. Hence, an +update in T will trigger the same update in the dependency tree. Observe that the update +operation SetDemand(v, d) triggers a change to ADP(v, ·). Then we can recompute the global +approximate DP table using Theorem 10. Since the DP is well-behaved, the tree has height +at most h and since we set δ = O(h/ϵ), the theorem implies that we need time ˜O(h3/ϵ2) to +recompute the ADP solution. Similarly, for SetCapacity((u, v), c) we can again update the + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +55 +rows ADP(u, ·) and ADP(v, ·) and we update the entire DP table using Theorem 10. By the +same arguments as above, this takes time ˜O(h3/ϵ2). For Remove(u, v), we remove the edge +(u, v) from T and by the same reasoning as before we get update time ˜O(h3/ϵ2). Finally, +consider Insert((u, v), c), where we assume that v becomes the child of u. Then we might +have the issue that before the update, v is not the root of its connected component. To +mitigate this issue, we run the same re-rooting procedure as described in Section D.5. As +described in Section D.5, this will only recompute the solutions of O(h) DP cells and thus +we again have a total update time of ˜O(h3/ϵ2). +Next, we prove the results from Theorem 5. First, consider the case in which all edge +capacities are set to 1 and where we want to obtain a bicriteria (1 + ϵ, no(1))-approximate +solution with amortized update time no(1)/ϵ2 and preprocessing time O(n2). Let G be +the dynamic input graph. We maintain the dynamic binarized Räcke tree T for G as per +Lemma 31 and remark that the dynamic Räcke tree from Theorem 16 is also a tree flow +sparsifier. We note that any update to G triggers an update operation on T that requires +amortized update time no(1). On T, we allow the leaves to act as sources but no other vertices. +Furthermore, we set the demands of the leaves in T to the demands of the corresponding +vertices in G; all other vertices have demand 0. Now we use the data structure for binary +trees from the previous paragraph to maintain a dynamic bicriteria (1 + ϵ, 1)-approximate +solution on T. That is, when a vertex demand changes in G, we update the corresponding +vertex demand in T. When an edge is inserted or deleted in T due to the subroutine from +Lemma 31, then we update the data structure from the previous paragraph that maintains +the DP solution on T. By the same argumentation as in Section E.3, we obtain that since T +has quality no(1), if we can exceed the edge capacities in G by a no(1) factor then any feasible +flow in T is also feasible in G. This implies the result claimed in the theorem. +If G is a tree of height h but (potentially) with unbounded degrees, we can maintain a +bicriteria (1 + ϵ, 1)-approximate solution with worst-case update time ˜O(h3/ϵ2) and prepro- +cessing time O(n2/ϵ2) similar to above. That is, we transform G into a binary tree using the +same procedure that we use in the proof of Lemma 31, where we replace each vertex u by a +subtree τu with root ru. Similar to what we argued in Section E.3, we only allow the vertices +ru as roots in T and obtain any flow in T corresponds to a flow in G. Then by applying the +dynamic data structure for binary trees on T, we obtain the result. +Finally, let us consider the case in which we wish to obtain bicriteria approximation +algorithms when we only allow the update operations SetDemand(v, d). In this case, we +observe that the underlying graph is static, since only the vertex demands change. Therefore, +for our input graph G, we can build the Räcke tree from Theorem 15 which is also a tree flow +sparsifier and we consider its binarized version T as per Lemma 31. Note that building this +tree with quality ˜O(log4 n) takes time ˜O(m). Given such a static Räcke tree, we can use our +dynamic data structure for binary trees from above to support the operations SetDemand(v, d) +on T. Since T has height ˜O(1), we obtain the result with the bicriteria (1 + ϵ, O(log4 n))- +approximation. To obtain the bicriteria (1 + ϵ, O(log2 n log log n))-approximation we do +exactly the same as above, but instead of using the Räcke tree from Theorem 15, we use the +Räcke tree from Harrelson, Hildrum and Rao [37] which can be used as a tree flow sparsifier. +As it has quality O(log2 n log log n) and can be constructed in time poly(n), we obtain the +result. + +56 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +F +Recourse Bounds +In this section discuss the recourse bounds we derive. To motivate these lower bounds, let us +note that “classic” dynamic algorithms with polylogarithmic update time maintain a single +explicit solution in memory; this is desirable in many practical scenarios. However, some +dynamic algorithms (like our DP algorithms above) only return the value of an approximate +solution in polylogarithmic time, which is sometimes referred to as implicit. To understand +whether for our problems implicit solutions are necessary, we consider algorithms which +maintain multiple explicit solutions, of which only one has to be feasible. We believe that this +is an interesting setting to look at, as it essentially interpolates between the two scenarios +above. If even algorithms with multiple solutions must have high recourse, this suggests that +implicit solutions are somehow inevitable. We show below that for fully dynamic knapsack +and fully dynamic k-balanced partitioning the latter is the case. +Here, we consider dynamic algorithms over inputs that are undergoing insertions and +deletions via an update operation. The algorithms are allowed to maintain multiple explicit +solutions and must ensure that after every time step, there exists a solution with certain +guarantees while minimizing the recourse for updating the solutions. More concretely, we +consider algorithms which explicitly maintain s solutions S(t) +1 , . . . , S(t) +s +for each time step t. +Here, we assume that after each time step a single update operation is performed, after +which an algorithm can make changes to its solutions. We say that an algorithm maintains +an α-approximate solution if for each time step t, there exists an index i = i(t) such that +S(t) +i +is a feasible and α-approximate solution for the problem we study. +Observe that in this setting, the algorithm might have much lower recourse, since for each +time t it may pick a different solution. Thus, it may not have to update any of the solutions +significantly after the update operations. Further note that this notion of ensuring that at +each time step there exists a feasible solution is somewhat reminiscent of list decoding in +coding theory, where the decoder can output a list of messages and only has to ensure that +the correct messages is contained in that list. +Measuring the recourse will be problem-specific, based on how the solutions for the +problems are stored. In general, given two solutions from consecutive time steps, we let +d(S(t) +i , S(t+1) +i +) denote the (problem-specific) recourse incurred by the i’th solution at time +step t (see below for how to set d(·, ·) for the problems we study). The total recourse of an +algorithm is given by +� +t +s +� +i=1 +d(S(t) +i , S(t+1) +i +). +Next, we will present the concrete recourse lower bounds that we derive. +Recourse Bounds for Knapsack. In knapsack, the solutions S(t) +i +simply correspond +to subsets of items which are contained in the knapsack. To measure the recourse, we set +d(S(t) +i , S(t+1) +i +) = +���S(t) +i △S(t+1) +i +���, i.e., we consider the cardinality of the symmetric difference +of the i’th solution at time steps t and t + 1. +Our main result shows that for a fixed accuracy ϵ, any dynamic (1 − ϵ)-approximation +algorithm must maintain Ω(1/ϵ) solutions or it must have recourse Ω( n +ϵ ), even when only a +single item is inserted. +▶ Theorem 34. Let ϵ ∈ (0, 1/2). Assume s < +1 +8ϵ(1+2ϵ) and n ∈ N is a sufficiently large +multiple of s. Then any dynamic randomized (1 − ϵ)-approximation algorithm for knapsack +with s solutions must have recourse Ω( n +s ). This holds even for a single item insertion. + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +57 +Recourse Bounds for k-Balanced Partitioning. In k-balanced partitioning, each +solution S(t) +i +consists of k clusters V (i,t) +1 +, . . . , V (i,t) +k +that partition the set of vertices V . To +measure the recourse, we set d(S(t) +i , S(t+1) +i +) = �k +j=1 +���V (i,t) +j +△V (i,t+1) +j +���, i.e., we consider the +total number of vertices that change their set V (i,·) +j +from time t to t + 1. +Our main result shows that for any C and fixed ϵ, any algorithm that maintains a +(C, 1 + ϵ)-approximate solution must use Ω(1/ϵ) solutions or it must have amortized recourse +Ω(ϵ2 n +k ), even when only O(1/ϵ) edges are inserted. Here, the amortized recourse refers to +the total recourse divided by the total number of update operations. +▶ Theorem 35. Let C > 0 be arbitrary and ϵ ∈ (0, 1/2). Assume k ≥ 4 and s < +1 +4ϵ. Then +any dynamic randomized (C, 1 + ϵ)-approximation algorithm for k-balanced partitioning with +s solutions must have amortized recourse Ω(ϵ2 n +k ). This holds even for O(1/ϵ) edge insertions. +F.1 +Proof of Theorem 34 +We prove Theorem 34. We use Yao’s principle [63], i.e., we consider a deterministic algorithm +and give a distribution over inputs, showing that in expectation the algorithm will have +recourse Ω( n +s ). +We consider an instance in which initially we have n items, and each item i has weight +wi = 1 and price pi = 1. We refer to these items as small items. We set the budget of our +knapsack to B = n. Note that in this instance, OPT = n because all small items fit into the +knapsack. +Now we sample an integer j uniformly at random from [2s − 1] = {0, . . . , 2s − 1}, and we +set k = i · n +2s + n +4s. We insert a single heavy item with p = n − k + 2ϵn and w = n − k. Note +that after inserting the heavy item, we have that OPT = n + 2ϵn since the optimal solution +consists of the heavy item and k small items. +We let S1, . . . , Ss denote the solutions maintained by the algorithm before the heavy item +was inserted and we let S′ +1, . . . , S′ +s denote the solutions after the heavy item was inserted. +We write small(Si) to denote the number of small items in solution Si. +In the following, we will show that any (1 − ϵ)-approximate solution S′ +i must contain +the heavy item and “almost” k small items. However, we will also show that with constant +probability all solutions Si had “much less” or “much more” than k small items initially. +This then gives that obtaining any (1 − ϵ)-approximate solution must encur high recourse. +We follow this proof strategy in reverse order. We start by showing that with constant +probability, all Si have “much less” or ”much more” than k small items. +▶ Lemma 36. With probability at least 1/2, it holds that |k − small(Si)| ≥ n +4s for all i ∈ [s]. +Proof. Suppose that we partition the set {1, . . . , n} into 2s consecutive intervals, each of +length +n +2s. Note that k is the middle point of one of these intervals. Furthermore, as there +are only s solutions and 2s intervals, at least half of the intervals do not contain a number +from the set {small(Si): i = 1, . . . , s}; we call these intervals empty. Thus, with probability +at least 1/2, k is the middle point of an empty interval. If the interval containing k is empty, +then k has distance at least +n +4s to small(Si) for all i = 1, . . . , s. +◀ +Next, recall that S′ +1, . . . , S′ +n are the solutions maintained by the algorithm after the +insertion of the heavy item. We show that any (1 − ϵ)-approximate solution must contain +the heavy item and some small items. +▶ Lemma 37. Suppose that S′ +i is a (1 − ϵ)-approximate solution. Then S′ +i contains the heavy +item and a positive number of small items. + +58 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +Proof. First, suppose that a solution S′ +i only contains small items. Then its total price is at +most n. However, we have that +(1 − ϵ) OPT = (1 − ϵ)(1 + 2ϵ)n > n, +where we used that ϵ < 1/2. Hence, S′ +i is not a (1 − ϵ)-approximate solution. +Second, suppose that S′ +i only contains the heavy item. Then we have that +(1 − ϵ) OPT = (1 − ϵ)(p + k) += p + k − ϵ(p + k) +≥ p + n +4s − ϵ(1 + 2ϵ)n +> p + 2ϵ(1 + 2ϵ)n − ϵ(1 + 2ϵ)n +> p, +where we used that k ≥ n +4s, p+k = n+2ϵn and s < +1 +8ϵ(1+2ϵ). Therefore, a solution containing +only the heavy item is not (1 − ϵ)-approximate. +◀ +Next, we show that any (1−ϵ)-approximate solution must contain “almost” k small items. +▶ Lemma 38. Suppose S′ +i is a (1 − ϵ)-approximate solution. Then +k − n +8s ≤ small(S′ +i) ≤ k. +Proof. The upper bound follows from the fact S′ +i must contain the heavy item (by Lemma 37) +of weight n − k and then it can only include k small items since the budget constraint is set +to B = n. +To prove the lower bound, note that since S′ +i is a (1 − ϵ)-approximate solution, we have +that its solution has value +p + small(S′ +i) ≥ (1 − ϵ) OPT = (1 − ϵ)(p + k). +Hence, we get that +small(S′ +i) ≥ k − ϵ(p + k) += k − ϵ(1 + 2ϵ)n +> k − n +8s, +where we used that s < +1 +8ϵ(1+2ϵ). +◀ +To finish the proof of the theorem, we condition on the event from Lemma 36, i.e., we +have that |k − small(Si)| ≥ n +4s for all i ∈ [s]. +Now consider any solution S′ +i after the insertion of the heavy item. If S′ +i is not (1 − ϵ)- +approximate, we can ignore S′ +i. If S′ +i is (1 − ϵ)-approximate then it satisfies k − +n +8s ≤ +small(S′ +i) ≤ k by Lemma 38. Since we are assuming the event from Lemma 36, the algorithm +had to insert/delete at least +n +8s small items into/from Si to obtain S′ +i. +Since the event from Lemma 36 occurs with probability at least 1/2 and the above +argument holds for all (1 − ϵ)-approximate solutions S′ +i, we have that the expected recourse +is Ω( n +s ). + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +59 +F.2 +Proof of Theorem 35 +We prove Theorem 35. Again, we apply Yao’s principle [63], i.e., we consider a deterministic +algorithm and give a distribution over inputs, showing that in expectation the algorithm will +have amortized recourse Ω(ϵ2 n +k ). +We consider a graph with n vertices. Our initial instance consists of +k +2ϵ star graphs, each +of which contains 2ϵ n +k vertices. Note that here an optimal solution places the vertices from +exactly 1 +2ϵ star graphs into each partition Vj; there are no edges between vertices from different +Vj and hence the optimal cut-value is zero. Hence, the solution of any (C, 1 + ϵ)-approximate +solution must also have cut-value zero. +In the update phase, we sample s edges between the central nodes of the star graphs +uniformly at random and insert them into the graph. Note that after the insertion of the +edges, we connected at most s star graphs and the largest connected component has size at +most s·2ϵ n +k ≤ 1 +2 +n +k , where we used that s ≤ +1 +4ϵ. Hence, the optimal solution still has cut-value +zero and thus any (C, 1 + ϵ)-approximate solution must have cut-value zero. +Next, let us analyze the recourse of an algorithm which starts with initial solutions +S(0) +1 , . . . , S(0) +s . In a first step, we show that solutions which at time 0 splits one of the star +graphs up “too much” must entail high recourse. In a second step, we consider all other +solutions and show that our insertions still trigger high recourse in expectation. +We say that a solution S(0) +i += {V (i,0) +1 +, . . . , V (i,0) +k +} is useful if for all star graphs H, it +holds that there exists an index j such that V (i,0) +j +contains at least ϵ n +k vertices from H and +at most ϵ n +k vertices are placed in � +j′̸=j V (i,0) +j′ +. Given a star graph H and a solution S(0) +i +, we +write j(H, i) to the denote the index j such that V (i,0) +j +contains at least at least ϵ n +k vertices +from H. If a solution is not useful, we call it useless. +First, consider solutions which are useless. There exist two cases. Case A: Suppose there +exists a star graph H such that for all indices j it holds that � +j′̸=j V (i,0) +j′ +contains more than +ϵ n +k vertices from H. Observe that if the algorithm wants to use this solution after the edge +insertions finished, it must ensure that the cut-value is zero. Thus it must move at least +ϵ n +k vertices to one of the V (i,0) +j +which requires +���� +j′̸=j V (i,0) +j′ +��� ≥ ϵ n +k vertex moves. Case B: +Suppose there exists a star graph H such that for all j it holds that V (i,0) +j +contains less than +ϵ n +k vertices from H. Using that ϵ ∈ (0, 1 +2), also in this case the algorithm must move at +least (1 − ϵ) n +k ≥ ϵ n +k vertices such that eventually all of H is contained in the same set V (i,s) +j +when the updates finished. We conclude that for useless solutions our theorem holds after +amortizing over s ≤ 1 +ϵ insertions. +Second, for the remainder of the proof consider only solutions S(0) +i += {V (i,0) +1 +, . . . , V (i,0) +k +} +which are useful. Observe that when we insert an edge between two star graphs H1 and H2, +then if j(H1, i) ̸= j(H2, i) the algorithm must move at least ϵ n +k vertices to ensure that after +the s insertions finished, all vertices from H1 and H2 are placed in the same set V (i,s) +j +for +some j. We call such an insertion expensive for solution i. +Observe that if our edge insertions are such that they contain an expensive insertion for +all solutions, then updating any solution S(0) +i +such that S(s) +i +is (C, 1 + ϵ)-approximate will +incur recourse at least ϵ n +k . The rest of our proof is devoted to showing that with constant +probability this event occurs. This will prove the theorem. +We start by considering a fixed solution S(0) +i +and a single random edge insertion between +randomly picked star graphs H1 and H2. Recall that there are +k +2ϵ star graphs in total. +Furthermore, we have that +���V (i,0) +j +��� ≤ (1 + ϵ) n +k for all j and thus for each j there can be at +most (1+ϵ) +ϵ +star graphs H with j = j(H, i). Hence, for the probability that the edge insertion + +60 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +is expensive we get that +Pr (j(H1, i) ̸= j(H2, i)) = 1 − Pr (j(H1, i) = j(H2, i)) +≥ 1 − (1 + ϵ)/ϵ +k/(2ϵ) += 1 − 2(1 + ϵ) +k +≥ 1 +2, +where we used that k ≥ 4. +Next, we consider a fixed solution S(0) +i +and s edge insertions between star graphs which +were picked independently and uniformly at random. Then with probability at least 1 − 2−s, +at least one of these edge insertions is expensive for solution i. +Finally, observe that probability that for all solutions i there exists an expensive edge +insertion is at least +� +1 − 2−s�s = exp +� +s ln(1 − 2−s) +� +≥ 1 + s ln(1 − 2−s) +≥ 1 − s2−s +≥ 1 +4, +where we used that exp(x) ≥ 1 + x for all x ∈ R, the Taylor expansion of ln(x) for x close +to 1 and the fact that s2−s ≤ 3 +4 for all s. +We conclude that with constant probability, for all solutions i there exists an expensive +edge insertion. In this case, the algorithm has total recourse at least ϵ n +k . Hence, the expected +total recourse of the algorithm is Ω(ϵ n +k ). Since we only performed s edge insertions, this +gives an amortized recourse of Ω(ϵ2 n +k ). +G +Non-Monotone Functions and ℓ∞-Necklace Alignment +So far we have only considered monotone piecewise constant functions. Now we will generalize +some of our results to piecewise constant functions with multiple non-monotonicities and +provide the details in Section G.1. We also derive new approximation algorithms for the +ℓ∞-necklace problem in Section G.2. In particular, for ℓ∞-necklace we present the first +approximation algorithm with near-linear running time with additive error ϵ. We also present +the first dynamic approximation algorithm for this problem which achieves additive error ϵ +and has update time O((1/ϵ)2 log(1/ϵ)); the algorithm has preprocessing time O(1) when +starting with empty vectors x and y and requires sublinear space O(1/ϵ). See Theorem 44 +for the details of our results. +G.1 +Piecewise Constant Functions With Non-Monotonicities +We now show that we can perform efficient operations on piecewise constant functions +even when these functions contain non-monotonicities. However, the running times of our +subprocedures will typically have some dependency on the number of non-monotonicities of +the function. +Let us formalize our notion of non-monotonicities. We say that a function f : [0, t) → +[0, W] ∪ {−∞, ∞} has k monotone segments if there exist values 0 = x0 < x1 < · · · < xk = t + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +61 +such that on each interval [xi, xi+1), f is monotone. Here, we require that either f is +monotonically decreasing on all segments or it is monotonically increasing on all segments. +Note that a monotone function has k = 1 monotone segments (by setting x0 = 0 and x1 = t) +and that the points x1, . . . , xk−1 can be viewed as the points in which f is non-monotone. +One crucial operations will again be rounding. However, unlike previously we will mostly +talk about rounding to multiples of δ instead of rounding to powers of 1 + δ. This will be +convenient for our applications to ℓ∞-necklace later. We will also briefly mention how to +extend our results from this subsection to the setting in which we round to powers of 1 + δ. +Next, let δ > 0 and consider a simple rounding function that rounds down to multiples +of δ. More concretely, for y ∈ R we set ⌊y⌋∗ +δ = max{i · δ: i · δ ≤ y, i ∈ Z} and we follow the +convention that ⌊−∞⌋∗ +δ = −∞ and ⌊∞⌋∗ +δ = ∞. We also extend the rounding operation to +functions f : [0, t) → [0, W] ∪ {−∞, ∞} by defining ⌊f⌋∗ +δ : [0, t) → [0, W] ∪ {−∞, ∞} to be +the function with ⌊f⌋∗ +δ(x) = ⌊f(x)⌋∗ +δ for all x ∈ [0, t). Next, we show that the function ⌊f⌋∗ +δ +can be computed efficiently and that it has only few pieces. +▶ Lemma 39. Let δ > 0 and let f : [0, t) → [0, W] ∪ {−∞, ∞} be a piecewise constant +function with p pieces and k monotone segments. Then we can compute the function ⌊f⌋∗ +δ in +time O(p log p) and ⌊f⌋∗ +δ has O(k · W/δ) pieces. +Proof. Let (x1, y1), . . . , (xp, yp) denote the list representation of f. We construct the list +representation (x′ +1, y′ +1), . . . , (x′ +p, y′ +p) of ⌊f⌋∗ +δ. For all i = 1, . . . , p, we set x′ +i = xi and y′ +i = ⌊yi⌋∗ +δ. +After that, we merge all consecutive pieces that have the same y′ +i-values; this can be done +exactly as in the pruning step described in the proof of Lemma 6. Since f takes values in +[0, W] ∪ {−∞, ∞}, there are O(W/δ) choices for multiples of δ in [0, W]. In particular, on +each monotone segment of f, ⌊f⌋∗ +δ has O(W/δ) pieces. Since f has k monotone segments +this implies that ⌊f⌋∗ +δ has O(k · W/δ) pieces in total. Note that all operations from above +can be performed in linear time and the running time bound stems from the fact that we +also need to store the pieces in a binary search tree. +◀ +Next, we show that we can compute the (min, +)-convolution of two piecewise constant +functions in time that is quadratic in the number of their pieces. The lemma generalizes the +result from Lemma 7 because we drop the assumption that one of the functions needs to be +monotone (but this comes at the cost of a more complicated proof). We prove the lemma in +Section G.1.1. +▶ Lemma 40. Let f1, f2 : [0, t) → [0, W] ∪ {−∞, ∞} be piecewise constant functions which +have at most p pieces. Then we can compute f = f1 ⊕ f2 in time O(p2 log p) and f has O(p2) +pieces. +By combining the two lemmas above, we can show that we can efficiently compute +additive approximations of (min, +)-convolutions even in the case of non-monotonicities. +More concretely, we say that f : [0, t) → [0, W] ∪ {−∞, ∞} is an additive ϵ-approximation of +g: [0, t) → [0, W] ∪ {−∞, ∞} if g(x) − ϵ ≤ f(x) ≤ g(x) for all x ∈ [0, t). Now we obtain the +following theorem. +▶ Theorem 41. Let f, g: [0, t) → [0, W] ∪ {−∞, ∞} be two functions with k monotone +segments and suppose we have already computed ⌊f⌋∗ +δ and ⌊g⌋∗ +δ. Then the function (⌊f⌋∗ +δ) ⊕ +(⌊g⌋∗ +δ) is an additive 2δ-approximation of f ⊕ g, has at most O((k · W/δ)2) pieces and can be +computed in time O((k · W/δ)2 log((k · W/δ)2)). +Proof. The approximation ratio follows from the triangle inequality. The claims about the +number of pieces and the running time follow from combining Lemma 39 and Lemma 40. +◀ + +62 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +We note that by stating Lemma 39 for the rounding operation ⌈·⌉1+δ that rounds to +powers of 1 + δ (see Lemma 6), we can obtain the following version of Theorem 41. +▶ Theorem 42. Let f, g: [0, t) → [0, W] ∪ {−∞, ∞} be two functions with k monotone +segments and suppose we have already computed ⌈f⌉1+δ and ⌈g⌉1+δ. Then (⌈f⌉1+δ)⊕(⌈g⌉1+δ) +is a (1+δ)-approximation of f ⊕g, has at most O((k·log1+δ(W)2) pieces and can be computed +in time O((k · log1+δ(W))2 log((k · log1+δ(W))2)). +This result generalizes our previous method of first rounding a monotone function via +Lemma 6 and then applying the efficient convolution from Lemma 7. More concretely, +observe that monotone functions have one monotone segment and, thus, after rounding +both functions, our algorithm from Lemma 7 computes the (min, +)-convolution in time +O(log2 +1+δ(W) log log1+δ(W)) which is the same running time that we obtain by combining +the two lemmas above. Hence, the algorithm from Theorem 42 matches this result for k = 1 +and it generalizes it when we apply it for k > 1. +G.1.1 +Proof of Lemma 40 +We assume that fi for i = 1, 2 is given as a doubly linked list (xi +1, yi +1), . . . , (xi +p, yi +p) such that +xi +j < xi +j+1 for all 1 ≤ j < p. We will output f in the same representation. +To compute f we will make use of the following non-overlapping interval data structure +(NOI). Let [a, b] and [a′, b′] be two subsets of the real line. We call each of them an interval +and say that they overlap if [a, b] ∩ [a′, b′] ̸= ∅. We say that an interval [a, b] is empty if a ≥ b. +The NOI data structure stores a set S of non-overlapping, non-empty intervals I = [a, b] and +supports the following operations: +ClosestLargerInterval(z), which given a number z returns the interval [a, b] together with +a Boolean value bool. If bool is true, then z ≤ b and there is no interval [a′, b′] in S with +z ≤ b′ < b. Note that it is possible that z belongs to [a, b]. If bool is false, then there +exists no interval [a, b] with z ≤ b and the returned values for a and b are undefined. +InsertInterval(a, b), which inserts the interval [a, b] into S, merging it with any interval +that it overlaps with and updating S accordingly. +There exists an efficient implementation of such a data structure as stated in the next +claim, which we prove at the end of this section. +▷ Claim 43. +There exists an implementation of the non-overlapping interval data structure +such that any sequence of q operations takes time O(q log q). +We compute f as follows. Note that the function values of f1 and of f2 are constant +over each 2-dimensional rectangle whose corners are (x1 +s, x2 +t), (x1 +s, x2 +t+1), (x1 +s+1, x2 +t), and +(x1 +s+1, x2 +t+1) for any 1 ≤ s ≤ p and 1 ≤ t ≤ p. We call this rectangle Rst and denote by +[x1 +s + x2 +t, x1 +s+1 + x2 +t+1] the range of the rectangle Rst and by y1 +s + y2 +t the function value of +the rectangle, where we assume that ∞ + y with y ∈ W∞ equals ∞. There are K2 such +rectangles. +Now note that for any value x with x1 +s +x2 +t ≤ x ≤ x1 +s+1 +x2 +t+1, i.e., x is in the range of the +rectangle Rst, the function value y1 +s + y2 +t is one of the sums that occurs in the computation +of f(x) = min¯x{f1(¯x) + f2(x − ¯x)}. We will compute f(x) (for all values x “simultaneously”) +by comparing the function values of all rectangles Rst to whose range x belongs. The main +observation that we exploit is the following: As we will consider the rectangles by decreasing +function values, the first rectangle (in this order) to whose range a value x belongs is the +rectangle whose function value equals f(x). + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +63 +Thus, when processing a rectangle, we need to determine all ranges, i.e, subintervals of +[0, t], to which no function value has yet been assigned. To do so, we use the NOI data +structure to store the intervals of all values x for which we have already assigned a function +value. Furthermore we use a balanced binary search tree B that stores at its leaves every +interval to which a function value has already been assigned, together with its (constant) +function value. Specifically, we will store these ranges in the leaves of B, ordered by their +smaller boundary value x′. The difference between the two is that the NOI data structures +merges overlapping intervals, no matter what their function value is, while every interval +stored as a leaf of B has the same function value, i.e., has a constant f-value. +To be precise we proceed as follows: We first generate all rectangles Rst by iterating +over the lists of f1 and f2 and sort them by non-decreasing order of their function value. +This takes time O(p2 log p). Then we process the rectangles in this order. To do so, we +first initialize an empty NOI data structure as well as an empty balanced binary search tree +B. Next we describe how to process the rectangles. Let Rst be the next rectangle to be +processed. We execute the following steps for Rst: +1. z = x1 +s + x2 +t +2. (a, b, bool) = ClosestLargerInterval(z) +3. while bool is true and b < x1 +s+1 + x2 +t+1 do +a. if z ̸∈ [a, b] then insert the interval [z, a] together with the function value of Rst into B. +b. z = b +c. (a, b, bool) = ClosestLargerInterval(z) +4. If bool is true then insert the interval [z, a] together with the function value of Rst into B; +else insert the interval [z, x1 +s+1 + x2 +t+1] together with the function value of Rst into B. +5. InsertInterval(x1 +s + x2 +t, x1 +s+1 + x2 +t+1). +Once all rectangles have been processed, we traverse the leaves of B in order and connect +them by a doubly linked list to create an (ordered) list representation of the function f. As +we process the rectangles in increasing order of function value this guarantees that for each +value x the smallest function value of any rectangle Rst is returned as f(x). +Note that each insertion into B takes time O(log p) and the number of calls to the NOI +data structure is proportional to the number of rectangles plus the number of intervals +merged in the NOI data structure. As processing a rectangle creates at most one new interval, +and merged intervals are never separated again, the number of interval merges is at most the +number of rectangles. Thus, there are at most p interval merges and at most 2p2 insertions +into B. Hence, the total running time for the above algorithm is O(p2 log p) plus the time for +the NOI data structure, which, by Claim 43, is also O(p2 log p) as q = O(p2). +We still have to prove Claim 43. +Proof of Claim 43. We implement the NOI data structure with a balanced binary search +tree. The leaves store the non-overlapping intervals, ordered by their upper endpoint. +The ClosestLargerInterval(z) operation searches for the interval [a, b] such that b is the +smallest upper endpoint of an interval that is at least z. If no such interval exists, bool is set +to false, otherwise it is set to true and [a, b] is returned as interval. Note that finding [a, b] +takes time O(log q), as q is the maximum number of intervals stored in the balanced binary +tree. +The InsertInterval(a, b) operation first executes a ClosestLargerInterval(a) operation. Let +(a′, b′, bool) be the result. If bool is false, then the interval [a, b] is inserted as new interval +and the procedure terminates. Otherwise the interval [a′, b′] is the interval with smallest +upper endpoint such that a ≤ b′. Note that [a′, b′] might overlap with [a, b] and we test for +this next. If b < a′ then a leaf with range [a, b] is inserted into the balanced search tree and + +64 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +InsertInterval(a, b) terminates. Otherwise (b ≥ a′), let L be the leaf of the balanced search +tree that stores [a′, b′]. If b ≤ b′, the two intervals are merged by updating L to store the +interval [min(a, a′), b′] and InsertInterval(a, b) terminates. If, however, b > b′, it is possible +that the new interval [a, b] overlaps with even more intervals in S. Thus, we execute the +following steps: +1. z = b′ +2. (a′′, b′′, bool) = ClosestLargerInterval(z) +3. while bool is true do +a. If b < a′′ then the leaf L is updated to store the interval [min(a, a′), b] and InsertInterval +terminates. +b. The leaf storing the interval [a′′, b′′] is removed from the balanced search tree. +c. If b ≤ b′′ then the leaf L is updated to store the interval [min(a, a′), b′′] and InsertIn- +terval terminates. +d. Otherwise, z = b′′ and (a′′, b′′, bool) = ClosestLargerInterval(z). +4. L is updated to store the interval [min(a, a′), b]. +Note that this algorithm merges all intervals that overlap with [a, b] into one interval and +updates the balanced search tree accordingly. +Let t be the number of iterations executed by InsertInterval(x, y). The running time is +O((t + 1) log q) as each iteration executes one call to ClosestLargerInterval, one deletion of a +leaf in the balanced binary tree, and at most one modification of a label at a leaf. Every such +iteration decreases the number of leaves in the balanced binary tree by 1. Furthermore, each +call to InsertInterval that does not execute any iterations of the above while-loop increases +the number of leaves by at most 1 and there is no other operation that modifies the number +of leaves. As there are at most q calls to InsertInterval, the while-loop can be executed +at most q times over all calls to InsertInterval, each taking time O(log q). Thus, the total +runnning time for q calls to InsertInterval is O(q log q). +◀ +G.2 +ℓ∞-Necklace Alignment +Using our techniques from above, we present a novel approximation algorithm for the +ℓ∞-necklace alignment problem [14, 58]. In this problem, the input consists of two neck- +laces represented as two sorted vectors of n real numbers, x = ⟨x0, x1, . . . , xn−1⟩ and +y = ⟨y0, y1, . . . , yn−1⟩, where the xi, yi ∈ [0, 1) represent points on the unit-circumference +circle. We will sometimes refer to the elements xi and yj as beads. +We define the distance between two beads xi and yj by the minimum of the clockwise and +counterclockwise distances along the circumference of the unit-perimeter circular necklaces, +i.e., we set +d◦(xi, yj) = min{|xi − yj| , 1 − |xi − yj|}. +In the ℓ∞-necklace alignment problem, we need to find an offset c ∈ [0, 1) and a shift +s ∈ [n + 1] that minimize +n−1 +max +i=0 (d◦((xi + c) +mod 1, y(i+s) +mod n)). +In the above definition, the offset c encodes how much we rotate the first necklace clockwise +relative to the second necklace. Additionally, the shift s defines a perfect matching between +the beads such that bead i of the first necklace is matched with bead (i + s) mod n of the +second necklace. + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +65 +Bremner et al. [14] showed that the ℓ∞-necklace alignment problem can be solved exactly +in time ˜O(n2). We complement this by showing that we can compute a solution with additive +error ϵ in time ˜O(n + ϵ−2). +We also consider the dynamic version of the problem in which beads are inserted and +deleted. More concretely, we assume that initially x and y are empty and we offer the +following update operations: +Insert(i, α, β) which inserts α ∈ [0, 1) into x at the i’th position and it further inserts +β ∈ [0, 1) into y at the i’th position. We require that after the insertion, x and y are still +ordered. +Delete(i) which deletes xi from x and yi from y. +Note that both of these operations change the number of entries in x and y but they ensure +that x and y always have the same length. We show that we can maintain a solution with +additive error ϵ using update time O(1/ϵ2 log(1/ϵ)). The preprocessing time is O(1) and the +space usage is only O(1/ϵ) which is sublinear in the size of the vectors x and y. +▶ Theorem 44. Let ϵ > 0. There exists a static algorithm for the ℓ∞-necklace alignment +problem that computes a solution with additive error ϵ in time O(n + (1/ϵ)2 log(1/ϵ)). Fur- +thermore, there exists a fully dynamic algorithm for the ℓ∞-necklace alignment problem that +maintains a solution with additive error ϵ with update time O(1/ϵ2 log(1/ϵ)) and preprocessing +time O(1); the space usage of the algorithm is O(1/ϵ). +To obtain the result for the dynamic algorithm, we show that for vectors A, B ∈ Rn +that are undergoing element insertions and deletions, we can dynamically maintain an +approximation of the (min, +)-convolution A ⊕ B. We expect that this result will have +further applications. The proof of the theorem follows from Propositions 45 and 48 below. +G.2.1 +The Static Algorithm +Now we consider our static algorithm and prove the following proposition. +▶ Proposition 45. There exists a static algorithm for the ℓ∞-necklace alignment problem +that computes a solution with additive error ϵ in time O(n + (1/ϵ)2 log(1/ϵ)). +We devote the rest of this subsection to the proof of the proposition. +The Algorithm. Our algorithm is rather simple and (up to the part in which we perform +the rounding) it is the same as the one used by Bremner et al. [14]. Consider the input ϵ +(as error parameter), x = ⟨x0, x1, . . . , xn−1⟩ and y = ⟨y0, y1, . . . , yn−1⟩. Now we set δ = ϵ/2 +and perform a single pass over x and y and apply the rounding function ⌊·⌋∗ +δ to each of the +entries. While doing so, we compute the list representations of x and y (where we interpret +x and y as functions from [0, n) to [0, 1)) which have at most O(1/δ) pieces (by applying +Lemma 39 with W = 1). Then we compute the vectors +x′ = ⟨x0, x1, . . . , xn−1, ∞, . . . , ∞ +� +�� +� +n times +⟩, +x′′ = ⟨x0, x1, . . . , xn−1, −∞, . . . , −∞ +� +�� +� +n times +⟩, +y′ = ⟨yn−1, yn−2, . . . , y0, yn−1, yn−2, . . . , y0⟩, +but we do not store them explicitly. Instead, we only store their list representations. We note +that x′ is a monotonically increasing vector, x′′ has two monotonically increasing segments +and y′ has two monotonically decreasing segments. + +66 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +Next, we set a to the (min, −)-convolution of x′ and y′ and we set b to the (max, −)- +convolution of x′′ and y′ (we show below in Lemma 46 that we can compute these functions +efficiently). +Finally, we set v = 1 +2(b − a) and return min{vs : s ∈ [n]} as the solution for our problem. +We note that v can be efficiently computed via the list representations of a and b and we can +also quickly find the minimum over the vs by iterating over the list representation of v. +Analysis. Now we turn to the analysis of the algorithm above. We adapt the proof of +Theorem 6 in Bremner et al. [14] for approximate solutions and argue how to implement it +using piecewise constant functions. +We start by showing that we can compute (min, −)-convolution and (max, −)-convolution +as efficiently as the classic (min, +)-convolution. +▶ Lemma 46. Let f and g be two piecewise constant functions with p pieces and suppose +that g has k monotonically decreasing segments. Suppose that we can compute the (min, +)- +convolution of f ′ and g′ in time t(p, k) if f ′ and g′ have k monotonically decreasing segments. +Then in time O(t(p, k) + p log p) we can compute: +The (max, −)-convolution of f and g if f has k monotonically increasing segments. +The (min, −)-convolution of f and g if f has k monotonically increasing segments. +Proof. First, suppose that we wish to compute the (max, −)-convolution of two functions +f and g. +We show that we can compute the (max, −)-convolution of f and g via the +(min, +)-convolution of −f and g. Indeed, for all x it holds that: +max +¯x∈[0,x]{f(¯x) − g(x − ¯x)} = max +¯x∈[0,x]{−(−f(¯x) + g(x − ¯x))} += − min +¯x∈[0,x]{−f(¯x) + g(x − ¯x)} += −(((−f) ⊕ g)(x)). +To see that the running time is correct, note that we can compute the list representation +of −f in time O(p) and it takes takes O(p log p) to update the binary search tree in which +we store the pieces of −f. Furthermore, −f has k monotonically decreasing segments since +f has k monotonically increasing segments. Thus, we can apply the efficient algorithm for +(min, +)-convolution in time t(p, k) on −f and g. +We can prove the result for (min, −)-convolution similarly by computing a (min, +)- +convolution of g and −f. More concretely, for all x it holds that +min +¯x∈[0,x]{f(¯x) − g(x − ¯x)} = min +¯x∈[0,x]{−f(¯x) + g(x − ¯x)} = ((−f) ⊕ g)(x), +where in the first step we used the symmetry of (min, −)-convolution. The running time +analysis is exactly as above. +◀ +In the proof of Proposition 45 we need the following lemma. We will use the lemma to +find the optimal offset c for a given shift s. +▶ Lemma 47 (Fact 5 in [14]). Let z = ⟨z0, z1, . . . , zn−1⟩. Then +min +c∈R +n−1 +max +i=0 |zi + c| = 1 +2 +� +n−1 +max +i=0 zi − +n−1 +min +i=0 zi +� +and the minimizer for this quantity is given by c = − 1 +2(minn−1 +i=0 zi + maxn−1 +i=0 zi). +Next, we can prove Theorem 44. + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +67 +Proof of Theorem 44. We prove the theorem in three steps. In Step 1, we will prove that +we compute the correct result in the exact case (i.e., when we perform no rounding). This +first part is essentially the same proof as in in Bremner et al. [14] but with more details. In +Step 2, we argue about approximation guarantee of our algorithm. In Step 3, we prove its +running time. +Step 1: The Exact Case. First, we use Theorem 2 of Bremner et al. [14] which states +that if +˜y = ⟨y0, y1, . . . , yn−1, y0, y1, . . . , yn−1⟩ +then +min +c,s +n−1 +max +i=0 d◦((xi + c) +mod 1, y(i+s) +mod n) = min +c,s +n−1 +max +i=0 d−(xi + c, ˜yi+s), +where d−(a, b) = |a − b| for all a, b ∈ R. Thus, instead of directly optimizing the origi- +nal objective function minc,s maxn−1 +i=0 d◦((xi + c) mod 1, y(i+s) mod n), we will consider the +more convenient objective function minc,s maxn−1 +i=0 d−(xi + c, ˜yi+s) which involves no modulo +operations. +Indeed, consider the new objective function and for all s ∈ [n] we define the vector +z(s) ∈ Rn such that z(s)i = xi − y(i+s) mod n. Now we obtain that for the new objective +function it holds that: +min +c,s +n−1 +max +i=0 d−(xi + c, ˜yi+s) = min +c,s +n−1 +max +i=0 +��xi + c − y(i+s) +mod n) +�� += min +s +min +c +max +i +|z(s)i + c| += min +s +1 +2 +� +max +i {z(s)i} − min +i {z(s)i} +� +, +where in the first step we used the definition of d−(·, ·) and that ˜yk = yk +mod n for all k ∈ [2n], +in the second step we substituted the definition of z(s)i and in the third step we applied +Lemma 47. +The above implies that we need to compute the quantities maxi{z(s)i} and mini{z(s)i} effi- +ciently. Even more, consider the vector v ∈ Rn with entries vs = 1 +2 (maxi{z(s)i} − mini{z(s)i}) +and observe that the calculation above shows that the optimal objective function value is the +same as the smallest entry in v. Therefore, in the following we show that we can compute v +efficiently using the vectors a and b that we computed in our algorithm. +Recall the definitions of the two vectors x′ and y′: +x′ = ⟨x0, x1, . . . , xn−1, ∞, . . . , ∞ +� +�� +� +n times +⟩, +y′ = ⟨yn−1, yn−2, . . . , y0, yn−1, yn−2, . . . , y0⟩. +Now we let a ∈ R2n be the vector resulting from the (min, −)-convolution of x′ and y′, i.e., +ak = mini{x′ +i −yk−i} for all k ∈ [2n]. Now we observe that for each entry an+s′ with s′ ∈ [n], +it holds that +an+s′ = +n+s′ +min +i=0 {x′ +i − y′ +n+s′−i} = +n−1 +min +i=0 {xi − y(i−s′−1) +mod n}, +where in the second step we used that x′ +i = ∞ for i ≥ n and that y′ +n+s′−i = y((n−1)−(n+s′−i)) mod n = +y(i−s′−1) mod n since in y′ we concatenated the entries of y twice but in reverse order. Now +observe that if s′ = n − 1 − s then +a2n−s−1 = an+s′ = +n−1 +min +i=0 {xi − y(i−s′−1) +mod n} = +n−1 +min +i=0 {xi − y(i+s) +mod n} = +n−1 +min +i=0 {z(s)i}. + +68 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +Next, we define the vector x′′ such that: +x′′ = ⟨x0, x1, . . . , xn−1, −∞, . . . , −∞ +� +�� +� +n times +⟩. +We let b denote the vector resulting from the (max, −)-convolution of x′′ and y′, i.e., +bk = maxi{x′′ +i − y′ +k−i} for all k ∈ [2n]. +Now a similar argument as above shows that +b2n−s−1 = maxn−1 +i=0 {z(s)i} for all s ∈ [n]. More concretely, for each entry bn+s′ with s′ ∈ [n] +it holds that +bn+s′ = +n+s′ +max +i=0 {x′ +i − y′ +n+s′−i} = +n−1 +max +i=0 {xi − y(i−s′−1) +mod n}, +where we used that x′′ +i = −∞ for i ≥ n and the same argument relating the entries of y′ and +y as above. Thus, if s′ = n − 1 − s then +b2n−s−1 = bn+s′ = +n−1 +max +i=0 {xi − y(i−s′−1) +mod n} = +n−1 +max +i=0 {xi − y(i+s) +mod n} = +n−1 +max +i=0 {z(s)i}. +Combining the results above we get that vs = 1 +2(b2n−s−1 −a2n−s−1) for all s ∈ [n]. There- +fore, we get that the optimal objective function value is given by mins vs = mins 1 +2(b2n−s−1 − +a2n−s−1). In other words, to compute the optimal objective function value it suffices to +compute the difference 1 +2(b−a) and then to return the smallest entry in v with index between +n and 2n − 1. +Step 2: Approximation Guarantees. We argue that the algorithm returns an additive +ϵ-approximation. First, observe that in the algorithm all computations are performed exactly +except for the rounding at the beginning. In the rounding process, we decrease each entry by +at most δ = ϵ/2. Therefore, the triangle inequality implies that when we match bead xi to +bead yi+s, the error that was introduced by the approximation is at most 2δ = ϵ. Since in +the objective function we are only interested in the maximum error over all matched beads, +this implies that we obtain an additive ϵ-approximation. +Step 3: Running Time Analysis. It is left to analyze the running time of our algorithm. +Iterating over the input vectors x and y, rounding the entries and computing the list +representation of x and y can be done in time O(n). Recall that x and y have O(1/δ) +pieces. Therefore, we can also compute the vectors x′, x′′ and y in time O(1/δ log 1/δ). +Then Lemmas 46 and 40 imply that we can compute the (min, −)-convolution and the +(min, +)-convolutions in time O(1/δ2 log(1/δ)) and the resulting vectors have O(1/δ2) pieces. +Finally, the vector v can be computed in time O(1/δ2 log(1/δ)) and the minimum that we +return can be found by simply iterating over the pieces of v. Since previously we have set +δ = ϵ/2, this finishes the proof. +◀ +G.2.2 +The Dynamic Algorithm +We now give our extension to the dynamic setting of the ℓ∞-necklace alignment problem in +which there are insertions and deletions from x and y. +▶ Proposition 48. Let ϵ > 0. There exists a fully dynamic algorithm for the ℓ∞-necklace align- +ment problem that maintains a solution with additive error ϵ with update time O(1/ϵ2 log(1/ϵ)) +and preprocessing time O(1); the space usage of the algorithm is O(1/ϵ). +Proof. In the preprocessing, we initialize x and y as empty vectors and store them as +piecewise constant functions (as per Section 2) and we do not store them explicitly as vectors. +Furthermore, we set δ = ϵ/2. These operations can be done in time O(1). + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +69 +Next, consider an operation Insert(i, α, β) which asks to insert α into x at the i’th +position and to insert β into y at the i’th position. Since we are in the approximate setting, +instead of inserting the exact values of α and β, we insert ⌊α⌋∗ +δ into the i’th position of x +and ⌊β⌋∗ +δ into the i’th position of y. We perform these insertions by manipulating the list +representations of x and y. We only describe how to perform the manipulations for x, as for +y they are essentially the same. +Denote the list representation of x as (X0, Y0), . . . , (Xp, Yp) where p is the number of +pieces of x. Now we iterate over all pieces of x and check whether there exists a piece with +value Yj = ⌊α⌋∗ +δ. If no such piece exists, we insert (i, ⌊α⌋∗ +δ) into the list representation at the +appropriate position. Then we find the smallest integer j such that Yj > ⌊α⌋∗ +δ and for all +k ≥ j, we increment Xk by 1. Intuitively, we are moving all pieces that are larger than ⌊α⌋∗ +δ +one unit to the right in order to make space for the element that was just inserted. +Once we have updated x and y as described above, we simply run the static algorithm +without the step in which we initialize x and y. Note that, since we assume that after each +insertion x and y are still ordered and since we only insert rounded entries into x and y, +we get that x and y never have have more than O(1/δ) pieces by Lemma 39. Now, since +above we have set δ = ϵ/2, the proof of Proposition 45 implies that we obtain a solution with +additive error ϵ in time O(1/ϵ2 log(1/ϵ)). Furthermore, note that since we do not store x and +y explicitly (we only store their rounded version represented by their list representations), +the space usage is O(1/ϵ). +Finally, we note that the operation Delete(i) can be implemented similar to above by +first manipulating the list representations of x and y to remove the i’th entries from x and y +and then running the static algorithm. +◀ +We remark that by storing two dynamic vectors x and y that are undergoing element +insertions and deletions as described in the proof of Proposition 48, we can also efficiently +maintain an approximation of their (min, +)-convolution x ⊕ y via Lemma 40. +H +Omitted Proofs +H.1 +Proof of Lemma 6 +Denote the list representations of g and h as (xg +1, yg +1), . . . , (xg +pg, yg +pg) and (xh +1, yh +1 ), . . . , (xh +ph, yh +ph), +respectively. Recall that both list representation are stored in doubly linked lists and that +the pieces of g and h are stored in a binary search tree such that for all x ∈ [0, t] we can +evaluate g(x) and f(x) in time O(log pg) and O(log ph), respectively. +We show how to construct each of the functions fmin, fshift, fadd and fround by showing +how to construct their list representations. +First, let us consider fmin. We construct the list representation (xmin +1 +, ymin +1 +), . . . of fmin. +The intuition of our approach is that each piece of fmin must start and end at one of the +start or end points of the pieces of g and h. Thus, we will evaluate the function min{g, h} at +all points xg +i and xh +j and set fmin accordingly; then if fmin contains multiple pieces with the +same ymin +i +-value, we will remove these duplicate pieces. More concretely, we consider the set +X = {xg +1, . . . , xg +pg, xh +1, . . . , xh +ph} and order it from small to large. Now we set xmin +i +to the i’th +smallest element in X for all i = 1, . . . , pg +ph. Observe that on the interval [xmin +i−1, xmin +i +), fmin +must take the value min{g(xmin +i−1), h(xmin +i−1)}. Therefore, we set ymin +i += min{g(xmin +i−1), h(xmin +i−1)}. +This gives an initial list representation of f min. Then we “prune” the list representation of +fmin, i.e., we iterate over all pairs (xmin +i +, ymin +i +) in increasing order of i and if ymin +i−1 = ymin +i +then we remove the pair (xmin +i−1, ymin +i−1) from the list representation of fmin. Observe that at + +70 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +the end of this process, all values of ymin +i +are pairwise disjoint (since the functions g and h +are monotone). +To see that fmin(x) = min{g(x), h(x)} for all x ∈ [0, t], we observe that for all x ∈ X +(where X is as in the paragraph above) we have set fmin(x) correctly by construction. +Furthermore, on all contiguous intervals in [0, t] \ X, g and h are constant and thus fmin is +constant. Therefore, for all x ∈ [0, t] \ X, fmin(x) is also set correctly. +Next, we observe that fmin has at most pg+ph pieces because X consisted of at most pg+ph +elements and after that we only removed pieces from fmin. Furthermore, ordering the elements +in X can be done in time O((pg + ph) log(pg + ph)) and evaluating min{g(xmin +i +), h(xmin +i +)} +can be done in time O(log(pg) + log(ph)). After that we only performed a single pass +over the list representations of fmin in time O(|X|) = O(pg + ph). Therefore, it took time +O((pg + ph) log(pg + ph)) to create the list representation of fmin. Finally, note that to store +the elements xmin +i +in the binary search tree, we need additional time O((pg +ph) log(pg +ph)). +Now we observe that fadd can be computed similarly to fmin: the function fadd only +changes its functions values at the points in X (where X is as above). Therefore, we let xadd +i +be the i’th smallest element in X and set yadd +i += g(xadd +i−1) + h(xadd +i−1), followed by the same +pruning step as above. The rest of the proof goes through as above. +Next, consider fshift. We construct the list representation (xshift +1 +, yshift +1 +), . . . , (xshift +pg +, xshift +pg +) +of fshift. For all i = 1, . . . , pg, we set xshift +i += xg +i + c and yshift +i += yg +i . The correctness is +straightforward and from the construction it is evident that there are only pg pieces and that +everything can be done in time O(pg log(pg)) (since we still need to construct the binary tree +for the pieces of fshift). +Finally, us consider fround. As before, we construct the list representation of fround, +(xround +1 +, yround +1 +), . . . , (xround +pg +, xround +pg +). For all i = 1, . . . , pg, we set xround +i += xg +i and yround +i += +⌈yg +i ⌉1+δ. After that, we perform the same pruning step as in the construction of fmin. Since +g takes values in W∞ = {0} ∪ [1, W] ∪ {+∞} and g is monotone, fround can take at most +2 + ⌈log1+δ(W)⌉ different values. Again, the running time bound stems from the fact that +we have to construct the binary search tree for the pieces of fround. +H.2 +Proof of Lemma 7 +Let (xs +1, ys +1), . . . , (xs +ps, ys +ps) be the list representation of fs for s = 1, 2, where ps ≤ p is the +number of pieces of fs. We create pairs (y1 +i , y2 +j ) for all (i, j) ∈ {1, . . . , p1} × {1, . . . , p2}, +and order them such that y1 +i + y2 +j becomes monotonically increasing. We iterate over all +pairs in this order, and in each iteration we set the function value f(x) for some x-values +to y := y1 +i + y2 +j , where (y1 +i , y2 +j ) is the pair considered during the iteration. Here, we start +with large x-values (at which f takes the smallest values) and keep on decreasing x (and +the function values increase); in other words, we construct f on its domain [0, t] from right +to left. More concretely, let xmax denote the highest x-value for which we did not yet set a +function value (or −∞ if all function values have been set). Let x′ = x1 +i−1 + x2 +j−1. We set +the function values for all x ∈ [x′, xmax) to y and then set xmax = x′. For each such new +piece of f, we store that we combined the indices i and j of the pieces that we used from f1 +and from f2. Then we proceed with next iteration until all function values have been set. +The following two statements show that this procedure is correct. +1. Each x is assigned a function value that is at most the correct value f(x). To see this +let x ∈ [0, t] and recall that +f(x) = min +¯x∈[0,x] f1(¯x) + f2(x − ¯x) . + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +71 +Let ¯x∗ denote the value of ¯x that attains the minimum in the above expression, and +let i∗ and j∗ denote the indices of the pieces that ¯x∗ and x − ¯x∗ fall into, w.r.t. the list +representations of f1 and f2, respectively. This means ¯x∗ ∈ [x1 +i∗−1, x1 +i∗) and x − ¯x∗ ∈ +[x2 +j∗−1, x2 +j∗). Hence, x ≥ x′ = x1 +i∗−1 +x2 +j∗−1. Therefore, either in the iteration for the pair +(y1 +i∗, y2 +j∗) or before, the procedure assigns a function value to x. Because the procedure +assigns function-values in monotonically increasing fashion we are guaranteed that the +function value that is assigned is at most the correct value. +2. The function value y that is assigned is at least the correct value f(x). Suppose that +during some iteration we assign the function value y = y1 +i + y2 +j to x ∈ [x′, . . . , xmax), +where x′ = x1 +i−1 + x2 +j−1. We have +f(x) = min +¯x∈[0,x] f1(¯x) + f2(x − ¯x) +(definition) +≤ f1(x1 +i−1) + f2(x − x1 +i−1) +(consider ¯x = x1 +i−1) +≤ f1(x1 +i−1) + f2(x′ − x1 +i−1) +(x′ ≤ x, f2 monotonically decreasing) += f1(x1 +i−1) + f2(x2 +j−1) += y1 +i + y2 +j +(y1 +i = f1(x1 +i−1) and y2 +j = f(x2 +j−1)) += y . +Hence, the assigned value is at least f(x). +Observe that we can implement the above procedure in time O(p2 log p): We first sort the +at most p2 pairs in time O(p2 log p). Then every iteration can be executed in constant time +because setting the function values for x ∈ [x′, xmax) to y can be performed by adding the +pair (xmax, y) to the list-representation of f and updating xmax to x′ takes time O(1). +Finally, suppose we already computed f and, given x ∈ [0, t], we shall return a value +¯x∗ ∈ [0, t] such that f(x) = f1(¯x∗) + f2(x − ¯x∗). First, let (x1, y1), . . . , (xp, yp) denote the list +representation of f. Then we can determine the piece ℓ of f such that x ∈ [xℓ, xℓ+1) in time +O(log p) since we store the pairs (xi, yi) of f in a binary search tree. Recall that for each +piece of f, we stored the indices i and j of the pieces from f1 and f2 that we combined. Now +observe that we have ¯x∗ ∈ [x1 +i−1, x1 +i ) and x − ¯x∗ ∈ [x2 +j−1, x2 +j), where i and j are such that +these pieces from f1 and f2 form the corresponding piece of f. Thus, to find ¯x∗ we can first +try to set ¯x∗ = x1 +i−1. If x − ¯x∗ = x − x1 +i−1 ∈ [x2 +j−1, x2 +j) then we are done. Otherwise, we must +have that x − x1 +i−1 ≥ x2 +j. Thus, we have to increase the value of ¯x∗ from x1 +i−1 until it is large +enough such that x − ¯x∗ ∈ [x2 +j−1, x2 +j). This can be achieved by setting ∆ = (x − x1 +x−1) − x2 +j +and ¯x∗ = x1 +i−1 + ∆ + 1 +2 min{x1 +i − (x1 +i−1 + ∆), x2 +j − x2 +j−1}. Note that this value of ¯x∗ can be +computed in time O(1). Thus, the total time to return ¯x∗ is O(log p). +H.3 +Proof of Theorem 9 +Recall that the dependency graph is a DAG. We call a vertex without any incoming edges a +leaf. The level of a vertex u is the length of the longest path from a leaf to u. Note that +since each node can only reach h other nodes, every vertex has level at most h. +We compute the DP bottom-up, starting at the leaves of the DAG and then recursively +computing the solutions for rows i for which the solutions of In(i) have already been computed. +We store the approximate solutions ADP(i, ·) using monotone piecewise constant functions. +We prove the theorem by induction over the level of i in the dependency graph. We show +the stronger statement that for every DP row i of level ℓ, ADP(i, ·) is an αℓ+1-approximation +of DP(i, ·). + +72 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +We start with leaf vertices (i.e., vertices of level 0). For a leaf i, we use Properties 4(b) +and 4(c) to obtain that ˜Pi returns ADP(i, ·) which is a monotonone piecewiese constant +function with at most p pieces and which is an α-approximation of DP(i, ·). +Next, consider a row i of level ℓ. We use ˜Pi to compute ADP(i, ·) = ˜Pi({ADP(i′, ·): i′ ∈ +In(i)}. By induction hypothesis, all solutions ADP(i′, ·), i′ ∈ In(i), are stored as monotone +piecewise constant functions and each of them has at most p pieces. Since we apply the +operations from Lemma 6 only O(1) times, the number of pieces only grows by a factor +O(1). Since we only apply the (min, +)-convolution from Lemma 7 at most a single time, +the number of pieces after the convolution is bounded by O(p2). Thus, we will never operate +on functions with more than O(p2) pieces. The bounds from Lemmas 6 and 7 imply that +all operations to compute ˜Pi can be performed in time at most O(p2 log(p)). Furthermore, +by induction hypothesis and since each i′ is at level ℓ′ ≤ ℓ − 1, we know that ADP(i′, ·) is +an αℓ-approximation of DP(i′, ·). Using Properties (3) and 4(a), we get that ADP(i, ·) is an +αℓ+1-approximation of DP(i, ·). +The theorem’s approximation guarantee follows from Property (2) which implies that +ℓ ≤ h for all DP rows i in the dependency graph. Furthermore, above we argued that each +solution ADP(i, ·) can be computed in time O(p2 log(p)) which gives a total running time of +O(|I| · p2 log(p)). +H.4 +Proof of Theorem 10 +Consider a row i for which DP(i, ·) changes. Note that we only have to compute DP solutions +for rows i′ which are reachable from i in the dependency graph. Since we assume that the +dependency graph is a DAG and Reach(i) ≤ h for all rows i, there can be at most h such +rows. In the proof of Theorem 9 we argued that each solution ADP(i, ·) can be computed in +time O(p2 log(p)). This gives the proof of the theorem. +H.5 +Property of the Räcke Tree +Let G = (VG, EG) be an undirected graph and let T = (VT , ET ) be a Räcke tree for G. We +prove that mincutT (A, B) ≥ mincutG(A, B) by showing that for any set of vertices ST ⊆ VT , +it holds that capT (ST ) ≥ capG(S) where S ⊆ VG is the set of leaf vertices in VT . +Let ST ⊆ VT and consider the cut (ST , ¯ST ) in T. We use S to denote the restriction of +ST to the leaf vertices and observe that (S, ¯S) forms a cut in G as well. Then: +capT (ST ) = +� +(xt,yt)∈ST × ¯ST +capT (xt, yt) +(definition of capT (ST )) += +� +(xt,yt)∈ST × ¯ST +capG(Vxt ∩ Vyt) +(definition of tree edge capacity) += +� +(xt,yt)∈ST × ¯ST +� +(x,y)∈Vxt× ¯Vxt +capG(x, y) +(w.l.o.g. assume Vxt ⊆ Vyt) += +� +{x,y}∈EG +capG(x, y) +� +(xt,yt)∈ST × ¯ST +1{x ∈ Vxt ∧ y ∈ ¯Vxt} +(change order of summation) +≥ +� +(x,y)∈S× ¯S +capG(x, y) = capG(S) . +Here the inequality follows because a pair (x, y) ∈ S × ¯S whose capacity is counted on the +right hand side corresponds to a graph edge {x, y} ∈ EG (between x ∈ S and y ∈ ¯S). This + +M. Henzinger, S. Neumann, S. Schmid, H. Räcke +73 +graph edge contributes to the capacity on every edge of the x-y path in T. One of these +edges must be cut by ST , i.e., 1{x ∈ Vxt ∧y ∈ ¯Vxt} = 1 for this tree edge. Hence, its capacity +is also counted on the left hand side. +H.6 +Proof of Lemma 18 +The lower bound is immediate since ˜Pi is an α-approximation of Pi. For the upper bound +we use induction over ℓ. For ℓ = 0 observe that +ADP(i, ·) = ˜Pi({ADP(i′): i′ ∈ In(i)}) +(definition) +≤ αPi({ADP(i′): i′ ∈ In(i)}) +( ˜Pi is α-approximate) += αPi(∅) +(vi is a leaf) += αDP(i) +(the DP is okay-behaved) +For ℓ > 0 we have +ADP(i, ·) = ˜Pi({ADP(i′): i′ ∈ In(i)}) +(definition) +≤ αPi({ADP(i′): i′ ∈ In(i)}) +( ˜Pi is α-approximate) += αPi({αℓDP(i′, ·): i′ ∈ In(i)}) +(induction hypothesis) += αℓ+1DP(i, ·) +(the DP is okay-behaved) +Here the induction hypothesis exploits the fact that all i′ ∈ In(i), have level strictly less than +ℓ in the dependency graph. +H.7 +Proof of Lemma 19 +The claim about the approximation guarantee follows immediately from Lemma 18 and +the fact that the root has level at most h (since the longest leaf-root path in the depen- +dency tree has length h). To obtain running time O(|VT | · t), we compute the solutions +ADP(v1), . . . , ADP(vn) in this order, i.e., based on the topological ordering of the dependency +DAG. Then by assumption on the ordering of the rows i and since all ˜Pi can be computed in +time t, the lemma follows. +H.8 +Proof of Lemma 20 +Suppose the inserted or deleted edge is incident upon a vertex i. Since the DPs we consider +are well-behaved, we only need to recompute DP solutions for those vertices j such that +there exists a directed path from i to j, j ≥ i, in the dependency graph. By construction of +the dependency graph, there can be at most h such vertices (since the longest leaf-root path +in the dependency graph has length h). Therefore, we can recompute all of these solutions in +time O(h · t). After we finished the recomputation, the guarantees on the approximation +ratio are implied by Lemma 19. +H.9 +Proof of Lemma 21 +We only prove the case if all functions fi are monotonically decreasing. +The case for +monotonically increasing functions is analogous. +Let P denote the set of all pieces in +functions fi. Consider a piece p ∈ P that starts at t1 ends at t2 and has value α. We +construction a piece-wise constant function fp : [0, t] → W∞ with two pieces that has value +∞ on [0, t1), and value α on [t1, t] (this means we extend the piece from t2 to t). + +74 +Dynamic Maintenance of Monotone Dynamic Programs and Applications +Because the functions are monotonically decreasing we can rewrite fmin as a minimum of +the piece-functions fp, i.e., +fmin(x) = min +p∈P fp(x) . +We now sort all pieces in P by there start-point. By processing the pieces in sorting order we +can build the result function step-by-step. Let fr−1 denote the piece-wise constant function +encoding the minimum over the first r −1 pieces. In order to compute fr we have to compare +the last piece of fr−1 to the r-th piece pr in P. If the value of pr is higher than fr−1(∞) +(the value of the last piece in fr−1) we ignore the piece pr. Otherwise, we end the current +last piece of fr−1 at the start time tr of piece pr and add the piece pr with its start time, +its value, and an end time of t. The running time is dominated by sorting the pieces and +inserting them into a binary search tree when adding them to the result function. +H.10 +Proof of Lemma 22 +We can assume w.l.o.g. that f2 is monotonically decreasing (this follows from the symmetry +of (min, +)-convolution). Now the lemma is implied by the following computation, where in +the third step we use the monotonicity of f2, i.e., we use that f2(x′) ≤ f2(x) for all x′ ≥ x: +f(x′) = +min +¯x∈[0,x′] f1(¯x) + f2(x′ − ¯x) +≤ min +¯x∈[0,x] f1(¯x) + f2(x′ − ¯x) +≤ min +¯x∈[0,x] f1(¯x) + f2(x − ¯x) += f(x). + diff --git a/INAzT4oBgHgl3EQfxv6_/content/tmp_files/load_file.txt b/INAzT4oBgHgl3EQfxv6_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4952682420f7728d1a5dd07bf728fe897cfda11b --- /dev/null +++ b/INAzT4oBgHgl3EQfxv6_/content/tmp_files/load_file.txt @@ -0,0 +1,4212 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf,len=4211 +page_content='Dynamic Maintenance of Monotone Dynamic Programs and Applications Monika Henzinger � � Faculty of Computer Science, University of Vienna Stefan Neumann � KTH Royal Institute of Technology, Stockholm, Sweden Harald Räcke � � TU Munich, Munich, Germany Stefan Schmid � � TU Berlin, Germany and Fraunhofer SIT, Germany Abstract Dynamic programming (DP) is one of the fundamental paradigms in algorithm design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, many DP algorithms have to fill in large DP tables, represented by two-dimensional arrays, which causes at least quadratic running times and space usages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This has led to the development of improved algorithms for special cases when the DPs satisfy additional properties like, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', the Monge property or total monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In this paper, we consider a new condition which assumes (among some other technical as- sumptions) that the rows of the DP table are monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Under this assumption, we introduce a novel data structure for computing (1 + ϵ)-approximate DP solutions in near-linear time and space in the static setting, and with polylogarithmic update times when the DP entries change dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To the best of our knowledge, our new condition is incomparable to previous conditions and is the first which allows to derive dynamic algorithms based on existing DPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Instead of using two-dimensional arrays to store the DP tables, we store the rows of the DP tables using monotone piecewise constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This allows us to store length-n DP table rows with entries in [0, W] using only polylog(n, W) bits, and to perform operations, such as (min, +)-convolution or rounding, on these functions in polylogarithmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We further present several applications of our data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For bicriteria versions of k-balanced graph partitioning and simultaneous source location, we obtain the first dynamic algorithms with subpolynomial update times, as well as the first static algorithms using only near-linear time and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Additionally, we obtain the currently fastest algorithm for fully dynamic knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For k-balanced partitioning, we show how to monotonize an existing non-monotone DP by Feldmann and Foschini (Algorithmica’15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' for simultaneous source location, we obtain an efficient algorithm by considering the inverse DP function of the one used by Andreev, Garrod, Golovin, Maggs, and Meyerson (TALG’09).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our result for fully dynamic knapsack improves upon a recent result by Eberle, Megow, Nölke, Simon and Wiese (FSTTCS’21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 2012 ACM Subject Classification Theory of computation → Dynamic programming;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Theory of computation → Dynamic graph algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Theory of computation → Packing and covering problems Keywords and phrases Dynamic programming, dynamic algorithms, data structures Funding Monika Henzinger: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 101019564 “The Design of Modern Fully Dynamic Data Structures (MoDynStruct)” and from the Austrian Science Fund (FWF) project “Fast Algorithms for a Reactive Network Layer (ReactNet)”, P 33775-N, with additional funding from the netidee SCIENCE Stiftung, 2020–2024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Stefan Neumann: This research is supported by the the ERC Advanced Grant REBOUND (834862) and the EC H2020 RIA project SoBigData++ (871042).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Stefan Schmid: Research supported by Austrian Science Fund (FWF) project I 5025-N (DELTA), 2020-2024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='01744v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='DS] 4 Jan 2023 erc EuropeanResearchCouncil EstablishedbytheEuropeanCommission2 Dynamic Maintenance of Monotone Dynamic Programs and Applications Contents 1 Introduction 4 2 Maintaining Monotone Dynamic Programming Tables 8 3 Fully Dynamic Knapsack 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 Knapsack via Convolution of Monotone Functions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 Dynamically Maintaining a Small Instance .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 14 4 Technical Overview 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 Monotonizing the DP of Feldmann and Foschini .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 Computing the DP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 Inverting the DP of Andreev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 31 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 DP Definition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 33 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 Computing the DP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 33 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 The Approximate DP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 49 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 Computing the DP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 51 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 The Approximate DP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 53 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='3 Extension to General Graphs (Proof of Theorem 4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 54 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='4 Extension to the Dynamic Setting (Proof of Theorem 5) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 54 F Recourse Bounds 56 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 Proof of Theorem 34 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': 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Constant Functions With Non-Monotonicities .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 60 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 Proof of Lemma 40 .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 68 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 3 H Omitted Proofs 69 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 Proof of Lemma 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 70 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='3 Proof of Theorem 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 71 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='4 Proof of Theorem 10 .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 72 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='5 Property of the Räcke Tree .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 72 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='6 Proof of Lemma 18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 73 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='7 Proof of Lemma 19 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 73 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='8 Proof of Lemma 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 73 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='9 Proof of Lemma 21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 73 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='10 Proof of Lemma 22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 74 4 Dynamic Maintenance of Monotone Dynamic Programs and Applications 1 Introduction Dynamic programming (DP) is one of the fundamental paradigms in algorithm design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the DP paradigm, a complex problem is broken up into simpler subproblems and then the original problem is solved by combining the solutions for the subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' One of the drawbacks of DP algorithms is that in practice they are often slow and memory-intensive: for inputs of size n their running time is typically Ω(n2), and when the DP table is stored using a two-dimensional array they also need space Ω(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This motivated researchers to develop more efficient DP algorithms with near-linear time and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Indeed, such improvements are possible under a wide range of conditions on the DP tables [2,12,19,22,31,35,45,48,49,64], such as the Monge property, total monotonicity, certain convexity and concavity properties, or the Knuth–Yao quadrangle-inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' we discuss these properties in more detail in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' When these properties hold, typically one does not have to compute the entire DP table but instead only has to compute O(n) DP entries which reveal the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, we are not aware of any property for DPs that yields efficient dynamic algorithms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', algorithms that provide efficient update operations when the input changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' One might find this somewhat surprising because, from a conceptual point of view, many dynamic algorithms hierarchically partition the input and maintain solutions for subproblems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' this is quite similar to how many DP schemes are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Indeed, this conceptual similarity is exploited by many “hand-crafted” algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', [26,38]) which start with a DP scheme and then show how to maintain it dynamically under input changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, such algorithms are often quite involved and their analysis often requires sophisticated charging schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, it is natural to ask whether there exists a general criterion which, if satisfied, guarantees that a given DP can be updated efficiently under input changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The main contribution of our paper is the introduction of a general criterion which allows to approximate all entries of a DP table up to a factor of 1 + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We show that if our criterion is satisfied by a DP (with suitable parameters) then: In the dynamic setting, we can maintain a (1 + ϵ)-approximation of the entire DP table using polylogarithmic update time (see Theorem 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the static setting, we can compute a (1+ϵ)-approximation of the DP table in near-linear time and space (see Theorem 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our criterion essentially asserts that the rows of the DP tables should be monotone and that the dependency graph of the DP should be a DAG, where the sets of reachable nodes are small, among some other technical conditions (see Definition 8 for the formal definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our criterion is incomparable to the Monge property, total monotonicity or other criteria from the literature (see Appendix B for a more detailed discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To obtain our results, we introduce a novel data structure for maintaining DPs which satisfy our criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our data structure is based on the idea of storing the DP rows using monotone piecewise constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The monotonicity of the DP rows will allow us to ensure that our functions only contain very few pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we show that we can perform operations on such functions very efficiently, with the running times only depending on the number of pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This is crucial because it allows us to compute an entire (1+δ)-approximate DP row in time just polylog(W), even when the DP has Ω(n) columns, assuming that the DP entries are from [0, W].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that if W ≤ poly(n) then this decreases the running time for computing an entire row from Ω(n) to just polylog(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Additionally, this also allows us to store each row using only polylog(W) space rather than storing it in an array of size Ω(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We present our criterion and the details of our data structure in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 5 As applications of our data structure, we obtain new static and dynamic algorithms for various problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We present new algorithms for k-balanced partitioning, simultaneous source location and for fully dynamic knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we describe these results in detail;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' we discuss more related work in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our Results for Fully Dynamic 0-1 Knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, we provide a novel algorithm for fully dynamic 0-1 knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In this problem, the input consists of a knapsack size B ∈ R+ and a set of n items, where each item i ∈ [n] has a weight wi ∈ R+ and a price pi ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The goal is to find a set of items I that maximizes � i∈I pi while satisfying the constraint � i∈I wi ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the dynamic version of the problem, items are inserted and deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, we consider the following update operations: insert(pi, wi), in which the price and weight of item i are set to pi ∈ [1, ∞) and wi ∈ R+, respectively, and delete(i), where item i is removed from the set of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our main result is a dynamic (1 + ϵ)-approximation algorithm with worst-case update time ϵ−2 · log2(nW) · polylog(1/ϵ, log(nW)), where W = � i pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our algorithm improves upon a recent result by Eberle, Megow, Nölke, Simon and Wiese [29] that also maintained a (1 + ϵ)-approximate solution with update time O(ϵ−9 log4(nW)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' There exists an algorithm for fully dynamic knapsack that maintains a (1 + ϵ)-approximate solution with worst-case update time 1 ϵ2 log2(nW) polylog � 1 ϵ log(nW) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will also show that we can return the maintained solution I in time O(|I|) and that we can answer queries whether a given item i ∈ [n] is contained in I in time O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This matches the query times of [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To obtain this result, we first derive a slightly slower algorithm as a simple application of our data structure for maintaining DPs with monotone rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we use this algorithm together with additional ideas to obtain the theorem (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since our dynamic algorithm is based on a DP, it is possible that the solution changes significantly after each update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, in the appendix (Theorem 34) we prove a lower bound, showing that every dynamic (1 + ϵ)-approximation algorithm for knapsack must either make a lot of changes to the solution after each update or store many (potentially substantially different) solutions between which it can switch after each update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This implies that maintaining a single explicit solution with polylogarithmic update times is not possible and the property of our algorithm cannot be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our Results for k-Balanced Partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our most technically challenging result is for k-balanced graph partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In this problem, the input consists of an integer k and an undirected weighted graph G = (V, E, cap) with n vertices, where cap : E → W∞ is a weight function on the edges with weights in W∞ := [1, W] ∪ {0, ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The goal is to find a partition V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk of the vertices such that |Vi| ≤ ⌈|V | /k⌉ for all i and the weight of the edges which are cut by the partition is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More formally, we want to minimize cut(V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk) := �k i=1 � {u,v}∈E∩(Vi×(V \\Vi)) cap(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note that this problem is highly relevant in theory [5,32–34] and in practice [18,28, 44,55], where algorithms for balanced graph partitioning are often used as a preprocessing step for large scale data analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Obtaining practical improvements for this problem is of considerable interest in applied communities [18] and, for instance, the popular METIS heuristic [44] has 1,400+ citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since the above problem is NP-hard to approximate within a factor of n1−ϵ for any ϵ > 0 even on trees [34], we consider bicriteria approximation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Given an undirected weighted graph G = (V, E, cap), a partition V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk of V is a bicriteria (α, β)-approximate solution if |Vi| ≤ β⌈n/k⌉ for all i and cut(V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk) ≤ α · cut(OPT), where OPT = 6 Dynamic Maintenance of Monotone Dynamic Programs and Applications (V ∗ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , V ∗ k ) is the optimal solution with |V ∗ i | ≤ ⌈n/k⌉ for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note that the previously mentioned hardness result implies that any algorithm that computes a bicriteria (α, 1 + ϵ)- approximation for any α ≥ 1 and whose running time depends only polynomially on n, must have a running time depending super-polynomially on 1/ϵ, unless P = NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 Our main result for the static setting is presented in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' It gives the first algorithm with polylogarithmic approximation ratio for this problem with near-linear running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, we compute a bicriteria (O(log4 n), 1 + ϵ)-approximation in near-linear time for constant k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For comparison, the best approximation ratio achieved by a polynomial-time algorithm [34] is a bicriteria (O(log1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='5 n log log n), 1 + ϵ)-approximation with running time Ω(n4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ > 0 and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let G = (V, E, cap) be an undirected weighted graph with n vertices and m edges and edge weights in W∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then for the k-balanced partition problem we can compute: An (O(log4 n), 1+ϵ)-approximation in time (k/ϵ)O(log(1/ϵ)/ϵ)·O′(m·log2(W))+(k/ϵ)O(1/ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 A (1 + ϵ, 1 + ϵ)-approximation in time (k/ϵ)O(log(1/ϵ)/ϵ) · O′(n · h2 · log2(W)) + (k/ϵ)O(1/ϵ2) if G is a tree of height h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A (1, 1 + ϵ)-approximation in time (k/ϵ)O(log(1/ϵ)/ϵ) · O′(n4 · log2(W)) + (k/ϵ)O(1/ϵ2) if G is a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, we extend our results to the dynamic setting in which the graph G is under- going edge insertions and deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the following theorem, we present the first dynamic algorithm with subpolynomial update time for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We again consider bicriteria approximation algorithms with update and query times depending super-polynomially on 1/ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' this cannot be avoided since if we computed (α, 1)-approximations for any α ≥ 1 or if we had a polynomial dependency on 1/ϵ, then the hardness result from above implies that our update and query times must be super-polynomial in n (unless P = NP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ > 0 and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let G = (V, E, cap) be an undirected weighted graph with n vertices that is undergoing edge insertions and deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then for the k-balanced partition problem we can maintain: An (no(1), 1 + ϵ)-approximate solution with amortized update time (k/ϵ)O(log(1/ϵ)/ϵ) · no(1) · O′(log2(W)) and query time (k/ϵ)O(1/ϵ2) if G is unweighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A (1+ϵ, 1+ϵ)-approximate solution with worst-case update time (k/ϵ)O(log(1/ϵ)/ϵ) ·O′(h3 · log2(W)) and query time (k/ϵ)O(1/ϵ2) if G is a tree of height h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our approach is inspired by the DP of Feldmann and Foschini [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, the DP rows in the algorithm of [34] are not monotone and, hence, their DP cannot directly be sped up by our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, we first simplify and generalize the exact DP of Feldmann and Foschini to make it monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The DP we obtain eventually is still slightly too complex to fit into our black-box framework, but we show that the ideas from our framework can still be used to obtain the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1, we provide a technical overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Again, it is possible that the solution maintained by our algorithm changes substantially after each update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Similar to above we show in the appendix (Theorem 35) that this cannot be avoided when considering subpolynomial update times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 1 If we had an algorithm that computes a bicriteria (α, 1 + ϵ)-approximation in time poly(n, 1/ϵ) then we could set ϵ = 1/(2n) which implies that all partitions have size ⌈n/k⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus we can compute a bicriteria (α, 1)-approximate solution in time poly(n) which contradicts the hardness result, unless P = NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 2 We use the notation O′(·) to suppress factors in poly(log n, k, log(1/ϵ), log log(W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 7 Our Results for Simultaneous Source Location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we provide efficient algo- rithms for the simultaneous source location problem by Andreev, Garrod, Golovin, Maggs and Meyerson [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In this problem, the input consists of an undirected graph G = (V, E, cap, d) with a capacity function cap: E → W∞ on the edges and a demand function d: V → W∞ on the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The goal is to select a minimum set S ⊆ V of sources that can simultaneously supply all vertex demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, a set of sources S is feasible if there exists a flow from the vertices in S that supplies demand d(v) to all vertices v ∈ V and that does not violate the capacity constraints on the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The objective is to find a feasible set of sources of minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will again consider bicriteria approximation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let S∗ be the optimal solution for the simultaneous source location problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we say that S is a bicriteria (α, β)-approximate solution if |S| ≤ α |S∗| and if S is a feasible set of sources when all edge capacities are increased by a factor β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The following theorem summarizes our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' It presents the first near-linear time algorithm for simultaneous source location that computes a (1+ϵ)-approximate solution while only exceeding the edge capacities by a O(log4 n) factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In comparison, the best algorithm with arbitrary polynomial processing time computes a bicriteria (1, O(log2 n log log n))- approximate solution in time Ω(n3) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let G = (V, E, cap, d) be an undirected weighted graph with n vertices and m edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then for the simultaneous source location problem we can compute: A (1 + ϵ, O(log4(n)))-approximation in time3 ˜O( 1 ϵ2 m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A (1 + ϵ, 1)-approximation in time ˜O( 1 ϵ2 h2 · n) if G is a tree of height h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we turn to dynamic versions of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We consider the following update oper- ations: SetDemand(v, d): updates the demand of vertex v to d(v) = d, SetCapacity((u, v), c): updates the capacity of the edge (u, v) to cap(u, v) = c, Remove(u, v): removes the edge (u, v), Insert((u, v), c): inserts the edge (u, v) with capacity cap(u, v) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We obtain the first dynamic algorithms with subpolynomial update times for this problem, which exceed the edge capacities only by a small subpolynomial factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let G = (V, E, cap, d) be a graph with n vertices and m edges that is undergoing the update operations given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then for the simultaneous source location problem we can maintain: A (1 + ϵ, no(1))-approximation with amortized update time no(1)/ϵ2 and preprocessing time O(n2/ϵ2) if all edge capacities are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A (1+ϵ, O(log4(n)))-approximation with worst-case update time ˜O(1/ϵ2) and preprocessing time ˜O(m) if we only allow the update operation SetDemand(v, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A (1 + ϵ, O(log2(n) log log(n)))-approximation with worst-case update time ˜O(1/ϵ2) and preprocessing time poly(n) if we only allow the update operation SetDemand(v, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A (1 + ϵ, 1)-approximate solution with worst-case update time ˜O(h3/ϵ2) and preprocessing time O(n2/ϵ2) if G is a tree of height h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To obtain these results, we use a similar DP approach as the one used by Andreev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Interestingly, the DP function that we use essentially computes the inverse function of the one used by Andreev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We sketch the details of this approach in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' After making these changes, the theorems become straightforward applications of our data structure for maintaining DPs with monotone rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 3 We write ˜O(f(n, ϵ, W)) to denote running times of the form f(n, ϵ, W) · polylog(n, ϵ, log W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 8 Dynamic Maintenance of Monotone Dynamic Programs and Applications Organization of Our Paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In Section 2 we provide the details of our condition for DPs with monotone rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In Section 3 we present our results for 0-1 Knapsack which nicely illustrate the applicability of our black-box framework from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We provide a technical overview of our more involved results for k-Balanced Graph Partitioning and for Simultaneous Source Location in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We give an overview of the appendix in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the appendix we also present more related work and the full proofs of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We present omitted proofs from the main text in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Open Problems and Future Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the future, it will be interesting to use our framework to obtain more dynamic algorithms based on existing DPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We believe that this is interesting both in theory and in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, it is intriguing to ask whether our criterion from Definition 8 can be generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Indeed, our approach was built around approximating monotone functions using piecewise constant functions, which can be viewed as piecewiese degree-0 polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' An interesting question is whether we can obtain a more general criterion if we approximate DP rows using pieces of higher-degree polynomials, such as splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Results in this direction might be possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' for example, in Appendix G we give a side result for the case when the functions contain a small number of non-monotonicities and derive a dynamic algorithm for the ℓ∞-necklace problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 2 Maintaining Monotone Dynamic Programming Tables In this section, we introduce our notion of DP tables with monotone rows and the additional technical assumptions that we are making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we present our data structure for efficiently maintaining DP tables that satisfy our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In our data structure, we will store the rows of the DP using piecewise constant functions, which we will introduce first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' List Representation of Piecewise Constant Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let t ∈ R, W ∈ [1, ∞) and set W∞ := {0}∪[1, W]∪{+∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A function f : [0, t] → W∞ is piecewise constant with p pieces if there exist real numbers 0 = x0 < x1 < x2 < · · · < xp = t and numbers y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , yp ∈ W∞ such that on each interval [xi−1, xi), f is constant and has value yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More formally, for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , p} we have f(x) = yi for all real numbers x ∈ [xi−1, xi) and f(xp) = yp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that we need the condition f(xp) = yp such that f is defined on the whole domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the list representation of a piecewise constant function f, we use a doubly linked list to store the pairs (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , (xp, yp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We also store the pairs (xi, yi) in a binary search tree that is sorted by the xi-values, which allows us to compute a function value f(x) in time O(log p) for all x ∈ [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the following, we assume that all piecewise constant functions we consider are stored in the list representation with an additional binary search tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' One of the main observations we use is that many operations on piecewise constant functions are efficient if there are only few pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The following lemma shows that several operations can be computed in time almost linear in the number of pieces of the function, rather than in time depending on the size of the domain of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='4 For δ > 0 and y ∈ W∞, we write ⌈y⌉1+δ to denote the smallest power of 1+δ that is at least y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', ⌈y⌉1+δ = min{(1+δ)i : (1 + δ)i ≥ y, i ∈ N};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' we follow the convention that ⌈0⌉1+δ = 0 and ⌈∞⌉1+δ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let t ∈ R and c ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let g, h : [0, t] → W∞ be monotone and piecewise constant functions with pg and ph pieces, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we can compute the following functions: fmin(x) := min{g(x), h(x)} with at most pg + ph pieces in time O((pg + ph) log(pg + ph));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 4 We note that computing the operations themselves can be done in linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, since we also store the pairs (xi, yi) of the list representations in a binary search tree, the running times in the lemma include an additional logarithmic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 9 fshift(x) := g(x − c) for x ≥ c, fshift(x) = g(0) for x < c with at most pg pieces in time O(pg log(pg));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' fadd(x) := g(x) + h(x), with at most pg + ph pieces in time O((pg + ph) log(pg + ph));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' fround(x) := ⌈g(x)⌉1+δ for δ > 0 with at most 2+⌈log1+δ(W)⌉ pieces in time O(pg log(pg)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that if we set ˜f = ⌈f⌉1+δ then ˜f is a (1 + δ)-approximation of f in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For α > 1, we say that a function ˜f : [0, t] → W∞ α-approximates a function f : [0, t] → W∞ if for all x ∈ [0, t], f(x) ≤ ˜f(x) ≤ α · f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (1) Furthermore, if f is monotone then the rounded function ˜f contains at most O(log1+δ(W)) pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This will be crucial later because this ensures that, if we perform a single rounding operation for each row of our DP table, the resulting function will have few pieces and operations on the function can be performed efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, consider functions f1, f2 : [0, t] → W∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A function f : [0, t] → W∞ is the (min, +)- convolution f1 ⊕ f2 if for all x ∈ [0, t], f(x) = (f1 ⊕ f2)(x) := min¯x∈[0,x] f1(¯x) + f2(x − ¯x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Such convolutions are highly useful for the computation of many DPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The following lemma shows that we can efficiently compute the convolution of piecewise constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let f1, f2 : [0, t] → W∞ be piecewise constant functions with at most p pieces and assume that one of them is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we can compute the function f : [0, t] → W∞ with f = f1 ⊕ f2 in time O(p2 log p) and f is a piecewise constant function with O(p2) pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, after computing f, for any x ∈ [0, t] we can return a value ¯x∗ ∈ [0, t] such that f(x) = f1(¯x∗) + f2(x − ¯x∗) in time O(log p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now observe that Lemma 7 has a drawback for our approach: The number of pieces (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', the complexity of the functions) grows quadratically with every application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' An important property which can be used to mitigate this issue is that the result of the convolution is still a monotone function, as we show in Lemma 22 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Later, to keep the number of pieces in our functions small, after each convolution that we perform via Lemma 7 (and that might grow the number of pieces quadratically), we perform a rounding operation ⌈·⌉1+δ (see Lemma 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This loses a factor 1 + δ in approximation but guarantees that the resulting function has O(log1+δ(W)) pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This will be crucial to ensure that our functions have only few pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Maintaining DPs With Monotone Rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we introduce our DP scheme formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We consider DP tables with a finite set of rows I and a set of columns J , with entries taking values in W∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will consider DP tables as functions DP: I × J → W∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='5 Further, we will associate the i’th row of the DP with a function DP(i, ·): J → W∞, and we store each such function DP(i, ·) using piecewise constant functions from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we introduce the dependency graph for the rows of our DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, the dependency graph D = (I, ED) is a directed graph that has the rows I as vertices and a directed edge (i′, i) between two rows if for some columns j, j′ ∈ J the entry DP(i′, j′) is required to compute DP(i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We write In(i) = {i′ ∈ I : (i′, i) ∈ ED} to denote the set of rows i′ that are required to compute row i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For the rest of the paper we will assume that the dependency graph is a DAG, which is the case for all applications that we study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will also write Reach(i) to denote the set of vertices that are reachable from row i in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 5 Even though our definition may suggest that we only consider two-dimensional DP tables, we do not require an order on I and we allow I to be any finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For example, in Section D we will set I to 3-tuples corresponding to the parameters of a four-dimensional DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 10 Dynamic Maintenance of Monotone Dynamic Programs and Applications Since we assume that the dependency graph is a DAG, we can compute the i’th DP row as soon as we have computed the solutions for the DP rows in In(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We assume that this is done via a procedure Pi that takes as input the DP rows DP(i′, ·) for all i′ ∈ In(i) and returns the row DP(i, ·) = Pi({DP(i′, ·): i′ ∈ In(i)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we come to our condition which encodes when our scheme applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the definition and for the rest of the paper, we write ADP to refer to an approximate DP table, which approximates the exact DP table DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let β > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We say that ADP β-approximates DP if DP(i, j) ≤ ADP(i, j) ≤ βDP(i, j) for all i ∈ I, j ∈ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A DP table is (h, α, p)-well-behaved if it satisfies the following conditions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (Monotonicity:) For all i ∈ I, the function DP(i, ·) is monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (Dependency graph:) The dependency graph is a DAG and |Reach(i)| ≤ h for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (Sensitivity:) Suppose β > 1 and for all i′ ∈ In(i), we obtain a β-approximation ADP(i′, ·) of DP(i′, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then applying Pi on the ADP(i′, ·) yields a β-approximation of DP(i, ·), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', DP(i, ·) ≤ Pi({ADP(i′, ·): i′ ∈ In(i)}) ≤ β · DP(i, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (Pieces:) For each procedure Pi there exists an approximate procedure ˜Pi such that: (a) ˜Pi({ADP(i′, ·): i′ ∈ In(i)}) is an α-approximation of Pi({ADP(i′, ·): i′ ∈ In(i)}), (b) ˜Pi can be computed as the composition of a constant number of operations from Lemma 6 and and at most one application of Lemma 7, and (c) ˜Pi returns a monotone piecewise constant function with at most p pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The definition is motivated in the following way: our operations on the piecewise constant functions have efficient running times when the functions are monotone and have few pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This is ensured by Properties (1), 4(b), and 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, rounding errors cannot compound too much if each row can only reach h other rows and the sensitivity condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This is ensured by Properties (2), (3), and 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Even though the definition might look slightly technical at first glance, it applies in many settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In particular, Property (2) is satisfied when the dependency graph is a rooted tree of height h in which all edges point towards the root;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' this is the case in all of our applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The other conditions are immediately satisfied by our DP for 0-1 Knapsack in Section 3 and the DP for simultaneous source location in Section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, our DP for balanced graph partitioning violates Property (4b) of Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, we will also consider a weaker assumption in Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 which, however, will not allow for nice black-box results, such as Theorems 9 and 10 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we state our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' They imply that we obtain static (1 + ϵ)-approximation algorithms running in near-linear time and space for ( ˜O(1), ln(1+ϵ)/ ˜O(1), ˜O(1))-well-behaved DPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' They also show that under this assumption, we can dynamically maintain (1 + ϵ)- approximate DP solutions with polylogarithmic update times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our main theorem for static algorithms is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consider an (h, α, p)-well-behaved DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we can compute an approximate DP table ADP which αh+1-approximates DP in time and space O(|I| · p2 log(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Later, we will apply the theorem to DPs with dependency trees of logarithmic heights h = O(log n), we will set the approximation ratio to α = ln(1 + ϵ)/(h + 1), and the number of pieces to p = polylog(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This will yield our desired algorithms with near-linear running time ˜O(|I|) and space usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that this is a big improvement upon the brute-force running times and space usages of Ω(|I| · |J |).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 11 The proof of the theorem follows from observing that when moving from one vertex to another in the dependency graph, we lose a multiplicative α-factor in the approximation ratio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' as each vertex can only reach h other vertices, this will compound to at most αh+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Combining the assumptions on the functions ˜Pi and the results from Lemmas 6 and 7, we get that each row ADP(i, ·) can be computed in time O(p2 log(p)) which gives O(|I| · p2 log(p)) total time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We also give the following extension to the dynamic setting which shows that if one of the DP rows changes, we can update the entire table efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consider an (h, α, p)-well-behaved DP and suppose that row i is changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we can update our approximate DP table ADP such that after time O(h · p2 log(p)) it is an αh+1-approximation of DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As before, we will typically use the theorem with h = O(log n), α = ln(1 + ϵ)/(h + 1) and p = polylog(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This will then result in our desired polylogarithmic update times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that this is a significant speedup compared to storing the DP tables using two-dimensional arrays: in that case even updating a single row would take time Ω(|J |), which in many applications would already be linear in the size of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The theorem follows from observing that after a row i changes, we only have to update those rows which can be reached from i in the dependency graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' But these can be at most h and each of them can be updated in time O(p2 log(p)) by Lemmas 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 3 Fully Dynamic Knapsack In 0-1 knapsack, the input consists of a knapsack size B ∈ R+ and a set of n items, where each item i ∈ [n] has a weight wi ∈ R+ and a price pi ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The goal is to find a set of items I that maximizes � i∈I pi while satisfying the constraint � i∈I wi ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For a set of items I ⊆ [n], we refer to the sum � i∈I wi as the weight of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For the rest of this section we set W = � i pi and t = � i∈[n] wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we first derive a dynamic algorithm with update time ˜O(log3(n) log2(W)/ϵ2) which is based on our framework for DPs with monotone rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we will use this algorithm as a subroutine to obtain a faster algorithm with update time ˜O(log2(nW)/ϵ2) in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' this will prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' There exists an algorithm for fully dynamic knapsack that maintains a (1 + ϵ)-approximate solution with worst-case update time 1 ϵ2 log2(nW) polylog � 1 ϵ log(nW) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Below we will also show that we can return the maintained solution I in time O(|I|) and that we answer queries whether a given item i ∈ [n] is contained in I in time O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This matches the query times of [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 Knapsack via Convolution of Monotone Functions First, we give a brief recap of the knapsack approach by Chan [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We consider the more general problem of approximating the function fJ : [0, t] → R+, where J ⊆ [n] is a set of items and fJ(x) = max �� i∈I pi : � i∈I wi ≤ x, I ⊆ J � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (2) 12 Dynamic Maintenance of Monotone Dynamic Programs and Applications Intuitively, the value fJ(x) corresponds to the best possible knapsack solution if we can only pick items which are contained in J and if the weight of the solution can be at most x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, f[n](B) corresponds to the optimum solution of the global knapsack instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that for each J ⊆ [n], fJ(x) is a monotonically increasing piecewise constant function: Indeed, consider x′ ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Any solution I ⊆ J that is feasible for x′ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', the weight of I is at most x′) is also a feasible solution for x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, fJ(x′) ≤ fJ(x) and, therefore, fJ is monotonically increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, fJ is piecewise constant since each function value fJ(x) corresponds to a solution I ⊆ J and the number of choices for I ⊆ J is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, note that if we have two disjoint subsets J1, J2 ⊆ [n] then it holds that fJ1∪J2 is the (max, +)-convolution of fJ1 and fJ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', for all x it holds that fJ1∪J2(x) = max ¯x fJ1(¯x) + fJ2(x − ¯x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This can be seen by observing that for each x, the optimum solution I for the instance J1 ∪J2 with weight at most x can be split into two disjoint solutions I1 ⊆ J1 and I2 ⊆ J2 such that I1 has weight ¯x and I2 has knapsack weight at most x − ¯x (for suitable choice of ¯x ∈ [0, x]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We conclude that if we have two knapsack instances over disjoint sets of items J1 and J2, then we compute the solution for the knapsack instance with items J1 ∪ J2 by computing the (max, +)-convolution of fJ1 and fJ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The Exact DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The previous paragraphs imply a simple way of computing the exact solution of a knapsack instance: For each item i ∈ [n], compute the function f{i} and then recursively merge the solutions for sets of size 2j, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , ⌈log n⌉, by computing (max, +)-convolutions until we have computed the global solution f[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We perform the recursive merging of the solutions using a balanced binary tree, resulting in a tree of height O(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, we build a rooted balanced binary tree T with n leaf nodes, where all edges point towards the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We have one leaf f{i} for each item i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Each internal node u in T is associated with a function fJu as per Equation (2), where Ju is the set of all items in the subtree rooted at u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To simplify notation, we will also refer to fJu as fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we consider the exact computation of the DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This will reveal the procedures Pi from Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As base case, for each i ∈ [n], the i’th leaf of T contains the function f{i}, which is a piecewise constant function that has value 0 on the interval [0, wi) and value pi on the interval [wi, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, in each internal node u of T with children u1 and u2, we set fu to the (max, +)- convolution of fu1 and fu2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By induction it can be seen that for every node u in T, it holds that Ju = Ju1 ∪ Ju2 and thus Ju is the set of all items whose corresponding leaf is contained in the subtree Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, for the root r of T it holds that fr = f[n] and fr(B) is the optimal solution for the global knapsack instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the following, we check that our DP satisfies Properties (1–3) of Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, note that the tree T from above is also the dependency graph of our DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, our DP has a row for every vertex of T and thus O(n) rows in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, since T has height O(log n) and all edges point towards the root, every vertex can reach at most h = O(log n) vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, Property (2) of Definition 8 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Second, we observe that in both cases above, the function f{i} and fu which correspond to the rows of our DP table are monotonically increasing (we argued this above for all functions fJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, Property (1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Third, observe that Property (3) is also satisfied since (max, +)-convolution satisfies our sensitivity condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 13 We conclude that the first three properties of Definition 8 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Unfortunately, this does not yet imply that we can obtain efficient algorithms: Note that if we compute the exact DP bottom-up then we compute one convolution per node and thus the total running time of this approach is O(n·t(p)), where p is an upper bound on the number of pieces in our functions and t(p) is the time it takes to compute a (max, +)-convolution of two functions with p pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, observe that computing the convolutions can potentially take a large amount of time because the number of pieces of the functions might grow quadratically after each convolution (see Lemma 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will resolve this issue below using rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The Approximate DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we consider approximations which will reveal the functions ˜Pi from Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, note that we need to compute (max, +)-convolutions of monotonically increasing functions efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We observe that this can be done efficiently using our subroutine from Lemma 7 for the (min, +)-convolution of monotonically decreasing functions: Indeed, suppose that f is the (max, +)-convolution of two monotonically increasing functions g and h, then for all x it holds that f(x) = max ¯x {g(¯x) + h(x − ¯x)} = − min ¯x {−g(¯x) + (−h(x − ¯x))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now observe that −g and −h are monotonically decreasing functions and, therefore, f = −((−g) ⊕ (−h)), where ⊕ denotes the (min, +)-convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we can use the efficient routine for (min, +)-convolution from Lemma 7 with the same running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='6 Now we can define the subroutines ˜Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let δ > 0 be a parameter that we set later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Whenever we compute a function fu via a (max, +)-convolution, we use the efficient subroutine from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' After computing the convolution, we set fu = ⌈fu⌉1+δ via the subroutine from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that this approach satisfies Property (4a) of Definition 8 with α = 1 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, Property (4b) is satisfied since we only use a single convolution and a single rounding step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, Property (4c) is also satisfied because the resulting function is monotone and has p = O(log1+δ(W)) after the rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The above arguments show that our DP is (h, α, p)-well-behaved for h = ⌈log n⌉, α = 1+δ, δ = ln(1+ϵ)/⌈log n⌉ and p = O(log1+δ(W)) = O(log(W)/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, Theorem 10 immediately implies the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' There exists an algorithm that computes a (1 + ϵ)-approximate solution for 0-1 knapsack in time n · 1 ϵ2 log2(n) log2(W) · polylog( 1 ϵ log(nW)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note that we can return our solution I in time |I| log(n) · polylog( 1 ϵ log(nW)) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recall that our global objective function value is achieved by fr(B) and that fr(B) = fu1(¯x∗) + fu2(B − ¯x∗), where u1 and u2 are the nodes below the root node r of the dependency tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now using the second part of Lemma 7 we can get the value of ¯x∗ in time O(log p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If fu1(¯x∗) > 0 we recurse on fu1(¯x∗) and if fu2(B − ¯x∗) > 0 we also recurse on fu2(B − ¯x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' At some point we will reach a leaf node i and we include i in the solution iff f{i}(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that since we only recurse for function values which are strictly larger than zero, for each item that we include into the solution we have to follow a single path in the dependency tree of height O(log n) and our work in each internal node is bounded 6 We note that, formally, Lemma 7 can only be applied on functions with non-negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, this can be achieved by adding a number C to −g and −h, which is an upper bound on the values taken by g and h, and at the end we subtract the constant function 2C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', we set f = −((−g+C)⊕(−h+C))−2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 14 Dynamic Maintenance of Monotone Dynamic Programs and Applications by O(log p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This gives the total time of O(|I| log(n) log(p)) and our claim follows from our choice of p above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Extension to the Dynamic Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we extend our result to the dynamic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For the sake of simplicity, we assume that n is an upper bound on the maximum number of available items (items in S) and given to our algorithm in the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='7 We consider update operations that insert and delete items from the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, we consider the following update operations: insert(pi, wi), in which i is added to S by setting the price and weight of item i to pi ∈ W∞ and wi ∈ R+, respectively, and delete(i), where item i is removed from the set of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our implementation is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the preprocessing phase, we build the same tree T as above and use the subroutine from above to compute the function f{i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For the operation delete(i), we set pi = 0 and wi = 0, which changes exactly one row of our DP table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For the operation insert(pi, wi), we set the price and weight of item i to pi and wi, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', which again changes a single row in our DP table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' After changing such a row, we recompute the global DP solution via Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since the DP is (h, α, p)-well-behaved with the same parameters as above, the theorem implies the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' There exists an algorithm for the fully dynamic knap- sack problem that maintains a (1 + ϵ)-approximate solution with worst-case update time 1 ϵ2 log3(n) log2(W) · polylog � 1 ϵ log(nW) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that with the same procedure as for the static algorithm, we can return our solution I in time |I| log(n) · polylog( 1 ϵ log(nW)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, given an item i ∈ [n], we can return whether i ∈ I in time log(n) · polylog( 1 ϵ log(nW)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This can be done by using the same query procedure as in the static setting, where we only recurse on the unique subtree in the depedency tree that contains the node for item i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note that the above proposition already improves upon the update time in the result of Eberle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [29] in terms of the dependency on 1 ϵ but it has a worse dependency on log(nW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, our query time is slower than the O(1)-time query operation in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will resolve these issues in the next subsection, where we will use the algorithm from Proposition 12 as a subroutine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 Dynamically Maintaining a Small Instance Next, we we obtain a faster dynamic algorithm with update time ˜O( 1 ϵ2 log2(nW)) by combin- ing the algorithm from Proposition 12 and with ideas from Eberle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our high-level approach is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, we partition the items into a small number of price classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we take a few items of small weight from each price class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This will give a very small knapsack instance X for which we maintain an almost optimal solution using the subroutine from Proposition 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' since this instance is very small (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', |X| ≪ n), the update time for maintaining this instance essentially becomes O( 1 ϵ2 log2(W)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', we lose the O(log3 n) term that made the update time in the proposition too costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For the rest of the items which are not contained in X, we show that we can compute a good solution using fractional knapsack, 7 It is possible to drop this assumption using an amortization argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, every time the number of items is less than n/2 or more than n, we rebuild the data structure with a new value of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Each rebuild can be done in time O(nt(n)), where t(n) is our update time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since this only happens after Ω(n) updates occured, we can amortize this cost over the updates that appeared since the last rebuild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 15 which can be easily solved using a set of binary search trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then it remains to show that the combination of the two solutions is a (1 + ϵ)-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The main differences of our algorithm and the one by Eberle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [29] are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Eberle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' also partition the items into a small number of price classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' They also combine solutions for a small set of heavy items X and solutions based on fractional knapsack for the other items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, they have to enumerate many different sets X and they also guess the approximate price of the fractional knapsack solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' more concretely, they enumerate Θ( 1 ϵ2 log(W)) choices for X and the number of guesses they have to make for the fractional knapsack solution is Θ( 1 ϵ log(W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus they have to consider Θ( 1 ϵ3 log2(W)) guesses and for each of them they have to compute approximate solutions, which takes time Θ( 1 ϵ4 ) for each X since they have to run a static algorithm from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In our approach, we only have to consider a single set X which we maintain in our data structure from Proposition 12, which saves us a lot of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, the piecewise constant function, in which we store the solution for X, essentially “guides” our Θ( 1 ϵ log(W)) guesses for the weight of fractional knapsack solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In our analysis we have to be slightly more careful to ensure that our guesses for the weight of the fractional knapsack solution guarantee the correct approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We assume that ϵ < 1 and that 1/ϵ is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, we run the algorithm with ϵ′ = max{ 1 i : 1 i ≤ ϵ, i ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Set L = ⌈log1+ϵ(W)⌉ and recall that we set W = � i pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We define the price classes Vℓ = {i: (1 + ϵ)ℓ ≤ pi < (1 + ϵ)ℓ+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the following, we assume that all items from price class Vℓ have price exactly (1 + ϵ)ℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We only lose a factor of 1 + ϵ by making this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, we set V 1/ϵ ℓ to the set of 1/ϵ items from Vℓ with smallest weights wi (breaking ties arbitrarily).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We also define V ′ ℓ = Vℓ \\ V 1/ϵ ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we set X = � ℓ≥0 V 1/ϵ ℓ and Y = � ℓ≥0 V ′ ℓ for all ℓ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that X and Y partition the set of items and |X| = 1 ϵ · L = O(ϵ−2 log(W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now our strategy is to use our algorithm from Proposition 12 to maintain a solution for the items in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we show how we can combine the solution for X with a solution for Y that is based on fractional knapsack and a charging argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Data Structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For each ℓ ∈ [L], we maintain Vℓ sorted non-decreasingly by weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We also maintain the set X in a binary search tree, in which we sort the items by their index, and we maintain our data structure from Proposition 12 on the items in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, let Uℓ = � ℓ′≤ℓ V ′ ℓ′ denote the set of all items that are not contained in X and of price class at most ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For each ℓ, we maintain the set Uℓ in a binary search tree T in which the items are stored as leaves and sorted by their density pi wi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In each internal node u of T, we store the total weight of the items in the subtree Tu rooted at u and the total profit of the items in Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that this allows us to answer queries of the type: “Given a budget b, what is the value of the optimal fractional8 knapsack solution in Uℓ with weight at most b?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' in time O(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now consider an item insertion or deletion and suppose that the updated item is of price class Vℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We first update the sets Vℓ, Uℓ′ for ℓ′ ≤ ℓ and the sets X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that for each of these sets at most one item can be removed and inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, these steps can be done in time O(ℓ · log(n)) = O(ϵ−1 log(W) log(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, if X changed in the previous step, then we also perform the corresponding updates 8 In fractional knapsack, we may use items fractionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' An optimal solution is achieved by sorting the items items by their density and greedily adding items to the solution until we have used up our budget b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This approach uses at most one item fractionally (namely, the one at which we use up our budget).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 16 Dynamic Maintenance of Monotone Dynamic Programs and Applications in the data structure from Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since |X| = O(ϵ−2 log(W)) holds by construction of X, the update operations for the data structure maintaing the knapsack solution for X take a total time of O � ϵ−2 log3(|X|) log2(W) · polylog �1 ϵ log(|X| W) �� = O � ϵ−2 log2(W) · polylog �1 ϵ log(nW) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, we can explicitly write down our solution IX for the items in X in time ϵ−2 log(W) · polylog( 1 ϵ log(nW)) since |X| = O(ϵ−2 log(W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Also, for each i ∈ IX, we can set a bit indicating that i ∈ IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that the time for writing down IX and setting the bits is subsumed by the update time above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Returning the value of a solution: We return the value of a global knapsack solution as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consider the data structure from Proposition 12 which maintains a solution for the items in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that this solution is stored as a piecewise constant function with p ≤ L pieces and consider the list representation (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , (xp, yp) of this function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our strategy is as follows: For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , p, we consider a solution which spends budget xi on items in X and budget B − xi on items in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we take the maximum over all of the solutions we have considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, for given i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , p, we obtain our solution as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We pick ℓi such that (1 + ϵ)ℓi = ⌈ϵ · yi⌉1+ϵ (see Lemma 13 below for a justification of this choice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we use the binary search tree for Uℓi to find the highest profit that we can obtain from fractional knapsack on items in Uℓi ⊆ Y if we can spend budget at most b = B − xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let y′ i be the value of this query after removing any profit that we gain from the (at most one) fractionally cut item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We also store the density of the final item that is contained in the fractional knapsack solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we return the maximum of yi + y′ i over all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that since the solution for X has at most L = O(ϵ−1 log(W)) pieces and for each of them we perform a single query in a binary search tree, the total time for return the solution value is O(ϵ−1 log(W) log(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that this time is subsumed by the update time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Returning the entire solution: Now we can return our global solution I in time O(|I|) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that I is composed of the solution IX for the items in X and of the items in the fractional knapsack solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' During our updates, we already stored the items in IX and can write them down in time O(|IX|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, to return the items from the fractional knapsack solution, recall that we stored the density of the final item in the fractional knapsack solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we only have to output the items ordered non-decreasingly by their density, while we are above the desired density-threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This can be done in time linear in the size of the fractional knapsack solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This is essentially the same query procedure as in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Returning whether an item is in the solution: Furthermore, observe that the above implies that we can answer whether an item i ∈ [n] is contained in our solution in time O(1): If i ∈ X then we already stored a bit whether i ∈ IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If i ̸∈ X then we can check whether i is in the fractional knapsack solution by checking whether its density is above or below the threshold given by the final item in the fractional knapsack solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We start by making some simplifications to OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We let OPT′ denote the version of OPT in which for each ℓ ∈ [L], we pick the |OPT ∩Vℓ| items of smallest weight from Vℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This only loses a factor of 1+ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, define OPT′ X = OPT′ ∩X and OPT′ Y = OPT′ ∩Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that by how we picked OPT′, it holds that OPT′ Y ∩Vℓ ̸= ∅ iff ��OPT′ ∩Vℓ �� > 1/ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let pX denote the total price of items in OPT′ X and let wX denote the total weight of the items in OPT′ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let f denote the piecewise constant function that stores the solution M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 17 for the items in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that by Proposition 12 we have that pX ≤ f(wX) ≤ (1 + ϵ)pX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Also, the function value f(wX) is part of a piece (xi∗, yi∗) with xi∗ ≤ wX and yi∗ = f(wX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The next lemma justifies why we set ℓi such that (1 + ϵ)ℓi = ⌈ϵ · yi⌉1+ϵ in our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To this end, let ℓi∗ be such that (1 + ϵ)ℓi∗ = ⌈ϵ · yi∗⌉1+ϵ and let ℓY be the price class of the most valuable item in OPT′ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the lemma we show that ℓi∗ ≥ ℓY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will use this to show that our solution for X of profit yi∗ is valuable enough such that we can charge a fractionally cut item from fractional knapsack onto the solution from X and only lose a factor of (1 + ϵ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' It holds that ℓi∗ ≥ ℓY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since OPT′ Y ∩V ′ ℓY ̸= ∅, ��OPT′ ∩VℓY �� > 1/ϵ and thus OPT′ X contains all 1/ϵ items from V 1/ϵ ℓY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, pX ≥ 1 ϵ · (1 + ϵ)ℓY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' From above we get f(wX) = yi∗ and f(wX) ≥ pX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By choice of ℓi∗, (1 + ϵ)ℓi∗ = ⌈ϵ · yi∗⌉1+ϵ = ⌈ϵ · f(wX)⌉1+ϵ ≥ ⌈ϵ · pX⌉1+ϵ ≥ � ϵ · 1 ϵ (1 + ϵ)ℓY � 1+ϵ = (1 + ϵ)ℓY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This implies ℓi∗ ≥ ℓY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ Next, consider the the fractional knapsack solution that we obtain from our query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that this solution has a profit that is at least as large as the profit of OPT′ Y (since fractional knapsack is a relaxation of 0-1 knapsack).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, the fractional solution uses at most one item fractionally and this item is from Uℓi∗ and has value at most (1 + ϵ)ℓi∗ = ⌈ϵ · yi∗⌉1+ϵ ≤ (1 + ϵ)ϵ · yi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we can charge this item on OPT′ X and lose a factor of at most (1 + ϵ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We conclude that the solution yi∗ +y′ i∗ is a (1+ϵ)O(1)-approximation of OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Combining this with our previous running time analysis, we obtain Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 4 Technical Overview We now present an overview of two techniques for making DPs fit our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will briefly discuss how we monotonized the DP for k-balanced partitioning and how we inverted the DP for simultaneous source location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Due to space constraints, we only present excerpts of our algorithms and we only consider special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, for both problems we will consider the special case when the input graph is a binary tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the appendix we will show that the results can be extended to general graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 Monotonizing the DP of Feldmann and Foschini We start by considering the k-balanced graph partitioning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recall that in this problem, the input is a graph G = (V, E, cap), where cap : E → W∞ is a weight function on the edges, and an integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As discussed in the introduction, we assume that we can violate the partition sizes by a (1 + ϵ)-factor and our goal is to find a partition V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk of the vertices such that |Vi| ≤ ⌈(1+ϵ) |V | /k⌉ for all i and such that we minimize cut(V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk) := �k i=1 � {u,v}∈E∩(Vi×(V \\Vi)) cap(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For the sake of better exposition, here we only consider the special case in which G is a binary tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='4 we show how to drop this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 18 Dynamic Maintenance of Monotone Dynamic Programs and Applications In the following we present a DP in which the rows are monotone and we show how to efficiently perform operations on these solution vectors using monotone piecewise constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our DP is related to the DP by Feldmann and Foschini [34] which is non-monotone and thus our DP can be viewed as the monotonization of the DP by Feldmann and Foschini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We believe that our technique to monotonize the DP will have further applications in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' High-Level Description of the DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our DP is computed bottom-up starting at the leaves of the tree and then moving up in the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For each vertex v, we will compute a DP solution of minimum cost that encodes whether the edge to the parent p of v is cut and which edges shall be cut inside the subtree Tv that is rooted at v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that the removal of the cut edges in our solution will decompose the tree into disjoint connected components and exactly one of them contains v’s parent p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Additionally, we store information about the number of vertices that are still connected to the parent p (and, therefore, to the outside of Tv) after the cut edges are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will assume that when we compute the DP cell for a vertex v, we have access to the solutions for both of its children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, when we have computed a solution for a subtree Tv, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', we know which edges incident to nodes in this subtree we are going to remove (note that the edge leading to the parent of v is incident to Tv and thus we consider it as part of this solution), we store the following information in the DP table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, we store its cost, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', the total capacity of all edges that are incident to vertices in Tv and that are cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As described above, we would also like to store the number of vertices that are connected to the parent of v and the sizes of connected components inside Tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, there are two difficulties: (1) We cannot store the number of vertices that are connected to the root exactly because this would result in a too large DP table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Instead, we store the cheapest solution in which vertices of at most some given number are still connected to the parent of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As we will see, this approach gives rise to monotonically decreasing functions and allows for a very efficient computation of the DP table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (2) We store implicitly the size of all connected components that are created after the cut edges are removed and that lie completely inside Tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As before, storing these sizes exactly would result in a very large DP table and, therefore, we store them concisely using the concept of a signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The signatures will help us to characterize the sizes of the components inside Tv very efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We call a connected component in Tv large if it contains at least ϵ⌈|V | /k⌉ vertices and otherwise we call it small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let t = ⌈log1+ϵ(1/ϵ)⌉ + 1, and let M = ⌈k/ϵ⌉ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A signature is a vector g = (g0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , gt−1) ∈ [M −1]t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that each Pi is an integer between 0 and M − 1 and hence there are M t = (k/ϵ)O(ϵ−1 log(1/ϵ)) different signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Intuitively, an entry Pi in g tells us roughly how many components of size (1 + ϵ)i · ϵ⌈|V | /k⌉ there are in the DP solutions that we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Due to space constraints, we refer to the appendix for the formal definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For x ∈ N, we let e(x) ∈ [M − 1]t denote the signature of a single component with x vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More precisely, we set e(x) to the vector that has e(x)j = 1 for j = arg min{j ∈ N: x ≤ (1 + ϵ)j · ϵ⌈|V | /k⌉} and e(x)j = 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If x < ϵ⌈|V | /k⌉, we define e(x) = ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Formal DP Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we describe the DP formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' An entry DP(v, g, cut, x) ∈ W∞ in the DP table for a vertex v is indexed by a signature g, a Boolean value cut and x ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will consider the tuples (v, g, cut) as the rows I of the DP table and x as the columns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' we associate each such row with a function DP(v, g, cut, ·): [n] → W∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that our DP has |V | · M t · 2 = (k/ϵ)O(ϵ−1 log(1/ϵ)) · n rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Also, note that it has columns n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' later, even though x only takes discrete values, we will allow x to take values in [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' An entry DP(v, g, cut, x) describes the optimum cost of cutting edges incident on the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 19 subtree Tv (including the cost of maybe cutting the edge to the parent of v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will refer to the set of vertices in Tv that are still connected to the parent of v after the cut edges are removed as the root component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We impose the following conditions on DP(v, g, cut, x): Once the cut edges are removed, the root component U ⊆ Tv has at most x vertices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', |U| ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If cut is set to true then the edge between v and its parent is cut, otherwise it is kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The vertices inside Tv that (once the cut edges are removed) are not connected to the parent of v form connected components that are consistent with the signature g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we observe that if we fix a vertex v, a signature g and a value for cut, then the resulting function DP(v, g, cut, ·) is monotonically decreasing in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Observation 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let v ∈ V , g ∈ [M − 1]t be a signature and cut ∈ {true, false}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then the function DP(v, g, cut, ·) : [0, ∞) → R+ is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By definition, DP(v, g, cut, x) stores the cost of the optimum solution in which there are at most x vertices in the root component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since x ≤ x′, the solution DP(v, g, cut, x) is also a feasible solution for DP(v, g, cut, x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, DP(v, g, cut, ·) is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ Comparison With the DP by Feldmann and Foschini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' When comparing our DP with the one by Feldmann and Foschini [34] then one of the crucial changes is that in our DP, x encodes an upper bound on the number of vertices in the root component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Previously, Feldmann of Foschini considered root components with exactly x vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This is why their DP was non-monotone and why one can view our DP as the monotonization of the DP in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, we also generalize the DP to the setting with vertex weights and, as we will see below, parts of our algorithm for computing the DP approximately are rather involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 Computing the DP We now give a flavor of what our algorithms for computing the DP look like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We start by showing how to compute the exact DP solution DP(v, ·, ·, ·) for a vertex v of the tree, where v has parent p and children vl, vr and it is connected to them via edges ep, el and er, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Computing the DP is based on several case distinctions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' here, we only consider the case in which v is an internal vertex of the tree we do not cut the edges el and er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' All other cases are presented in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' When computing a DP row given by DP(v, ·, ·, ·), we will only require access to the DP rows DP(vl, ·, ·, ·) and DP(vr, ·, ·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This implies that the dependency tree of the DP is a tree and has the same height as our input graph G (recall that here we assume that G is a binary tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that the height of the tree also implies an upper bound on the number of reachable nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Exact Computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We start with the exact computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Here, we can afford to iterate over all values x ∈ [n] and g ∈ [M − 1]t to compute DP(v, ·, ·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, we consider the values for x and g as input to our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since we assume that we do not cut the edges el and er, we have to select subsolutions for Tvl and Tvr, where each subsolution is characterized by the upper bound xl (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' xr) and its signature gl (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' gr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, suppose that we cut the edge ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If we let xl and xr denote the number of vertices of the root components for the subsolutions, then the vertex v will be included in a component of size xl + xr + 1 afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, we can combine the subsolutions to a solution for 20 Dynamic Maintenance of Monotone Dynamic Programs and Applications signature g as long as gl + gr + e(xl + xr + 1) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consequently we set for every x ∈ [0, ∞), DPB(v, g, true, x) = cap(v, p)+ min xl,xr,gl+gr=g−e(xl+xr+1) DP(vl, gl, false, xl)+DP(vr, gr, false, xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Second, suppose that we do not cut ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Again we have to set DPB(v, g, false, x) = ∞ for all signatures g and all x ∈ [0, 1), because v can reach p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For x ≥ 1 we have to select xl and xr such that they sum to x − 1 as this guarantees that at most x vertices can reach the parent p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consequently, we set for all x ∈ [1, ∞) DPB(v, g, false, x) = min gl+gr=g,xl+xr=x−1 DP(vl, gl, false, xl) + DP(vr, gr, false, xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Here, we can afford to exhaustively enumerate all O(M tn2) possibilities in the min- operations above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Approximate Computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now let us consider the approximate computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We denote the approximate DP solution by ADP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We assume that we have already computed the children solutions ADP(vl, g, cut, ·) and ADP(vr, g, cut, ·) and that they are stored using our data structure from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will maintain as an invariant that each of these functions has at most p = O(log1+δ(W)) pieces and we will ensure this by rounding our solution at the end of every step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', by setting ADP(v, g, cut, ·) = ⌈ADP(v, g, cut, ·)⌉1+δ using the rounding procedure from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This will ensure the following two properties: (1) The functions ADP(v, g, cut, ·) never have more than O(log1+δ(W)) pieces by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we can perform all of our operations very efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (2) For the function at the root of the tree, the approximation error is at most (1 + δ)h, where h is the height of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By picking δ = O(ϵ/h), we will achieve that we obtain a (1 + ϵ)-approximate solution at the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we proceed to the explanation of our computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If we do not cut the edge to the parent of v, we proceed similar to the exact DP above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We start by setting ADPB(v, g, false, x) = ∞ for all x ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, for x ∈ [1, ∞) we wish to set ADPB(v, g, false, x) = min gl+gr=g,xl+xr=x−1 ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr) (3) = min gl+gr=g min xl+xr=x−1 ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (4) Note that for fixed gl and gr, the inner min-operation in the second line describes a (min, +)- convolution due to the constraint xl + xr = x − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, in the inner min-operation we compute a convolution ADP(vl, gl, false, ·) ⊕ ADP(vr, gr, false, ·) and shift the result by 1 via the shift operation from Lemma 6 (where for x ∈ [0, 1) we set ADPB(v, g, false, x) = ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We need time O(p2 log p) for computing the convolution according to Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To compute the outer minimum in Equation (4), we iterate over all gl ∈ [M − 1]t using Lemma 6 and thus perform O(M t) minimum computations over piecewise constant functions with at most p2 pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, we need time O(M tp2 log(M tp2)) according to Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By Lemma 22, ADPB(v, g, false, ·) is monotonically decreasing since it is the minimum over convolutions of two monotonically decreasing functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If we cut the edge to the parent of v, then for all x ∈ [0, ∞) we would like to set ADPB(v, g, true, x) = cap(v, p) + min xl,xr,gl+gr=g−e(xl+xr+1) ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that here we need to be careful as the range of gl and gr depends on the choice of xl +xr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since there are Ω(n) possible values for xl +xr, we cannot afford to iterate over all values that M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 21 xl + xr can take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Instead, we will show that we only need to consider O(log(k/ϵ)/ϵ) different pairs (xl, xr) by exploiting the monotonicity of ADP(vl, gl, false, ·) and ADP(vr, gr, false, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, observe that we can assume xl ≤ |Tvl| and xr ≤ |Tvr|: increasing the upper bounds on the number of vertices of the root component further would mean that the root component contains than all vertices inside the sub-tree, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, xl + xr + 1 ∈ [1, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Second, we partition the interval [1, n] into O(log(k/ϵ)/ϵ) intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We have intervals Ij = (ξj−1, ξj] with ξj = (1 + ϵ)jϵ⌈n/k⌉ for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , log1+ϵ(k/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In addition, we add an “interval” I0 := [ϵ⌈n/k⌉, ϵ⌈n/k⌉] and the interval I−1 := [1, ϵ⌈n/k⌉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We set ξ0 = ϵ⌈n/k⌉ and we set ξ−1 to the largest integer that is less than ϵ⌈n/k⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that for all j ≥ −1 and x ∈ Ij, we have e(x) = e(ξj), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', the value of e(x) does not change inside the interval Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Below, this property will allow us to separate the conditions on xl + xr and on gl + gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we can rewrite the above expression as ADPB(v, g, true, x) = cap(v, p) + min j min xl+xr+1∈Ij min gl+gr=g−e(ξj) ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Third, note that now the two min-operations only depend on the choice of j and, importantly, the minimum over gl and gr does not depend on the choice of xl + xr any- more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, we can swap the order of the two min-operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, since ADPB(v, g, false, x) is monotonically decreasing with x, we can restrict the choice of xl and xr such that xl + xr + 1 is the largest number in the corresponding interval Ij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', xl + xr + 1 = ξj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, ADPB(v, g, true, x) = cap(v, p) + min j min gl+gr=g−e(ξj) min xl+xr+1=ξj ADP(vl, gl, false, xl) + ADP(vr, gr, false, ξj − xl − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we explain how the above expression can be computed efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let us first argue how we can efficiently compute the inner min-operation of the above expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We start by observing that this min-operation is not a convolution since in the constraint we sum up to ξi which is a constant (rather than to the variable x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now recall that ADP(vl, gl, false, ·) and ADP(vr, gr, false, ·) are piecewise constant functions with O(p) pieces by our invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since xl, xr ≥ 0 this implies that there are only O(p2) choices for xl and xr such that xl, xr ∈ Ij and either a new piece starts in ADP(vl, gl, false, xl) or in ADP(vr, gr, false, xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we can iterate over all these pairs (xl, xr) and evaluate ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr), where xr = ξj − xl − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we can compute the inner min-operation in time O(p2 log p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note that since this min-operation is considering a super-constant number of terms, this DP is not well-behaved (it violates Property (4b) of Definition 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This is why in our analysis we will use the more general notion from Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we can compute the outer two min-operations by simply iterating over j and all choices for gl and setting gr = g − e(ξj) − gl as above in O(M t · log(k/ϵ)/ϵ) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, we obtain a running time of O(M tp2 log p · log(k/ϵ)/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, we note that as ADPB(v, g, true, x) is independent of x, it is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, ADPB(v, g, true, x) is a piecewise constant function with a single piece and it is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Rounding Step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As noted earlier, after computing the solutions ADPB(v, g, false, ·) and ADPB(v, g, true, ·), we also round the solution by setting ADPB(v, g, cut, ·) = ⌈ADPB(v, g, cut, ·)⌉1+δ for cut ∈ {true, false} to ensure that we only have p = O(log1+δ(W)) pieces in the result- ing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that this is the only approximate operation we perform and all other operations above have been exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 22 Dynamic Maintenance of Monotone Dynamic Programs and Applications 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 Inverting the DP of Andreev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we briefly describe our DP for simultaneous source location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recall that in this problem, the input consists of an undirected graph G = (V, E, cap, d) with a capacity function cap: E → W∞ and a demand function d: V → W∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The goal is to select a minimum set S ⊆ V of sources that can simultaneously supply all vertex demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, a set of sources S is feasible if there exists a flow from the vertices in S that supplies demand d(v) to all vertices v ∈ V and that does not violate the capacity constraints on the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The objective is to find a feasible set of sources of minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Here, we will again assume the special case in which G is a binary tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' we show in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='3 how to drop this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' DP Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Given a vertex v and a value x ∈ R, we let DP(v, x) denote the minimum number of sources that we need to place in the subtree Tv such that when v receives flow at most x from its parent then all demands in Tv can be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note that x can take positive and negative values: for x ≥ 0 this corresponds to the setting in which flow is sent from the parent of v into Tv and for x < 0 this corresponds to the setting in which flow is sent from Tv towards the parent of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We further follow the convention that when the demands in Tv cannot be satisfied when v receives flow x from its parent, then we set DP(v, x) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that this DP has rows I = V and columns J = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, DP(v, ·) is monotonically decreasing since for x < x′, any solution in which Tv receives flow at most x from the parent of v is also feasible when Tv receives flow at most x′ from the parent of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This satisfies Property (1) of Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The Inverse DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Interestingly, our DP is very related to the one by Andreev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' They defined a function f(v, i) which, given a vertex v and an integer i ∈ N, denotes the minimum amount of flow that v needs to receive from its parent if all demands in Tv need to be satisfied and if we can place i sources in the subtree Tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Similar to above, f(v, i) takes positive values if the demand in Tv can only be satisified by receiving flow from the parent of v and it takes negative values if the demand in Tv is already satisfied by the sources in the subtree Tv and v can send flow to its parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now observe that our DP can essentially be viewed as the “inverse” of f(v, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More formally, observe that DP(v, x) = f −1(v, x) := min{i: f(v, i) ≤ x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The reason why we chose the inverse formulation for our DP is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To ensure that our algorithms are efficient, we have to make sure that our monotone piecewise constant functions have only few pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' One natural way to do is using rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, since the function values of f are positive and negative, it is not clear how we should perform the rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For example, to only use a small number of pieces for representing f, we would have to use different rounding mechanisms for those function values in [−1, 1] and those in [−W, W]\\[−1, 1], where W is the largest edge capacity: Indeed, if we rounded the values of f to powers of (1 + δ)j then there are only O(log1+δ(W)) function values in [−W, W] \\ [−1, 1] but there are infinitely many function values in [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Similarly, if we rounded to multiples of δ then there are only O(1/δ) function values in [−1, 1] but this would lead to O(W/δ) function values in [−W, W] \\ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In both cases, our functions would have too many pieces and we would have to pick a rounding function which provides a tradeoff between these two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, we would have to find an analysis that shows that this “more involved” rounding function does not introduce much too error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In our DP we bypass these issues because we move the negative numbers into the domain of the function DP(v, ·): R → [n + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then in the codomain we only have non-negative numbers to which we can apply the standard rounding function ⌈·⌉1+δ in a straightforward way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This also has the positive side effects that instead of getting factors of polylog(W) in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 23 our running times, we only get factors of polylog(n) because our codomain became [n + 1] rather than some potentially large interval [−W, W].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We believe this technique of considering inverse DPs will be useful in the future to compute approximate solutions for DPs that can take positive and negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Due to lack of space, we present the details for computing the DP in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' References 1 Amir Abboud, Vincent Cohen-Addad, and Philip N Klein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' New hardness results for planar graph problems in p and an algorithm for sparsest cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In STOC, pages 996–1009, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 2 Alok Aggarwal, Maria M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Klawe, Shlomo Moran, Peter W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Shor, and Robert E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Wilber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Geometric applications of a matrix-searching algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Algorithmica, 2:195–208, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 3 Greg Aloupis, Thomas Fevens, Stefan Langerman, Tomomi Matsui, Antonio Mesa, Yurai Nuñez, David Rappaport, and Godfried Toussaint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Algorithms for computing geometric measures of melodic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Computer Music Journal, 30(3):67–76, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 4 Konstantin Andreev, Charles Garrod, Daniel Golovin, Bruce M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Maggs, and Adam Meyerson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Simultaneous source location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Algorithms, 6(1):16:1–16:17, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 5 Konstantin Andreev and Harald Räcke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Balanced graph partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Theory of Computing Systems, 39(6):929–939, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 6 Ali Aouad and Danny Segev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' An approximate dynamic programming approach to the incremental knapsack problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Operations Research, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 7 Kouji Arata, Satoru Iwata, Kazuhisa Makino, and Satoru Fujishige.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Locating sources to meet flow demands in undirected networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Journal of Algorithms, 42(1):54–68, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 8 Yoan José Pinzón Ardila, Raphaël Clifford, and Manal Mohamed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Necklace swap problem for rhythmic similarity measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In International Symposium on String Processing and Information Retrieval, pages 234–245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Springer, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 9 Kyriakos Axiotis and Christos Tzamos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Capacitated dynamic programming: Faster knapsack and graph algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In ICALP, pages 19:1–19:13, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 10 Arturs Backurs, Piotr Indyk, and Ludwig Schmidt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Better approximations for tree sparsity in nearly-linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In SODA, pages 2215–2229, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 11 MohammadHossein Bateni, MohammadTaghi Hajiaghayi, Saeed Seddighin, and Cliff Stein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Fast algorithms for knapsack via convolution and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In STOC, pages 1269–1282, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 12 Wolfgang W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Bein, Mordecai J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Golin, Lawrence L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Larmore, and Yan Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The knuth–yao quadrangle-inequality speedup is a consequence of total monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Algorithms, 6(1):17:1–17:22, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 13 Marcin Bienkowski, Miroslaw Korzeniowski, and Harald Räcke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A practical algorithm for constructing oblivious routing schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In SPAA, pages 24–33, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 14 David Bremner, Timothy M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Chan, Erik D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Demaine, Jeff Erickson, Ferran Hurtado, John Iacono, Stefan Langerman, Mihai Pătraşcu, and Perouz Taslakian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Necklaces, convolutions, and X+Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Algorithmica, 69(2):294–314, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 15 Karl Bringmann and Alejandro Cassis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Faster knapsack algorithms via bounded monotone min-plus-convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In ICALP, volume 229, pages 31:1–31:21, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 16 Karl Bringmann, Fabrizio Grandoni, Barna Saha, and Virginia Vassilevska Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Truly subcubic algorithms for language edit distance and RNA folding via fast bounded-difference min-plus product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', 48(2):481–512, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 17 Karl Bringmann, Marvin Künnemann, and Karol Wegrzycki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Approximating APSP without scaling: equivalence of approximate min-plus and exact min-max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In STOC, pages 943–954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ACM, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 18 Aydin Buluç, Henning Meyerhenke, Ilya Safro, Peter Sanders, and Christian Schulz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recent advances in graph partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In Algorithm Engineering - Selected Results and Surveys, volume 9220 of Lecture Notes in Computer Science, pages 117–158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 24 Dynamic Maintenance of Monotone Dynamic Programs and Applications 19 Rainer E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Burkard, Bettina Klinz, and Rüdiger Rudolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Perspectives of monge properties in optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Discret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', 70(2):95–161, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 20 Michael Bussieck, Hannes Hassler, Gerhard J Woeginger, and Uwe T Zimmermann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Fast algorithms for the maximum convolution problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Operations research letters, 15(3):133–141, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 21 Timothy M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Chan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Approximation schemes for 0-1 knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In SOSA, volume 61, pages 5:1–5:12, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 22 Timothy M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Chan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Near-optimal randomized algorithms for selection in totally monotone matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In SODA, pages 1483–1495, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 23 Timothy M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Chan and Moshe Lewenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Clustered integer 3sum via additive combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In STOC, pages 31–40, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 24 Shucheng Chi, Ran Duan, Tianle Xie, and Tianyi Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Faster min-plus product for monotone instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In STOC, pages 1529–1542, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 25 Justin Colannino, Mirela Damian, Ferran Hurtado, John Iacono, Henk Meijer, Suneeta Ramaswami, and Godfried Toussaint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' An o(n log n)-time algorithm for the restriction scaffold assignment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Journal of Computational Biology, 13(4):979–989, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 26 Spencer Compton, Slobodan Mitrovic, and Ronitt Rubinfeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' New partitioning techniques and faster algorithms for approximate interval scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' CoRR, abs/2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='15002, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' URL: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='org/abs/2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='15002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 27 Marek Cygan, Marcin Mucha, Karol Wegrzycki, and Michal Wlodarczyk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' On problems equivalent to (min, +)-convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Algorithms, 15(1):14:1–14:25, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 28 Yihe Dong, Piotr Indyk, Ilya P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Razenshteyn, and Tal Wagner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Learning space partitions for nearest neighbor search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In ICLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 29 Franziska Eberle, Nicole Megow, Lukas Nölke, Bertrand Simon, and Andreas Wiese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Fully Dynamic Algorithms for Knapsack Problems with Polylogarithmic Update Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In FSTTCS, volume 213, pages 18:1–18:17, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 30 Eduard Eiben, Daniel Lokshtanov, and Amer E Mouawad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Bisection of bounded treewidth graphs by convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In ESA 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 31 David Eppstein, Zvi Galil, and Raffaele Giancarlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Speeding up dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In FOCS, pages 488–496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' IEEE Computer Society, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 32 Guy Even, Joseph Naor, Satish Rao, and Baruch Schieber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Fast approximate graph partitioning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', 28(6):2187–2214, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 33 Uriel Feige and Robert Krauthgamer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A polylogarithmic approximation of the minimum bisection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', 31(4):1090–1118, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 34 Andreas Emil Feldmann and Luca Foschini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Balanced partitions of trees and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Algorithmica, 71(2):354–376, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 35 Zvi Galil and Kunsoo Park.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Dynamic programming with convexity, concavity, and sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', 92(1):49–76, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 36 Gramoz Goranci, Harald Räcke, Thatchaphol Saranurak, and Zihan Tan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The expander hierarchy and its applications to dynamic graph algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In SODA, pages 2212–2228, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 37 Chris Harrelson, Kirsten Hildrum, and Satish Rao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A polynomial-time tree decomposition to minimize congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In SPAA, pages 34–43, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 38 Monika Henzinger, Stefan Neumann, and Andreas Wiese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Dynamic approximate maximum independent set of intervals, hypercubes and hyperrectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In SoCG, volume 164 of LIPIcs, pages 51:1–51:14, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 39 Dorit S Hochbaum and David B Shmoys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Using dual approximation algorithms for scheduling problems theoretical and practical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Journal of the ACM, 34(1):144–162, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 40 Yiding Hua, Rasmus Kyng, Maximilian Probst Gutenberg, and Zihang Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Maintaining expander decompositions via sparse cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' CoRR, abs/2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='02519, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 41 Piotr Indyk, Fabian Kuhn, and Anca Muscholl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' On the fine-grained complexity of one- dimensional dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In ICALP 2017, volume 80, page 21, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 25 42 Klaus Jansen and Lars Rohwedder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' On integer programming, discrepancy, and convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' arXiv preprint arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='04744, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 43 Ce Jin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' An improved FPTAS for 0-1 knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In ICALP, volume 132, pages 76:1–76:14, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 44 George Karypis and Vipin Kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Metis: A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 45 Donald E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Knuth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Optimum binary search trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Acta Informatica, 1:14–25, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 46 Soh Kumabe and Yuichi Yoshida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Average sensitivity of dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In SODA, pages 1925–1961, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 47 Eduardo S Laber, Wilfredo Bardales, and Ferdinando Cicalese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' On lower bounds for the maximum consecutive subsums problem and the (min,+)-convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In ISIT, pages 1807– 1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' IEEE, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 48 Gary L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Miller, Richard Peng, Russell Schwartz, and Charalampos E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Tsourakakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Approximate dynamic programming using halfspace queries and multiscale monge decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In SODA, pages 1675–1682, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 49 Gaspard Monge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Mémoire sur la théorie des déblais et des remblais.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Royale Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', pages 666–704, 1781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 50 Marcin Mucha, Karol Węgrzycki, and Michał Włodarczyk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A subquadratic approximation scheme for partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In SODA, pages 70–88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' SIAM, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 51 Richard Peng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Approximate undirected maximum flows in O(mpolylog(n)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In SODA, pages 1862–1867, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 52 Harald Räcke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Minimizing congestion in general networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In FOCS, pages 43–52, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 53 Harald Räcke and Chintan Shah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Improved guarantees for tree cut sparsifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In ESA, pages 774–785, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 54 Harald Räcke, Chintan Shah, and Hanjo Täubig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Computing cut-based hierarchical decompo- sitions in almost linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In SODA, pages 227–238, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 55 Peter Sanders and Christian Schulz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Think locally, act globally: Highly balanced graph partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In SEA, volume 7933, pages 164–175, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 56 Hiroshi Tamura, Masakazu Sengoku, Shoji Shinoda, and Takeo Abe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Location problems on undirected flow networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' IEICE TRANSACTIONS (1976-1990), 73(12):1989–1993, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 57 Hiroshi Tamura, Masakazu Sengoku, Shoji Shinoda, and Takeo Abe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Some covering problems in location theory on flow networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 75(6):678–684, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 58 Godfried Toussaint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The geometry of musical rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In Japanese Conference on Discrete and Computational Geometry, pages 198–212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Springer, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 59 Godfried T Toussaint et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A comparison of rhythmic similarity measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In ISMIR, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 60 Nithin Varma and Yuichi Yoshida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Average sensitivity of graph algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In SODA, pages 684–703, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 61 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Ryan Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Faster all-pairs shortest paths via circuit complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', 47(5):1965–1985, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 62 Virginia Vassilevska Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' On some fine-grained questions in algorithms and complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In Proceedings of the ICM, volume 3, pages 3431–3472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' World Scientific, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 63 Andrew Chi-Chih Yao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Probabilistic Computations: Toward a Unified Measure of Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In FOCS, pages 222–227, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 64 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Frances Yao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Efficient dynamic programming using quadrangle inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In STOC, pages 429–435, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 26 Dynamic Maintenance of Monotone Dynamic Programs and Applications A Organization of the Appendix Our appendix is organized as follows: In Appendix B we discuss more related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Appendix C introduces preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Appendix D presents our results for k-balanced partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Appendix E presents our results for simultaneous source location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Appendix F presents our recourse lower bounds for algorithms which only maintain few solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Appendix G presents our generalization to functions with non-monotonicities and our results for ℓ∞-necklace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Appendix H presents missing proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' B Further Related Work Speeding up DP algorithms is a well-studied topic, which has received attention for several decades [2,12,19,22,31,35,45,48,49,64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This line of work has led to several conditions, which, if satisfied, imply that the underlying DP can be solved more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' These conditions include, for example, the Monge property, total monotonicity, certain convexity and concavity properties, or the Knuth–Yao quadrangle-inequality, which are often related to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For example, it is known that DP tables which satisfy the Monge property are also totally monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' One of the most popular methods in this area is the SMAWK algorithm [2] which runs in near-linear time in the number of columns of the DP table if the DP table is totally monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, a DP table is totally monotone if for each submatrix A of the DP table and for every pair of consecutive rows i and i + 1 in A, the minimum entry for row i + 1 appears in a column that is equal to or greater than the minimum entry for row i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, these conditions are quite different from our conditions in Definition 8 and they are essentially incomparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For the purpose of illustration, we will briefly argue this for total monotonicity and Definition 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' similar arguments can also be made for the Monge property and other criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' On one hand, the totally monotone matrices do not imply that the rows of the DP table are monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Indeed, when the rows are monotone then finding the columns with the minimum entries is trivial (they are always in the first or last column, depending on whether we consider monotonically increasing or decreasing rows, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, total monotonicity does not imply our condition from Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' On the other hand, the ordering of the rows is highly important for the conditions above: just swapping two rows of a totally monotone DP table can break total monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In our case, the rows can be ordered arbitrarily in the DP table, as long as their dependency graph has good properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, our property does not imply total monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This shows that these definitions are incomparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recently, Varma and Yoshida [60] and Kumabe and Yoshida [46] studied the sensitivity of graph algorithms and of DP algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' They studied how much the solutions of such algorithms change when a random element from the input is deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For several problems including knapsack they showed that these algorithms have small sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, we show in Section F that when insertions are allowed, dynamic algorithms must have high recourse or they have to maintain many different solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The k-balanced graph partitioning problem has received a lot of attention in the theory community [5,32–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The problem is also highly relevant in practice [18,28,44,55], where algorithms for balanced graph partitioning are often used as a preprocessing step for large scale data analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For the special case of k = 2, this corresponds to the minimum M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 27 bisection problem and Feige and Krauthgamer [33] presented polynomial-time algorithms with polylogarithmic approximation ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For k ≥ 3, Andreev and Räcke [5] showed that no polynomial-time algorithm can achieve a finite approximation ratio unless P = NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' They also showed how to compute a bicriteria (O(log1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='5(n)/ϵ2), 1 + ϵ)-approximate solution in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Feldmann and Foschini [34] obtained a polynomial-time bicriteria (O(log1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='5(n) log log n), 1 + ϵ)-approximation algorithm which has the advantage that the approximation ratio does not depend on the parameter ϵ of the partition sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Even et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [32] showed that one can compute a bicriteria (O(log n), 2)-approximation in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The simultaneous source location problem that we study is closely related to the source location problem introduced by Tamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [56,57], in which a minimum number of sources must be selected to be able to satisfy any single demand in an undirected edge-capacitated graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Arata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [7] showed that the problem is NP-hard and presented an exact algorithm for the variant with uniform vertex costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the simultaneous source location problem that was introduced by Andreev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [5] and that we study in this paper, all demands must be satisfied simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Andreev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' provide an O(log D)-approximation algorithm, where D is the sum of demands, and a matching hardness result for this problem in general graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' They also present an exact polynomial-time algorithm when the input graph is a tree and show that this result can be extended to general graphs when the edge capacities can be violated by a O(log2 n log log n)-factor, where n is the number of vertices in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Chan [21] showed that one can consider the solutions for the 0-1 knapsack as monotone piecewise constant functions and used this insight to obtain faster algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recently, these results were improved by Jin [43] who showed how to compute a (1+ϵ)-approximation for 0-1 knapsack with n items in time ˜O(n + ϵ−9/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Bringmann and Cassis [15] derived faster exact algorithms for 0-1 knapsack using bounded monotone min-plus-convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Aouad and Segev [6] study the incremental knapsack problem, where the capacity constraint is increased over time and the goal is to find nested subsets of items which maximize the average profit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' we note that this is different from our setting, where the goal is to obtain efficient update times, while the solutions may change arbitrarily over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' An ℓ1-necklace alignment problem was first considered by Toussaint [58], motivated by computational music theory and rhythmic similarity [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Toussaint focused on a scenario where the beads lie at integer coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Ardila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [8] studied the problem for binary strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' There also exist results for different distance measures between two sets of points on the real line in which not every points needs to be matched [25], as well as for computing the similarity of two melodies when they are represented as closed orthogonal chains on a cylinder [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Bremner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [14] showed that ℓ2-necklace alignment can be solved in time O(n log n), where n is the number of beads, using FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' They also showed that ℓ∞-necklace alignment can be solved using a constant number of (min, +)-operations and obtained subquadratic-time algorithms for ℓ1- and ℓ∞-necklace alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A common subroutine that is employed when solving DPs is (min, +)-convolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' note that this subroutine is also of high importance in all of our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The complexity of (min, +) convolution has received significant attention in the literature [9–11,14,17,20,23,24, 27,41,42,47,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' It was shown that naive algorithm with running time O(n2) can be improved to time n2/2Ω(√ log n) [14,61] by a reduction to All Pairs Shortest Path [14] using Williams’ algorithm for the latter [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, so far, no O(n2−ϵ)-time algorithm was found, which led to the MinConv hardness conjecture in fine-grained complexity theory [27, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The conjecture is particularly appealing because it implies other conjectures such as the 3-SUM and the All-Pairs Shortest Paths conjectures, and dozens of lower bounds that follow from 28 Dynamic Maintenance of Monotone Dynamic Programs and Applications them (see [27,62]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' There further exist many conditional lower bounds from the MinConv conjecture and several MinConv-equivalent problems are known, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', related to the knapsack problem or to subadditive sequences [27,41], among others [1,10,27,30,41,42,47,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' There have also been improvements for efficiently approximating the (min, +)-convolution in the case of large weights [17] for the exact (min, +)-matrix product with bounded differences [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' C Preliminaries We introduce some preliminaries that we will use in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For the sake of better readability, we present some of the proofs in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We write [m] to denote the set {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Throughout the paper, we will consider input graphs G = (VG, EG, capG) with n vertices and m edges, where capG : EG → W∞ ∪ {∞} is a weight function that for an edge e ∈ EG describes the capacity of the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To simplify notation we extend capG to all vertex pairs and define capG(x, y) = � capG({x, y}) {x, y} ∈ EG 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Additionally, for disjoint sets A, B ⊆ VG, we set capG(A, B) := � (a,b)∈A×B capG(a, b) and capG(A) := capG(A, V \\ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We drop the subscript G of the capacity function cap whenever the graph is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let (VT , ET , r) be a rooted tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For a vertex v ∈ VT we use Tv to denote the subtree rooted at v and we say that the degree of v is its number of children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The height h of T is the length of the longest path from the root to a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 Räcke Tree A Räcke tree [52] (or tree cut sparsifier) T = (VT , ET ) for an undirected graph G = (VG, EG) is a weighted, rooted tree in which the leaf nodes correspond to vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For a vertex v ∈ VT , we write Vv ⊆ VG to denote the set of leaf vertices in Tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Naturally, an edge e = (u, v) of T corresponds to a cut in G, namely to the cut formed by the set Vu ∩ Vv in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The capacity capT of the tree edge (u, v) is set to the capacity of this cut, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', to capG(Vu ∩ Vv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For a graph H = (VH, EH) and two disjoint subsets A, B ⊆ VH, we write mincutH(A, B) := min S⊆VH:A⊆S,B⊆ ¯S capH(S) to denote the minimum capacity of a cut that separates A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By definition of the edge capacities in T we have mincutT (A, B) ≥ mincutG(A, B) for any two disjoint subsets A, B ∈ VG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For the sake of completeness, we prove this property in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The goal of a Räcke tree T is to approximate the cut-structure of G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', to guarantee that for all disjoint sets of vertices A, B ⊆ VG, mincutG(A, B) ≤ mincutT (A, B) ≤ q · mincutG(A, B) , for a small value q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The parameter q is called the quality of the Räcke tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the static setting, Räcke trees with polylogarithmic quality guarantees can be computed in nearly linear time [51,54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' When larger running times are allowed, better qualities can be achieved [13,37,53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 29 ▶ Theorem 15 (Peng [51]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let G be a connected undirected graph with n vertices and m edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then there exist an algorithm that computes a Räcke tree of height O(log n) for G with quality O(log4 n) in time ˜O(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, there has recently been interest in maintaining Räcke trees dynamically [36, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Here, we will use a result by Goranci, Räcke, Saranurak and Tan who showed that one can maintain Räcke trees for unweighted graphs dynamically with subpolynomial update time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Theorem 16 (Goranci, Räcke, Saranurak and Tan [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let G be an undirected, unweighted graph with n vertices that is undergoing edge insertions and deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' There exists a deterministic algorithm with amortized update time no(1) that maintains a Räcke tree for G with quality no(1) and height O(log1/6 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 Okay-Behaved DPs We introduce a more general DP condition compared to the one in Definition 8 which, however, will not allow us to obtain results like Theorems 9 or 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will consider the same type of DP tables as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A DP is okay-behaved if it fulfills the sensitivity condition of well-behaved DPs: Suppose β > 1 and for all i′ ∈ In(i), we obtain a β-approximation ADP(i′, ·) of DP(i′, ·) (as per Equation (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then applying Pi on the ADP(i′, ·) yields a β-approximation of DP(i, ·), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', DP(i, ·) ≤ Pi({ADP(i′, ·): i′ ∈ In(i)}) ≤ β · DP(i, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We also use routines ˜Pi to compute the DP rows ADP(i, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Again, if for all i it holds that ˜Pi({ADP(i′, ·): i′ ∈ In(i)}) is an α-approximation of Pi({ADP(i′, ·): i′ ∈ In(i)}), we say that ADP(1, ·), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , ADP(n, ·) is an α-approximate DP solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the dependency graph, we call a vertex without any incoming edges a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The level of a vertex u is the length of the longest path from a leaf to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Similar to the proof of Theorem 9 we can show the following approximation guarantee for the approximate solutions ADP(i, ·) and the exact solutions DP(i, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let i be a vertex of the dependency graph with level ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then the entry ADP(i, ·) in the α-approximate ADP-solution for a okay-behaved DP problem fulfills DP(i, ·) ≤ ADP(i, ·) ≤ αℓ+1 · DP(i, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, suppose the dependency graph of the DP that we consider is derived from a tree as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let T = (VT , ET , r) be a rooted tree with root r and height h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We assume that the children of a vertex are ordered from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The dependency graph that we associate with T is simply a directed copy of T in which we direct each edge towards the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More precisely, the dependency graph contains copies of all vertices in VT and for each vertex v (except for r) an edge to its parent p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Clearly, this set of edges induces a DAG in which the longest path has at most h edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The following lemma summarizes the properties of approximate DP solutions when using this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consider a rooted tree T = (VT , ET , r) with height h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consider an okay- behaved DP and the ADP-solution ADP(i, ·) corresponding to the dependency graph described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Assume that each ˜Pi is an α-approximation of Pi and can be computed in time at most t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then ADP(r, ·) is an αh+1-approximation of DP(r, ·) and can be computed in time O(|VT | · t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 30 Dynamic Maintenance of Monotone Dynamic Programs and Applications The main difference of this lemma together with the definition of okay-behaved DPs and Theorem 9 with well-behaved DPs is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' When applying Theorem 9, we only have to consider how many pieces our functions have and we do not have to bother about deriving running times bound for computing the operations on our functions (because the additional conditions from the well-behaved DPs imply good running time bounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Here, we have to check less conditions for okay-behaved DPs (in particular, we do not have to bound the number of pieces or operations) but we have to provide our own running time analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Later, when we consider dynamic algorithms, we will have to consider the scenario when the underlying tree T changes due to edge insertions and deletions (and therefore might become a forest).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In that case, the dependency graph and the DP solutions DP(i, ·) and ADP(i, ·) change over time as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The following lemma asserts that when a vertex i is affected by an edge insertion or deletion, we only have to recompute the solutions DP(j, ·) and ADP(j, ·) for vertices j that are reachable from i in the dependency graph and that there are at most h such vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consider a rooted tree T = (VT , ET , r) with height h that is undergoing edge insertions and deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then after each insertion or deletion, we can recompute an ADP-solution with the same guarantees as in Lemma 19 in time O(h · t), where t is the time it takes to compute the functions ˜Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Lemma 6 already provided a way to compute the minimum of two monotone piecewise constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' When more than two functions are involved in the minimum computation, the following version gives improved guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let fi : [0, t] → W∞, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , k} be piecewise constant functions that are either all monotonically increasing or all monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then fmin(x) := mini{fi(x)} can be computed in time O(� i pi · log(� i pi)), where pi denotes the number of pieces of function fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We also note the following well-known lemma for sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let f1, f2 : [0, t] → W∞ and suppose that one of f1 and f2 is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then f = f1 ⊕ f2 is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' D Balanced Graph Partitioning In this section, we provide an algorithm for the k-balanced graph partitioning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In this problem, the input consists of a graph G = (V, E, cap), where cap : E → W∞ is a weight function on the edges, and an integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The goal is to find a partition V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk of the vertices such that |Vi| ≤ ⌈|V | /k⌉ for all i and the weight of the edges which are cut by the partition is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More formally, we want to minimize cut(V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk) := � i cap(Vi), where cap(Vi) = � {u,v}∈E∩(Vi,V \\Vi) cap(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since the above problem is NP-hard to approximate within any factor n1−ϵ for any ϵ even on trees [34], we consider bicriteria approximation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Given a weighted graph G = (V, E, cap), we say that a partition V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk of V is an (α, β)-approximate solution if |Vi| ≤ β⌈n/k⌉ for all i and cut(V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk) ≤ α · cut(OPT), where OPT = (V ∗ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , V ∗ k ) is the optimal solution with |V ∗ i | ≤ ⌈n/k⌉ for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our first main result in this section is summarized in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We use the notation O′(·) to suppress factors in poly(log n, k, log(1/ϵ), log log(W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 31 ▶ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ > 0 and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let G = (V, E, cap) be an undirected weighted graph with n vertices and m edges and edge weights in W∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then for the k-balanced partition problem we can compute: An (O(log4 n), 1+ϵ)-approximation in time (k/ϵ)O(log(1/ϵ)/ϵ)·O′(m·log2(W))+(k/ϵ)O(1/ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='9 A (1 + ϵ, 1 + ϵ)-approximation in time (k/ϵ)O(log(1/ϵ)/ϵ) · O′(n · h2 · log2(W)) + (k/ϵ)O(1/ϵ2) if G is a tree of height h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A (1, 1 + ϵ)-approximation in time (k/ϵ)O(log(1/ϵ)/ϵ) · O′(n4 · log2(W)) + (k/ϵ)O(1/ϵ2) if G is a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, we can also extend our results to the dynamic setting in which the graph G is undergoing edge insertions and deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our second main result in this section is summarized in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ > 0 and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let G = (V, E, cap) be an undirected weighted graph with n vertices that is undergoing edge insertions and deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then for the k-balanced partition problem we can maintain: An (no(1), 1 + ϵ)-approximate solution with amortized update time (k/ϵ)O(log(1/ϵ)/ϵ) · no(1) · O′(log2(W)) and query time (k/ϵ)O(1/ϵ2) if G is unweighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A (1+ϵ, 1+ϵ)-approximate solution with worst-case update time (k/ϵ)O(log(1/ϵ)/ϵ) ·O′(h3 · log2(W)) and query time (k/ϵ)O(1/ϵ2) if G is a tree of height h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our DP approach is inspired by the DP of Feldmann and Foschini [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, the DP cells in the algorithm of Feldmann and Foschini are not monotone and, therefore, their DP cannot directly be sped up by the fast convolution of monotone functions approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, we first simplify and generalize their DP to make it monotone such that we can apply the fast convolution of monotone functions approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note that in our static and dynamic algorithms, we can output the corresponding solutions similarly to what we descriped after Proposition 12 for knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To obtain these results, we will first describe an exact DP in Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 for the special case of binary trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we will show how to compute the DP more efficiently by introducing approximation in Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='3 we show how to return a solution based on our DP table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Sections D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='4 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='5 provide extensions from binary trees to more general graphs and to the dynamic setting, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 The Exact DP When describing the DP, we will make two assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, we assume that the input graph T = (V, E) is a binary tree (we show in Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='4 how to remove this assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Second, we consider a slight generalization of the k-balanced partition problem on trees;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' we note that we did not mention this generalization in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In this generalization, we suppose that each vertex is assigned a weight by a weight function w: V → {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='10 For convenience we set w(U) = � u∈U w(u) for all U ⊆ V and refer to w(U) as the weight of the vertices in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now our goal will be to find a partition V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk of V such that w(Vi) ≤ (1 + ϵ)⌈w(V )/k⌉ for all i and we will compare against OPT = (V ∗ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , V ∗ k ), where OPT is the optimal solution with w(V ∗ i ) ≤ ⌈w(V )/k⌉ for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that by setting w(v) = 1 9 We use the notation O′(·) to suppress factors in poly(log n, k, log(1/ϵ), log log(W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 10 We note that our proofs and algorithms also work for more general weight functions w: V → R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, in that case the functions DP(v, g, cut, ·) that we will introduce later will become more complicated to compute and, therefore, we stick with the simpler case of vertex weights in {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 32 Dynamic Maintenance of Monotone Dynamic Programs and Applications for all v ∈ V , we obtain the standard k-balanced partition problem and, therefore, our variant is a strict generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The reason for considering the above generalization is that later we want to use our algorithm to find a balanced partitioning of general graphs G = (V ′, E′) using a Räcke tree T = (V, E) (see Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, the vertices V ′ of G are just a subset of the vertices V of the Räcke tree T (since the vertices of G correspond to leaves in T and the internal nodes of T do not correspond to any vertices in G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, if we assigned weight w(v) = 1 to all vertices in T and computed a balanced partitioning of T, this would not necessarily correspond to a balanced partitioning of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Instead, later we will consider the weight function which assigns weight 1 to all leaves in T (corresponding to the vertices in G) and weight 0 to all internal nodes of T (which can be ignored when deriving a partitioning of G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then each set Vi in T will correspond to a set V ′ i in G with w(Vi) = |V ′ i |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In particular, if w(Vi) ≤ (1 + ϵ)⌈w(V )/k⌉ then we will obtain that |V ′ i | ≤ (1 + ϵ)⌈|V ′| /k⌉ and, therefore, the sets V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk imply a balanced partition V ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , V ′ k of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' High-Level Description of the DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We start by giving a high-level description of the DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The DP is computed bottom-up starting at the leaves of the tree G and then moving up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For each vertex v, we will compute a DP solution of minimum cost that encodes whether the edge to the parent p of v is cut and which edges shall be cut inside the subtree Tv that is rooted at v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that the removal of the cut edges in our solution will decompose the tree into disjoint connected components and exactly one of them contains v’s parent p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Additionally, we store information about the weight of the vertices that are still connected to the parent p (and, therefore, to the outside of Tv) after the cut edges are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will assume that when we compute the DP cell for a vertex v, we have access to the solutions for both of its children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, when we have computed a solution for a subtree Tv, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', we know which edges incident to nodes in this subtree we are going to remove (note that the edge leading to the parent of v is incident to Tv and thus we consider it as part of this solution), we store the following information in the DP table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, we store its cost, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', the total capacity of all edges that are incident to vertices in Tv and that are cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As described above, we would also like to store the weight of the vertices that are connected to the parent of v and the sizes of connected components inside Tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, there are two difficulties: (1) We cannot store the weight of the vertices that are connected to the root exactly because this would result in a too large DP table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Instead, we store the cheapest solution in which vertices of at most some given weight are still connected to the parent of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As we will see, this approach gives rise to monotonically decreasing functions and allows for a very efficient computation of the DP table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (2) We store implicitly the size of all connected components that are created after the cut edges are removed and that lie completely inside Tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As before, storing these sizes exactly would result in a very large DP table and, therefore, we store them concisely using the concept of a signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The signatures will help us to characterize the sizes of the components inside Tv very efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We call a connected component in Tv large if it contains vertices of total weight at least ϵ⌈w(V )/k⌉ and otherwise we call it small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let t = ⌈log1+ϵ(1/ϵ)⌉ + 1, and let M = ⌈k/ϵ⌉ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A signature is a vector g = (g0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , gt−1) ∈ [M − 1]t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that each gi is an integer between 0 and M − 1 and hence there are M t = (k/ϵ)O(ϵ−1 log(1/ϵ)) different signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Intuitively, an entry gi in g tells us roughly how many components of weight (1 + ϵ)i · ϵ⌈w(V )/k⌉ there are in the DP solutions that we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The precise definition is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let S = {S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Sr} be a set of connected components inside Tv (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', think of S as the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 33 components that are created after removing the cut edges in the DP solution for vertex v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We say that a signature vector g = (g0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , gt−1) ∈ [M − 1]t is consistent for S if we can match the connected components in S to entries in g as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For each large component Sj we let ℓ(Sj) = arg min{i ∈ [t]: w(Sj) ≤ (1 + ϵ)i · ϵ⌈w(V )/k⌉}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', ℓ(Sj) is the smallest number i such that Sj has weight at most (1 + ϵ)i · ϵ⌈w(V )/k⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let si ∈ [M − 1] denote the number of times the value i ∈ [t] has been chosen in this process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', si = |{j : ℓ(Sj) = i}|, and let s = (s0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , st−1) denote the resulting vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We say that g is consistent with the set of components S if g = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, the above matching process can be viewed as rounding up the component sizes and counting the number of components of each size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For x ∈ N, we let e(x) ∈ [M − 1]t denote the signature of a single component with total weight x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More precisely, we set e(x) to the vector that has e(x)j = 1 for j = arg min{j ∈ N: x ≤ (1 + ϵ)j · ϵ⌈w(V )/k⌉} and e(x)j = 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If x < ϵ⌈w(V )/k⌉, we define e(x) = ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 DP Definition Now we describe the DP formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' An entry DP(v, g, cut, x) ∈ W∞ in the DP table for a vertex v is indexed by a signature g, a Boolean value cut and x ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will consider the tuples (v, g, cut) as the rows I of the DP table and x as the columns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' we associate each such row with a function DP(v, g, cut, ·): [n] → W∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that our DP has |V |·M t·2 = (k/ϵ)O(ϵ−1 log(1/ϵ))·n rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Also, note that it has columns n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' later, even though x only takes discrete values, we will allow x to take values in [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' It describes the optimum cost of cutting edges incident on the subtree Tv (including the cost of maybe cutting the edge to the parent of v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will refer to the set of vertices in Tv that are still connected to the parent of v after the cut edges are removed as the root component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We impose the following conditions on DP(v, g, cut, x): Once the cut edges are removed, the root component U ⊆ Tv has total weight at most x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', w(U) ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If cut is set to true then the edge between v and its parent is cut, otherwise it is kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The vertices inside Tv that (once the cut edges are removed) are not connected to the parent of v form connected components that are consistent with the signature g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We observe that if we fix a vertex v, a signature g and a value for cut, then the resulting function DP(v, g, cut, ·) is monotonically decreasing in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This will be the crucial property for the rest of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Observation 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let v ∈ V , g ∈ [M − 1]t be a signature and cut ∈ {true, false}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then the function DP(v, g, cut, ·) : [0, ∞) → R+ is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By definition, DP(v, g, cut, x) stores the cost of the optimum solution in which the vertices in the root component have weight at most x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now observe that for x ≤ x′, the solution DP(v, g, cut, x) is also a feasible solution for DP(v, g, cut, x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, DP(v, g, cut, ·) must be monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ Since the DP cells are monotonically decreasing in x, we will use the shorthand notation DP(v, g, cut, ∞) to denote the solution minx DP(v, g, cut, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that this minimum is obtained for the largest x-value at which DP(v, g, cut, ·) changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 Computing the DP In the following, we describe how to compute DP(v, ·, ·, ·) exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For computing DP(v, ·, ·, ·) we simply iterate over all possible choices of x, g and cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that since each vertex has 34 Dynamic Maintenance of Monotone Dynamic Programs and Applications weight in {0, 1}, the function DP(v, g, cut, ·) only changes for x ∈ [n + 1] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', when x is an integer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we only need to consider n + 1 choices for x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We conclude that to compute DP(v, ·, ·, ·) for a fixed vertex v, there are O(M t ·n) parameter choices that we need to iterate over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In our descriptions we use p to denote the parent of v, and vl and vr to denote v’s left and right child, respectively, if these exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Case 1: v is a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If we cut the edge to the parent of v, then the cost is cap(v, p), there are no vertices in the root component and v forms its own connected component with signature e(w(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we set DP(v, e(w(v)), true, x) = cap(v, p) for all x ∈ [0, ∞) and we set DP(v, g, true, x) = ∞ for all x ∈ [0, ∞) and for all signatures g ̸= e(w(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now suppose we do not cut the edge (v, p) to the parent of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we do not have to pay any cost since we are not cutting any edge, the weight of vertices in the root component is w(v) and the signature is g = 0 since there are no connected components in Tv that are not connected to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, for all x ∈ [0, w(v)) we set DP(v, 0, false, x) = ∞ and for all x ∈ [w(v), ∞) we set DP(v, 0, false, x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For all signatures g ̸= 0 and all x ∈ [0, ∞), we set DP(v, g, false, x) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Case 2: v is not a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If v is not a leaf then we assume that it has exactly two children vl and vr (if it has only one child, we can add a second child v′ with w(v′) = 0, cap(v, v′) = 0 and then v′ has no impact on the solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We assume that for both vl and vr, we have already computed the solutions DP(vl, g, cut, x) and DP(vr, g, cut, x) for all possible values of x, g and cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let el = (v, vl) and er = (v, vr) denote the edges to the respective child and let ep = (p, v) denote the edge to the parent p of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the following we distinguish four cases (A, B, C, D) depending on which of these edges we decide to cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For each case, we compute DPcase(v, g, cut, x)-values, case ∈ {A, B, C, D}, which are the optimum values under the condition that we cut el and er according to the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The final entry DP(v, g, cut, x) is then obtained by minimizing over all cases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', by setting DP(v, g, cut, x) = min case∈{A,B,C,D} DPcase(v, g, cut, x) for all x, g, cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Case A: cut el and er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Suppose we cut el and er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then, given x and g, we have to select subsolutions for the left and right sub-tree such that the weight of vertices that can reach p is at most x and the connected components inside are consistent with g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, assume we cut the edge ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then the cost for cutting this edge is cap(v, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, the weight of vertices inside Tv that can reach p is zero and, hence, the value of x is irrelevant by the monotonicity of DP(v, g, cut, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, if we have a solution with signatures gl and gr in the left and right subtree, respectively, we can combine these solutions as long as gl + gr + e(w(v)) = g (as the vertex v forms a single component of weight w(v) since we cut both edges el and er).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that in the subsolution for the child vl, the value of x does not play a role for the feasibility of the solution DP(v, g, cut, x) since the size of the root component in Tvl is already encoded in gl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, to obtain minimum cost we consider DP(vl, gl, true, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' by symmetry, the same holds for vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, we set for all x ∈ [0, ∞), DPA(v, g, true, x) = cap(v, p)+ min gl+gr=g−e(w(v)){DP(vl, gl, true, ∞)+DP(vr, gr, true, ∞)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (5) Second, assume we do not cut the edge to the parent p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then there will be at least one vertex (namely v) that can reach p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, DPA(v, g, false, x) = ∞ for all signatures g and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 35 x ∈ [0, w(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For x ∈ [w(v), ∞), we can combine the solutions as above and we set DPA(v, g, false, x) = min gl+gr=g{DP(vl, gl, true, ∞) + DP(vr, gr, true, ∞)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (6) Case B: cut neither el nor er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, suppose we cut neither el nor er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In this case we have to select subsolutions for Tvl and Tvr, where each subsolution is characterized by the upper bound xl (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' xr) and its signature gl (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' gr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, suppose that we cut the edge ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If we let xl and xr denote the exact weight of the root components for the subsolutions, then the vertex v will be included in a component of size xl + xr + w(v) afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, we can combine the subsolutions to a solution for signature g as long as gl + gr + e(xl + xr + w(v)) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consequently we set for every x ∈ [0, ∞), DPB(v, g, true, x) = cap(v, p) + min xl,xr,gl+gr=g−e(xl+xr+w(v)) DP(vl, gl, false, xl) + DP(vr, gr, false, xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Second, suppose that we do not cut ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then again we have to set DPB(v, g, false, x) = ∞ for all signatures g and all x ∈ [0, w(v)), because the vertex v of weight w(v) can reach p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For x ≥ w(v) we have to select xl and xr such that they sum to x − w(v) as this guarantees that vertices of weight at most x can reach the parent p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consequently, we set for all x ∈ [w(v), ∞) DPB(v, g, false, x) = min gl+gr=g,xl+xr=x−w(v) DP(vl, gl, false, xl) + DP(vr, gr, false, xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Case C: cut el but not er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now suppose we cut the edge to the left child vl but we do not cut the edge to the right child vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In this case, v stays connected to the root component of vr and we need to choose a subsolution with parameters xr and gr for Tvr and a subsolution with parameter gl for Tvl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that since we cut el, the upper bound on the weight of the root component of vl is irrelevant as this is implicitly encoded in gl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, suppose we cut ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If we let xr denote the exact weight of the root component for the subsolution in Tvr then v will be included in a component of size xr +w(v) afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, we can combine the subsolutions to a solution for signature g as long as gl+gr+e(xr+w(v)) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consequently, for every x ∈ [0, ∞) we set DPC(v, g, true, x) = cap(v, p)+ min xr,gl+gr=g−e(xr+w(v)) DP(vl, gl, true, ∞)+DP(vr, gr, false, xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (7) Second, suppose we do not cut ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we have to set DPC(v, g, false, x) = ∞ for all signatures g and all x ∈ [0, w(v)), because vertex v with weight w(v) can reach p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For x ∈ [w(v), ∞), we have to select xr ≤ x−w(v) as this guarantees that vertices of total weight at most x can reach the parent p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Due to the monotonicity of DP(vr, gr, false, ·) we can just choose xr = x − w(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consequently, for all x ∈ [w(v), ∞) we set DPC(v, g, false, x) = min gl+gr=g,xr=x−w(v) DP(vl, gl, true, ∞) + DP(vr, gr, false, xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (8) Case D: cut er but not el.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Symmetric to Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we argue that this DP is okay-behaved, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', it satisfies Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In particular, we note that this DP is not well-behaved because it does not satisfy Property (4b) of 36 Dynamic Maintenance of Monotone Dynamic Programs and Applications Definition 8 since in Case 2, Step B below we will have to perform too many min-operations (see Equation (11)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will also show that the DP’s dependency graph is exactly the input tree and hence the conditions of Lemma 19 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, all entries for a DP cell DP(v, ·, ·, ·) can be computed in time O(M 2tn3) by simply enumerating all choices in the different min-operations above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The DP is okay-behaved and the dependency tree and the input tree T are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, given a vertex v, we can compute all entries in DP(v, ·, ·, ·) in time O(M 2tn3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, note that in the DP each cell DP(v, ·, ·, ·) only depends on the solutions of its two children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that these are exactly the edges which are present in the dependency graph and also in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, the dependency graph and T are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, when the input for a child solution is a β-approximation, the output of the DP will also be an β-approximation because we perform all computations exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, the DP is also okay-behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Second, let us consider the running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recall that for fixed x, g and cut, we set DP(v, g, cut, x) = mincase∈{A,B,C,D} DPcase(v, g, cut, x) and this quantity can be computed in time O(1) by a simple table lookup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we only have to consider the time it takes to compute DPcase(v, g, cut, x) for each case ∈ {A, B, C, D} and for fixed x, g and cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For Case A, observe the min-operations can be computed by iterating over all M t choices of gl and setting gr = g − e(w(v)) − gl as long as gr is a non-negative vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then the expressions inside the min-term can be computed by table lookup in constant time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, the time is O(M t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For Case B, in case cut = true note that we can iterate over all choices of xl, xr and iterate over gl as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This takes time O(M tn2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the case cut = false we can again iterate over the gl as above and we can iterate over all xl ∈ [n + 1] and set xr = x − w(v) − xl as long as xr ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' thus, the case can be solved in time O(M tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For Cases C and D, we can iterate over all choices of xr and then iterate over the gl as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This gives a total running time of O(M tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We conclude that for fixed x, g and cut, the time to compute DPcase(v, g, cut, x) for all case ∈ {A, B, C, D} is O(M tn2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since there are O(n) choices of x, M t choices for g and two choices for cut, we conclude that the total running time to compute DP(v, ·, ·, ·) is O(M 2tn3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 The Approximate DP In this section we show how to construct the approximate DP table in an efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For this we essentially perform the same computations as above, but instead of computing the exact solution DP(v, ·, ·, ·) by computing exact solutions to the cases DPcase(v, ·, ·, ·), we compute an approximate solution ADP(v, ·, ·, ·) which will be the minimum of approximate solutions ADPcase(v, ·, ·, ·), where case ∈ {A, B, C, D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, there are a few crucial differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, for fixed v, g, cut and case ∈ {A, B, C, D}, we interpret ADPcase(v, g, cut, ·) as a piecewise constant function which is stored in an efficient list representation (as per Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' After we computed the solutions ADPcase(v, g, cut, ·), we compute the function ADP(v, g, cut, ·) := ⌈min{ADPA(v, g, cut, ·), ADPB(v, g, cut, ·), ADPC(v, g, cut, ·), ADPD(v, g, cut, ·), }⌉1+δ, (9) M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 37 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', instead of just taking the minimum over the different cases, we also perform a rounding step to multiples of 1 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This rounding step introduces an approximation error of α = 1 + δ but reduces the number of pieces within the piecewise constant function DP(v, g, cut, ·) to p := O(log1+δ(W)) according to Lemma 6 (for this to work we need to guarantee that the function to be rounded is monotone and therefore we will show that ADPcase(v, g, cut, ·) is monotone for each case ∈ {A, B, C, D}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The second crucial difference is, of course, that we perform the above computations with values that already have been rounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', with entries from ADP instead of entries from DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note that Equation (9) is the only place in the approximate DP which is not exact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' all other computations are done precisely (without any rounding) and, therefore, the approximate DP only loses a factor 1 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In order to guarantee a highly efficient implementation we rely on the following invariants for entries in the approximate DP: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For all v, g, and cut, the function ADP(v, g, cut, ·) is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For all v, g, and cut, the function ADP(v, g, cut, ·) is piecewise constant with at most p := O(log1+δ(W)) pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that the first property resembles the fact that for the exact DP, DP(v, g, cut, ·) is monotonically decreasing as per Observation 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, here we state this property as an invariant because there could exist approximations of DP(v, g, cut, ·) which are non- monotone and, therefore, we need to prove that each of our functions ADP(v, g, cut, ·) is indeed monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note the second property follows immediately from the monotonicity and the rounding step in Equation (9) and thus we will not need to prove it in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Similar to the description of the exact DP, we will now go through each of the cases and, given v, describe how to compute ADP(v, g, cut, ·) in time ˜O(1) for all g and cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The cases are exactly the same as for the exact DP and thus for the sake of brevity we do not repeat the correctness argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Case 1: v is a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then, we do the same in the exact case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We set ADP(v, e(w(v)), true, x) = cap(v, p) for all x ∈ [0, ∞) and we set ADP(v, g, true, x) = ∞ for all x ∈ [0, ∞) and all sig- natures g ̸= e(w(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, we set ADP(v, 0, false, x) = ∞ for all x ∈ [0, w(v)) and ADP(v, 0, false, x) = 0 for all x ∈ [w(v), ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For all signatures g ̸= 0 and all x ∈ [0, w(v)), we set ADP(v, g, false, x) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that in all cases, the corresponding functions ADP(v, g, cut, ·) are monotonically decreasing and have O(1) pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Case 2: v is not a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We distinguish the same four cases as for the exact DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Again, we will assume that v has exactly two children vl and vr and we let el = (v, vl), er = (v, vr) and ep = (p, v), where p is the parent of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Case A: cut el and er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, suppose we cut el and er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then, as in the exact DP, if we cut the edge to the parent of v, we wish to set ADPA(v, g, true, x) = cap(v, p) + min gl+gr=g−e(w(v)){ADP(vl, gl, true, ∞) + ADP(vr, gr, true, ∞)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' for all x ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that in the equation above, the quantities cap(v, p), ADP(vl, gl, true, ∞) and ADP(vr, gr, true, ∞) are simply numbers and can be viewed as a piecewise constant function with a single piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, ADPA(v, g, true, ·) is a piecewise constant function with a single piece and, therefore, it is also monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, the invariants are satisfied for ADPA(v, g, true, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, ADPA(v, g, true, ·) can be computed via a sum and a minimum over monotonically decreasing piecewise functions via Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that the minimum takes O(M t) different values because it is computed by iterating over all 38 Dynamic Maintenance of Monotone Dynamic Programs and Applications gl ∈ [M − 1]t and setting gr = g − e(w(v)) − gl as long as all entries in gr are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since each function ADP(vl, gl, true, ·) has O(p) pieces according to our invariants, we can compute the value ADP(vl, gl, true, ∞) in time O(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' the same holds for ADP(vr, gl, true, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, computing ADPA(v, g, true, ·) takes time O(M t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, suppose we do not cut the edge to the parent of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then, as in the exact DP, we wish to set: ADPA(v, g, false, x) = min gl+gr=g{ADP(vl, gl, true, ∞) + ADP(vr, gr, true, ∞)} for all x ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then by the same arguments as above, ADPA(v, g, false, ·) is a piecewise constant monotonically decreasing function with a single piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' It can be computed in time O(M t) as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Case B: cut neither el nor er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now suppose we do not cut any edge to the children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If we do not cut the edge to the parent of v, we proceed similar to the exact DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We start by setting ADPB(v, g, false, x) = ∞ for all x ∈ [0, w(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, for x ∈ [w(v), ∞) we wish to set ADPB(v, g, false, x) = min gl+gr=g,xl+xr=x−w(v) ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr) = min gl+gr=g min xl+xr=x−w(v) ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (10) Note that for fixed gl and gr, the inner min-operation in the second line describes a (min, +)- convolution due to the constraint xl+xr = x−w(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, in the inner min-operation we compute a convolution ADP(vl, gl, false, ·)⊕ADP(vr, gr, false, ·) and shift the result by w(v) via the shift operation from Lemma 6 (where for x ∈ [0, w(v)) we set ADPB(v, g, false, x) = ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We need time O(p2 log p) for computing the convolution according to Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To compute the outer minimum in Equation (10), we iterate over all gl ∈ [M −1]t and thus perform O(M t) minimum computations over piecewise constant functions with at most p2 pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, we need time O(M tp2 log(M tp2)) according to Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By Lemma 22, ADPB(v, g, false, ·) is monotonically decreasing since it is the minimum over convolutions of two monotonically decreasing functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If we cut the edge to the parent of v, then for all x ∈ [0, ∞) we would like to set ADPB(v, g, true, x) = cap(v, p) + min xl,xr,gl+gr=g−e(xl+xr+w(v)) ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that here we need to be careful as the range of gl and gr depends on the choice of xl +xr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since there are Ω(n) possible values for xl +xr, we cannot afford to iterate over all values that xl + xr can take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Instead, we will show that we only need to consider O(log(k/ϵ)/ϵ) different pairs (xl, xr) by exploiting the monotonicity of ADP(vl, gl, false, ·) and ADP(vr, gr, false, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, observe that we can assume xl ≤ w(Tvl) and xr ≤ w(Tvr): increasing the upper bounds on the weight of the root component further would mean that the root component contains more weight than all vertices inside the sub-tree, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, xl + xr + w(v) ∈ [1, w(V )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Second, we partition the interval [1, w(V )] into O(log(k/ϵ)/ϵ) intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We have intervals Ij = (ξj−1, ξj] with ξj = (1+ϵ)jϵ⌈w(V )/k⌉ for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , log1+ϵ(k/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In addition, we add an “interval” I0 := [ϵ⌈w(V )/k⌉, ϵ⌈w(V )/k⌉] and the interval I−1 := [1, ϵ⌈w(V )/k⌉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We set ξ0 = ϵ⌈w(V )/k⌉ and we set ξ−1 to the largest integer that is less than ϵ⌈w(V )/k⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 39 that for all j ≥ −1 and x ∈ Ij, we have e(x) = e(ξj), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', the value of e(x) does not change on in the interval Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Below, this property will allow us to separate the conditions on xl + xr and on gl + gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we can rewrite the above expression as ADPB(v, g, true, x) = cap(v, p) + min j min xl+xr+w(v)∈Ij min gl+gr=g−e(ξj) ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Third, note that now the two min-operations only depend on the choice of j and, importantly, the minimum over gl and gr does not depend on the choice of xl + xr any- more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, we can swap the order of the two min-operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, since ADPB(v, g, false, x) is monotonically decreasing with x, we can restrict the choice of xl and xr such that xl + xr + w(v) is the largest number in the corresponding interval Ij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', xl + xr + w(v) = ξj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, ADPB(v, g, true, x) = cap(v, p)+ min j min gl+gr=g−e(ξj) min xl+xr+w(v)=ξjADP(vl, gl, false, xl) + ADP(vr, gr, false, ξj − xl − w(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (11) Next, we explain how the above expression can be computed efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let us first argue how we can efficiently compute the inner min-operation of the above expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We start by observing that this min-operation is not a convolution since in the constraint we sum up to ξi which is a constant (rather than to the variable x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now recall that ADP(vl, gl, false, ·) and ADP(vr, gr, false, ·) are piecewise constant functions with O(p) pieces by our invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since xl, xr ≥ 0 this implies that there are only O(p2) choices for xl and xr such that xl, xr ∈ Ij and either a new piece starts in ADP(vl, gl, false, xl) or in ADP(vr, gr, false, xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we can iterate over all these pairs (xl, xr) and evaluate ADP(vl, gl, false, xl) + ADP(vr, gr, false, xr), where xr = ξj −xl −w(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we can compute the inner min-operation in time O(p2 log p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we can compute the outer two min-operations by simply iterating over j and all choices for gl and setting gr = g − e(ξj) − gl as above in O(M t · log(k/ϵ)/ϵ) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, we obtain a running time of O(M tp2 log p · log(k/ϵ)/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note that this is the step which makes the okay-behaved rather than well-behaved (since it violates Property (4b) of Definition 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, we note that as ADPB(v, g, true, x) is independent of x, it is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, ADPB(v, g, true, x) is a piecewise constant function with a single piece and it is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Case C: cut el but not er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now suppose we cut the edge to the left child but not to the right child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First assume that we cut the edge to the parent of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As in the exact DP, for all x ∈ [0, ∞) we want to set ADPC(v, g, true, x) = cap(v, p) + min xr,gl+gr=g−e(xr+w(v)) ADP(vl, gl, true, ∞) + ADP(vr, gr, false, xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As in the previous case, observe that in the minimum the constraint gl +gr = g −e(xr +w(v)) depends on the choice of xr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' we rewrite the above equation analogously to the previous 40 Dynamic Maintenance of Monotone Dynamic Programs and Applications case: ADPC(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' true,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' x) = cap(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' p)+ min j min xr+w(v)∈Ij min gl+gr=g−e(ξj) ADP(vl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' gl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' true,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ∞) + ADP(vr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' gr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' false,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' xr) = cap(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' p)+ min j min gl+gr=g−e(ξj) min xr+w(v)∈Ij ADP(vl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' gl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' true,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ∞) + ADP(vr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' gr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' false,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' xr) = cap(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' p)+ min j min gl+gr=g−e(ξj) ADP(vl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' gl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' true,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ∞) + ADP(vr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' gr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' false,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ξj − w(v)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' where in the last step we used that ADP(vr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' gr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' false,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ·) is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The evaluation of the function values of the two piecewise constant functions with O(p) pieces can be done in time O(log(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, by exhaustively enumerating all choices for j and proceeding for gl and gr as above, we obtain O(M t log(k/ϵ)/ϵ) iterations giving a total running time of O(M t log p log(k/ϵ)/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As before, ADPC(v, g, true, ·) is a constant (since the computation does not depend on x) and therefore it has only a single piece and it is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, suppose we do not cut the edge to the parent of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we set DPC(v, g, false, 0) = ∞ for all signatures g and all x ∈ [0, w(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For all x ∈ [w(v), ∞), we set ADPC(v, g, false, x) = min gl+gr=g ADP(vl, gl, true, ∞) + ADP(vr, gr, false, x − w(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that inside the min-operation, the first term is a constant and the second term is a piecewise constant function that is shifted by w(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, the minimum is taken over O(M t) piecewise constant functions (one for each choice of gl by the same argument as above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We can perform the addition and shift operation via Lemma 6 (time O(p log p) per application).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we perform a minimum operation over M t functions where each function has just p pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This can be done in time O(M tp log(M tp)) by Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In total we get a running time of O(M tp log(M tp)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Case D: cut er but not el.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Symmetric to Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We conlucde this subsection with the following lemma which summarizes the properties of the approximate DP computation The lemma follows immediately from the above discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The approximate DP computes a (1 + δ)-approximate DP solution and the dependency tree and the input tree T are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Given a vertex v, a signature g and value cut ∈ {true, false}, we can compute the corresponding approximate DP entry ADP(v, g, cut, ·) in time O(M tp2 log(M tp) log(k/ϵ)/ϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The approximation ratio of the approximate DP is (1 + δ)-approximate because, as we pointed out earlier, we only use exact computations except in the rounding step in Equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we only use (1 + δ)-factor in the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The claim about the running time follows immediately from the discussion above the lemma, where we already analyzed the running times for all steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='3 Computing the Result In this section we describe how the previously described DPs can be used to extract the result for the k-balanced partition problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recall that we consider the generalized version of the k-balanced partition problem, where each vertex v has a weight w(v) ∈ {0, 1} (see Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 for the definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 41 We focus on the value version of the problem in which we only need to output an approximation of the value of the optimal cut OPT but we do not have to return the actual partition V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk that obtains this cut value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note, however, that by analyzing the DP solution from top to bottom, we could also construct a concrete partition V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk in time ˜O(n) that achieves the cut value which is returned by the value version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Feasible Signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Before we describe our algorithm, we first need to introduce the notion of feasible signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, recall that in Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 we introduced signatures as a succinct way of storing the sizes of connected components in a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now, feasible signatures will refer to signatures in which the connected components can be partitioned such that we obtain a nearly k-balanced partitioning of the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We make this intuition more formal below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For every signature g = (g0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , gt−1) ∈ [M − 1]t, we say that its associated machine scheduling instance11 I(g) is the instance which contains exactly gi jobs of size (1 + ϵ)i · ϵ⌈w(V )/k⌉ of all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We say that g is a feasible signature if the jobs in I(g) can be scheduled on k machines with makespan at most (1 + ϵ)⌈w(V )/k⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Later, we will identify the machines of the scheduling problems with partitions in the k-balanced partitioning solution and the jobs with connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In this way, we will be able to ensure the balance constraints of the k-balanced partitioning solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We now describe our two static algorithms for binary trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The only difference between the algorithms is whether to use the exact DP from Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 or the approximate DP from Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' we will refer to these algorithms as the exact and the approximation algorithm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We assume that the input is an error parameter ϵ > 0 and a rooted, weighted tree T = (V, E, cap) with root r and vertex weights w(v) ∈ {0, 1} for which we wish to solve the k-balanced partitioning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, our algorithm augments T by adding a fake root r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We make r′ the parent of r and set w(r′) = 0 and cap(r, r′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we compute the DP bottom-up as described in Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2, where we interpret T as its own dependency graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the exact algorithm, we use the DP from Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1, and in the approximation algorithm, we use the DP from Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Second, we compute the set of all nearly feasible signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To obtain this set, we enumerate all M t signatures and for each of them, we check whether it is nearly feasible or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We do this as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For each signature g, we construct the machine scheduling instance I(g) and run the PTAS by Hochbaum and Shmoys [39] for this problem with approximation ratio 1 + ¯ϵ and running time (N/¯ϵ)O(1/¯ϵ2), where N denotes the total number of jobs in I(g) and we will see later that N is a constant if k, ϵ and ¯ϵ are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We add a signature g to the set of nearly feasible signatures if the returned makespan for I(g) is at most (1 + ¯ϵ)(1 + ϵ)⌈w(V )/k⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note that by using the PTAS, the set that we compute can potentially contain some signatures which are infeasible but they still do not violate the balance constraint too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Third, we consider the entries in the DP table at the (true) root r of the tree for the case that the edge to its (artificial) parent is not cut (recall that we added an edge of weight 0 from the true root r to the fake root r′ and so cutting it does not incur any cost), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', we consider the DP entries DP(r, ·, true, w(V )) or ADP(r, ·, true, w(V )) depending on whether we are in the approximate or in the exact case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We iterate over all feasible signature vectors 11 Recall that in the makespan minimization problem with identical machines, the input consists of a set of N jobs of sizes s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , sN and an integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The goal is to find an assignment of the jobs to k machines such that the makespan is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Here, the makespan refers to maximum load of all k machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 42 Dynamic Maintenance of Monotone Dynamic Programs and Applications g and then take the minimum value that we have seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We conclude the algorithms’ guarantees in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note that for constant k, ϵ, ¯ϵ and W, the running time of the exact algorithm is ˜O(n4) and the running time of the approximation algorithm simplifies to ˜O(n · h2), where h is the height of the input tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, for trees of height ˜O(1), the approximation algorithm is very efficient and runs in time ˜O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Proposition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ, ¯ϵ > 0 and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let T = (V, E, cap) be a rooted binary tree that has edge weights cap(e) and vertex weights w(v) ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then: The exact algorithm obtains a bicriteria (1, (1+¯ϵ)(1+ϵ))-approximation for the k-balanced partitioning problem on T in time O(M 2tn4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The approximation algorithm obtains a bicriteria (1+ϵ, (1+¯ϵ)(1+ϵ))-approximation for the k-balanced partitioning problem on T in time O � nh2·M 2t log2(W) log(k/ϵ) log(M th log(W)/ϵ)/ϵ3� + M t(k/(ϵ¯ϵ))O(1/¯ϵ2), where h denotes the height of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To prove the proposition, we need to argue about the approximation ratios of the algorithms and we also need to prove that the partitioning does not violate the balance constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will also need to analyze the running times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We start by analyzing the balance constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We show that in the solution returned by the algorithm, the connected components V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk can be partitioned such that w(Vi) ≤ (1 + ¯ϵ)(1 + ϵ)⌈w(V )/k⌉ for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consider the DP entry DP(r, g, true, w(V )) for the (true) root r, where the edge to the parent is cut and any signature vector g that is in the set of nearly feasible signatures that we computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then this corresponds to the cost of some partition of T = Tr where, after removing the cut edges, the large connected components S in Tr can be matched to entries in g such that: a component S ∈ S is matched to entry gi with |S| ≤ (1 + ϵ)iϵ⌈w(V )/k⌉ and exactly gi components are matched to gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, we can obtain a partitioning V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, we compute the (1 + ¯ϵ)- approximate solution of I(g) in which (by assumption on g) the makespan is at most (1 + ¯ϵ)(1 + ϵ)⌈w(V )/k⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This gives us an assignment of jobs to machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we identify components with jobs and the sets Vi with machines and obtain an assignment of the large components to the Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In particular, each Vi receives large components for which the (rounded) weights sum to at most (1 + ¯ϵ)(1 + ϵ)⌈w(V )/k⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we need to assign the small components in the algorithm’s solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' These can be assigned greedily by always assigning a small component (of weight less than ϵ⌈n/k⌉) to set Vi of (currently) smallest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the end, all Vi will have weight at most (1 + ¯ϵ)(1 + ϵ)⌈w(V )/k⌉ (this follows from the standard argument that, when considering exact component weights, on average each server has makespan at most w(V )/k and thus there will always be a server of makespan at most w(V )/k to which the current small component can be assigned without violating the capacity constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This means if the algorithm returns an objective function value then there is a partition V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk with the same objective function value that is nearly feasible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', that satisfies w(Vi) ≤ (1 + ¯ϵ)(1 + ϵ)⌈w(V )/k⌉ for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, let us consider the approximation ratios of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consider the optimum partition OPT = (V ∗ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , V ∗ k ) that minimizes cut(V ∗ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , V ∗ k ) such that w(V ∗ i ) ≤ ⌈w(V )/k⌉ for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We first argue that OPT gives rise to a DP entry with a feasible signature and cost OPT in the exact DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To see this, take the optimum partition V ∗ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , V ∗ k and round up the weight of every large connected component to the next value of the form (1 + ϵ)i · ϵ⌈n/k⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let g = (g0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , gt−1) ∈ [M − 1]t be the signature where gi denotes the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 43 number of large components in OPT whose rounded weight is (1 + ϵ)i · ϵ⌈n/k⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that since w(V ∗ i ) ≤ ⌈w(V )/k⌉ for all i, the total rounded weight of components in V ∗ i is at most (1 + ϵ) · ⌈w(V )/k⌉ as component weights are increased at most by a (1 + ϵ)-factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, the constructed signature vector g is feasible because the partition V ∗ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , V ∗ k gives rise to a feasible solution for I(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, the rounding did not have any effect on the objective function value and, thus, OPT gives rise to a DP entry with a feasible signature and cost OPT in the exact DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This implies that the optimum value for the exact DP is at most cut(V ∗ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , V ∗ k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Together with the above claim that the DPs approximately satisfy the balance constraint, we obtain that the exact algorithm computes a bicriteria (1, (1 + ¯ϵ)(1 + ϵ))-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now let us turn to the approximation ratio of the approximation algorithm from Sec- tion D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recall that by Lemma 24 the exact DP is okay-behaved and in Lemma 25 we show that in each step the approximation algorithm loses a factor of at most 1 + δ at every level of the tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now, we can apply Lemma 19 to obtain that the approximation in the root is (1 + δ)h+1, where h is the height of the tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, the approximation ratio of the approxi- mate DP is 1 + ϵ if we set δ = ln(1 + ϵ)/(h + 1) since then (1 + δ)h+1 ≤ exp(δ(h + 1)) = 1 + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since the notion of approximation from Lemma 19 holds for all functions of the form ADP(r, g, true, ·) and all possible values of x, we obtain that the approximation algorithm computes a bicriteria (1 + ϵ, (1 + ¯ϵ)(1 + ϵ))-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We conclude the proof of the proposition by considering the running times of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' running time, both algorithms only differ by how long it takes to fill the DP cells and the time for computing the solution is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let us first consider the time for computing the solution as per Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, let us consider the time for solving the PTAS which is (N/¯ϵ)O(1/¯ϵ2), where N denotes the total number of jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that in our case there are at most N ≤ k(1 + 1/ϵ) jobs: each job has size at least ϵ⌈n/k⌉ and therefore a machine can take at most 1 + 1/ϵ jobs in an optimum solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, if we have more than k(1+1/ϵ) jobs, a PTAS can directly reject the instance and declare it infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, the time for running the PTAS a single time is (k/(ϵ¯ϵ))O(1/¯ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since we have to run the PTAS for each of the M t signatures, the total time for finding the nearly feasible configurations is M t(k/(ϵ¯ϵ))O(1/¯ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, let us consider the time for filling the DP cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For the exact DP, Lemma 24 states that filling a cell DP(v, ·, ·, ·) takes time O(M 2tn3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then, by applying Lemma 19, the total time to compute all DP cells is O(M 2tn4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For the approximate DP, it takes time O(M tp2 log(M tp) log(k/ϵ)/ϵ)) to fill a single DP cell ADP(v, g, cut, ·) by Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since there are M t choices for g and by again applying Lemma 19, we obtain that the total running time for filling the approximate DP table is O(nM 2tp2 log(M tp) log(k/ϵ)/ϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since in Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 we picked the number of pieces to be p = O(log1+δ(W)) and above we picked δ = O(ϵ/h), the running time is upper bounded by O � nM 2t · � 1/ϵ · h log W �2 · log(k/ϵ)/ϵ · log(M th log(W)/ϵ) � = O � nh2 · M 2t log2(W) log(k/ϵ) log(M th log(W)/ϵ)/ϵ3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='4 Extension to General Graphs Now we generalize the results of Proposition 26 from binary trees to general graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We start with the generalization to general graphs in which we will make use of Räcke trees (see Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since Räcke trees might be non-binary, we now introduce the notion of binarized Räcke trees which essentially describe a way of turning a non-binary Räcke tree into a binary tree that is very similar to a Räcke tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Later, the binarized Räcke trees will allow us to apply Proposition 26 on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 44 Dynamic Maintenance of Monotone Dynamic Programs and Applications ▶ Definition 27 (Binarized Räcke Tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let G = (VG, EG, capG) be a weighted graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We say that a weighted, rooted tree T = (VT , ET , capT ) is a binarized Räcke tree for G if the following properties hold: T is a rooted binary tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' VG ⊆ VT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' All edges in T have weights in W∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let T ′ be the tree that is obtained by contracting all edges with weight ∞ in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then T ′ is a Räcke tree for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We call the tree T ′ from the last bullet point the corresponding (non-binarized) Räcke tree of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We say that T has quality q if the corresponding Räcke tree T ′ has quality q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we observe that each cut in T of finite cost corresponds to a cut in the corresponding Räcke tree T ′ and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, cuts of finite cost in T ′ approximate the cut structure of the initial graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We make this more formal in following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Observation 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let G = (VG, EG, capG) be a weighted graph and let T = (VT , ET , capT ) be a binarized Räcke tree for G with quality q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then for all disjoint subsets A, B ⊆ VG it holds that mincutG(A, B) ≤ mincutT (A, B) ≤ q · mincutG(A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let T ′ be the corresponding (non-binarized) Räcke tree of T and consider two disjoint subsets of vertices A, B ⊆ VG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We show that minT (A, B) = mincutT ′(A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then the observation follows immediately since T has quality q (by assumption) and, therefore, T ′ is a Räcke tree for G with quality q which satisfies the property from the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since T ′ can be obtained from T only by contracting edges, we have mincutT ′(A, B) ≥ mincutT (A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, let us argue that mincutT (A, B) ≥ mincutT ′(A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, note that mincutT ′(A, B) ≤ q · mincut(A, B) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since we contract only edges with weight ∞ to go from T to T ′, T does not contain any cut with finite cost that is not contained in T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, mincutT (A, B) ≥ mincutT ′(A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ Additionally, we show that we can compute a binarized Räcke tree of good quality in nearly-linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let G = (VG, EG, capG) be a weighted graph with n vertices and m edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We can compute a binarized Räcke tree T = (VT , ET , capT ) with O(n) vertices, height O(log2 n) and quality O(log4 n) in time ˜O(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let T ′ = (VT ′, ET ′, capT ′) be the Räcke tree for G from Theorem 15 that can be computed in time ˜O(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, note that T ′ has nT ′ := O(n) vertices and height O(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Second, note that T ′ can have unbounded degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, we will show how to compute a binarized Räcke tree T that has T ′ as its corresponding (non-binarized) Räcke tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We do so replacing in T ′ each vertex u by a balanced binary tree τu with deg(u) leaves, where deg(u) denotes the number of children of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The internal edges of τu will have weight ∞ and the edges connecting subtrees τu and τv, u ̸= v, in T will correspond to the edges in T ′ and will have the same (finite) weight as in T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will see that by contracting all edges with weight ∞ in T, we will obtain T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We now elaborate on this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We construct T as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, we compute T ′ as per the algorithm from Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we construct T as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For each vertex u ∈ VT ′, we add a balanced rooted binary tree τu with deg(u) leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We refer to the root of τu as ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We identify each leaf of τu with a child of u and denote the leaf of τu that corresponds to the child v by cu,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We set the weight of edges inside τu to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that for each vertex u, the tree τu has O(deg(u)) vertices, and, therefore, T has O(nT ′) = O(n) vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, for each edge (u, v) ∈ ET ′ (where we assume M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 45 that v is a child of u), we insert the edge (cu,v, rv) in T and set capT (cu,v, ru) = capT ′(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, if u is the root of T ′ then we set ru to the root of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' It is left to show that T is a binarized Räcke tree of height O(log2 n) and quality O(log4 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Clearly, T is a binary tree since all vertices inside each subtree τu have at most two child nodes and, additionally, each vertex cu,v has at most one child node (namely rv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, T has height O(log2 n) since T ′ has height O(log n) and the subtrees τu have height O(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, let T ′′ be the tree obtained from T by contracting all edges with weight ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We argue that T ′ = T ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Indeed, consider any vertex u ∈ VT ′ and its subtree τu in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then after contracting the edges in τu, we are left with a subtree that only contains ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, all edges between vertices of different subtrees τu and τv, u ̸= v, have finite weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, T ′ = T ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This implies that T is binarized Räcke tree for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since T ′ has quality O(log4 n), the quality of T is also O(log4 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ We conclude the subsection by proving Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We can obtain the proof for the claim about general graphs as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let G = (VG, EG, capG) be a weighted graph with n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We compute a binarized Räcke tree T = (VT , ET , capT ) with O(n) vertices as per Lemma 29 in time ˜O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In T, we assign weight w(v) = 1 to all vertices v ∈ VG ∩VT (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', to the leaves in T that correspond to vertices in G) and weight w(v) = 0 to all vertices v ∈ VT \\ VG (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', to the internal nodes of T that do not correspond to any vertex in G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now observe that w(V ) = n and thus a balanced partitioning V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk of T with w(Vi) ≤ (1 +ϵ)⌈w(V )/k⌉ for all i corresponds to a balanced partitioning V ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , V ′ k of G with |V ′ i | ≤ (1+ϵ)⌈n/k⌉ for all i, where V ′ i = {v ∈ Vi : w(v) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now by combining Observation 28, Proposition 26 and the fact that T has quality O(log4 n), we obtain the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To obtain the result about general trees T ′ (with unbounded degrees), we proceed similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We construct a binarized tree T exactly as in the proof of Lemma 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now, in T we set w(rv) = 1 for all root vertices of the subtrees τv and we set w(v) = 0 for all other vertices of the subtrees τv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Similar to before, observe that w(VT ) = n and thus a balanced partitioning V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Vk of T with w(Vi) ≤ (1 +ϵ)⌈w(V )/k⌉ for all i corresponds to a balanced partitioning V ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , V ′ k of T ′ with |V ′ i | ≤ (1 + ϵ)⌈n/k⌉ for all i, where V ′ i = {v ∈ VT ′ : rv ∈ Vi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then by Proposition 26, this implies the proof for trees with unbounded degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='5 Extension to the Dynamic Setting Next, we provide new dynamic algorithms in which edges are inserted and deleted from the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We give new algorithms for trees and for general graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Extension to Dynamic Trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let us start with the case when T is a binary tree that is undergoing edge insertions and deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will use Lemma 20 to make the result from Proposition 26 dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, there is a slight technical difficulty: due to edge deletions, T will become a forest and fall apart into several connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This becomes an issue, when an edge (u, v) is inserted for which both u and v already have parents in their respective components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In that case, we cannot immediately make u the root of v (or vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, we need to find an efficient way of re-rooting the tree containing v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', we need to make v the root of its component and we need to ensure that we do not have to recompute the DP solution for all vertices in the component of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We now describe our dynamic algorithm in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, suppose that an edge (u, v) is removed from T and assume that (before the edge deletion) u is closer to the root of T than v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then T becomes a forest with multiple connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In that case, we make v the root of its component and recompute the DP 46 Dynamic Maintenance of Monotone Dynamic Programs and Applications solution for v (since v does not have a parent, we only have to recompute the DP cell for v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, for u and all of its ancestors we recompute the DP solution as per Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, suppose an edge (u, v) is inserted, where u and v are in different connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Further suppose that after the edge insertion, u is the parent of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we distinguish two cases whether v is the root of its component or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, suppose that v is the root of its component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we simply insert the edge (u, v) into T and recompute the solution for v and all of its ancestors (including u) as per Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Second, suppose that neither u nor v is the root of its component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now, we first have to re-root the component containing v such that it has v as its root and such that all DP solution are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We do this as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let v = v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , vℓ denote the vertices on the path from v to the root vℓ of its component (before the edge insertion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we first remove all edges (vℓ, vℓ−1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , (v2, v1) from T (in this order) as per the edge deletion routine described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that after the deletions, none of the vi has a parent and, therefore, each vi is the root of its own component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, by how we picked the order of the edge deletions, after the i’th deletion we only have to recompute the DP cells for the vertices vℓ−i and vℓ−i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we insert the edges again but with flipped direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', we insert the edges (vℓ−1, vℓ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , (v1, v2) (in this order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, v = v1 becomes the root of the component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To insert the edges, we use the subroutine from the paragraph above, where we exploit that each vi is the parent of its own component, which implies that the DP solutions can be updated efficiently: by how we picked the order of the edge insertions, after the i’th edge insertion we only need to recompute the DP cells for vertices vℓ−i−1 and vℓ−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' After the rebalancing of the component containing v is done, v has become the parent of its component and, therefore, we can use the routine from above to insert the edge (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This concludes the edge insertion procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, when we want to output the value of the DP solution, we simply use the subroutine described in Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We summarize the guarantees of our dynamic algorithm in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that when the parameters ϵ, ¯ϵ, k and W are constants, the update time becomes ˜O(h3) and the query time is just O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, the algorithm is very efficient for trees that have polylogarithmic or subpolynomial height in the number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Proposition 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ, ¯ϵ > 0 and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let T = (V, E, cap) be a rooted binary tree with edge weights cap(e) ∈ W∞ and vertex weights w(v) ∈ {0, 1}, that is under- going edge insertions and deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let h be an upper bound on the height of the tree T at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then there exists a fully dynamic algorithm that maintains a bicriteria (1 + ϵ, (1 + ¯ϵ)(1 + ϵ))-approximation for the k-balanced partition problem on T with update time O � h3 · M 2t log2(W) log(k/ϵ) log(M th log(W)/ϵ)/ϵ3� and query time M t(k/(ϵ¯ϵ))O(1/¯ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The fact that the algorithm maintains a bicriteria (1+ϵ, (1+¯ϵ)(1+ϵ))-approximation follows immediately from Lemma 20 and the same arguments as in the proof of Proposition 26, where we argued that the approximate DP satisfies the conditions of Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' It is left to analyze the update and query times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For the query times, note that all we do is run the subroutine from Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This subroutine runs in time M t(k/(ϵ¯ϵ))O(1/¯ϵ2) as we argued in the proof of Proposition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This proves the claim about the query time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For the update times, let us first consider edge deletions (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In this case, we need to update the DP cell for v and the DP solutions for u and all of its ancestors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By Lemma 20, Lemma 25 and by our choice of p = O(h log W/ϵ), this can be done in time O � h3 · M 2t log2(W) log(k/ϵ) log(M th log(W)/ϵ)/ϵ3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 47 Next, consider the case in which (u, v) is inserted and v is the root of its component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we need to recompute the DP solutions for v and all of its ancestors (including u) which, by Lemma 20, Lemma 25 and by our choice of p = O(h log W/ϵ), can be done in the time claimed in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the case that we need to re-root the component of v, note that we have to recompute the solutions for all ancestors of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since the height of T is bounded by h, there are at most h such ancestors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, we have picked the order of edge deletions such that whenever we delete or insert an edge in the re-rooting process then we only need to recompute two DP cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, in total we only need to recompute the solutions for O(h) DP cells in the re-rooting process and thus by Lemma 25, the total time for this process is O � h3 · M 2t log2(W) log(k/ϵ) log(M th log(W)/ϵ)/ϵ3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ Extension to Dynamic General Graphs and Non-Binary Trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now suppose that our input is a dynamic (general) graph G that is undergoing edge insertions and deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Essentially we will solve this problem by maintaining a dynamic Räcke tree and running the algorithm from Proposition 30 on top of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, the dynamic Räcke tree from Theorem 16 is non-binary and, therefore, we start by arguing that we can maintain a binarized Räcke tree dynamically in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let G = (VG, EG) be a dynamic unweighted graph with n vertices that is undergoing edge insertions and deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We can maintain a binarized Räcke tree T = (VT , ET , capT ) with O(n2) vertices, height O(log7/6 n) and quality no(1) in amortized update time no(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The preprocessing time is O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let T ′ = (VT ′, ET ′, capT ′) be the fully dynamic Räcke tree for G from Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, note that T ′ has nT ′ := O(n) vertices and height O(log1/6 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Second, note that T ′ can have unbounded degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, similar to the proof of Lemma 29, we will show how to maintain a binarized Räcke tree T that has T ′ as its corresponding (non-binarized) Räcke tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We do so by taking T ′ and replacing each vertex u in T ′ by a balanced binary tree τu with nT ′ leaves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' the internal edges of τu will have weight ∞ and the edges connecting subtrees τu and τv, u ̸= v, in T will correspond to the edges in T ′ and will have the same (finite) weight as in T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will see that by contracting all edges with weight ∞ in T, we will obtain again T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We now elaborate on this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' During the preprocessing, we first build T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that this takes time O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we construct T as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For each vertex u ∈ VT ′, we add a balanced rooted binary tree τu with nT ′ leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We refer to the root of τu as ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We identify each leaf of τu with a vertex v ∈ VT ′ and denote the leaf of τu that corresponds to v by cu,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We set the weight of the edges inside τu to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that T has O(n2 T ′) = O(n2) vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, for each edge (u, v) ∈ ET ′ (where we assume that v is a child of u), we insert the edge (cu,v, rv) in T and set capT (cu,v, ru) = capT ′(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, if u is the root of T ′ then we set ru to the root of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, suppose that G is changed due to an edge insertion or deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we first update the tree T ′ via the algorithm from Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now, whenever an edge (u, v) is inserted (deleted) in T ′, we insert (delete) the edge (cu,v, rv) into (from) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Each of these insertions and deletions in T can be done in time O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since it takes amortized time no(1) to update T ′ (via Theorem 16), the total update time is no(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' It is left to show that T is a binarized Räcke tree of height O(log7/6 n) and quality no(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Clearly, T is a binary tree since all vertices inside each subtree τu have at most two child nodes and, additionally, each vertex cu,v has at most one child node (namely rv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, T has height O(log7/6 n) since T ′ has height O(log1/6 n) and the subtrees τu have height O(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, let T ′′ be the tree obtained from T by contracting all edges with weight ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We argue that T ′ = T ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Indeed, consider any vertex u ∈ VT ′ and its subtree τu in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 48 Dynamic Maintenance of Monotone Dynamic Programs and Applications Then after contracting the edges in τu, we are left with a subtree that only contains ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, all edges between vertices of different subtrees τu and τv, u ̸= v, have finite weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, T ′ = T ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This implies that T is binarized Räcke tree for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since T ′ has quality no(1), the quality of T is also no(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ Given the lemma above, our dynamic algorithm for dynamic general graphs G works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We maintain the dynamic binarized Räcke tree T as per Lemma 31 on our input graph G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', whenever an edge is inserted or deleted in G, we update the data structure from the lemma as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that this causes edge insertions and deletions in T as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As before, we set the vertex weights in T such that w(v) = 1 if v corresponds to a vertex in G and w(v) = 0 if v is an internal node of T that does not correspond to any vertex in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, we run our dynamic algorithm from Proposition 30 for binary trees on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In particular, whenever T gets updated, we also update the DP solution as per Proposition 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We also use the same query procedure as in the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We conclude the subsection by proving Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We prove the result for general graphs first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since the dynamic binarized Räcke tree T that we maintain has quality no(1), the same argumentation as in the proof of Theorem 2 implies that we maintain a bicriteria (no(1), (1 + ¯ϵ)(1 + ϵ))-approximation for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since by Lemma 31 we can maintain T with amortized update time no(1), the amortized number of edge insertions and deletions into T is no(1) per update operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since T has height O(log7/6 n) and by Proposition 30, the total amortized update time no(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This implies the claim about dynamic general graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To obtain our result for non-binary trees, we can proceed similar to above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consider a non-binary T ′ that is undergoing edge insertions and deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We can maintain the same data structure as in the proof of Lemma 31 to obtain a binary tree T with O(n2) vertices with worst-case update time O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we assign weight w(rv) = 1 to all vertices rv that are roots of the subtrees τv in T and weight w(v) = 0 to all other vertices of the subtrees τv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By the same arguments as in the proof of Theorem 2, we obtain the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ E Simultaneous Source Location In this section, we provide efficient algorithms for the simultaneous source location problem as studied by Andreev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recall that in this problem, the input consists of a graph G = (V, E, cap, d) with a capacity function cap: E → W∞ on the edges and a demand function d: V → W∞ on the vertices of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The goal is to select a minimum set S ⊆ V of sources that can simultaneously supply all vertex demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, a set of sources S is feasible if there exists a flow from the vertices in S that supplies demand d(v) to all vertices v ∈ V and that does not violate the capacity constraints on the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Here, we assume that each source vertex can potentially send an infinite amount of flow that is only constrained by the edge capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The objective is to find a feasible set of sources of minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we summarize our main results for the simultaneous source location problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, we introduce our notion of bicriteria approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let S∗ be the optimal solution for the simultaneous source location problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we say that a solution S is a bicriteria (α, β)-approximate solution if |S| ≤ α |S∗| and if S is a feasible set of sources after all edge capacities are increased by a factor β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The following theorem summarizes our main result for static algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 49 ▶ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let G = (V, E, cap, d) be an undirected weighted graph with n vertices and m edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then for the simultaneous source location problem we can compute: A (1 + ϵ, O(log4(n)))-approximation in time12 ˜O( 1 ϵ2 m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A (1 + ϵ, 1)-approximation in time ˜O( 1 ϵ2 h2 · n) if G is a tree of height h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we turn to our dynamic algorithms which support the following update operations: SetDemand(v, d): updates the demand of vertex v to d(v) = d, SetCapacity((u, v), c): updates the capacity of the edge (u, v) to cap(u, v) = c, Remove(u, v): removes the edge (u, v) from the graph, Insert((u, v), c): inserts the edge (u, v) into the graph with capacity cap(u, v) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The next theorem summarizes our main results for dynamic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let G = (V, E, cap, d) be a graph with n vertices and m edges that is undergoing the update operations given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then for the simultaneous source location problem we can maintain: A (1 + ϵ, no(1))-approximation with amortized update time no(1)/ϵ2 and preprocessing time O(n2/ϵ2) if all edge capacities are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A (1+ϵ, O(log4(n)))-approximation with worst-case update time ˜O(1/ϵ2) and preprocessing time ˜O(m) if we only allow the update operation SetDemand(v, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A (1 + ϵ, O(log2(n) log log(n)))-approximation with worst-case update time ˜O(1/ϵ2) and preprocessing time poly(n) if we only allow the update operation SetDemand(v, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' A (1 + ϵ, 1)-approximate solution with worst-case update time ˜O(h3/ϵ2) and preprocessing time O(n2/ϵ2) if G is a tree of height h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note that in our static and dynamic algorithms, we can output the corresponding solutions similarly to what we descriped after Proposition 12 for knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We start by presenting an exact DP for the special case of binary trees in Section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 and then present an approximate DP in Section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' After that, we generalize the result from binary trees to general graphs in Section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='3 and then also to the fully dynamic setting in Section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 The Exact DP We consider the special case of the simultaneous source location problem on binary trees and provide a DP that solves this problem exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We let T = (V, E, cap, d) denote the rooted binary tree with root r that we obtain as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Additionally, we assume that for each vertex v ∈ V we obtain as input whether we are allowed to make v a source or not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' note that this only generalizes the problem (as in the original problem all vertices can be made sources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Later in Section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='3, this generalization will be helpful when we apply Räcke trees because then we only want to allow leaves to act as sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 DP Definition We now define our exact DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will also discuss its relationship with the DP by Andreev el al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [4] and why we did not use the DP of Andreev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Given a vertex v and a value x ∈ R, we denote by DP(v, x) the minimum number of sources to place in Tv such that when v receives flow at most x from its parent then all demands in Tv can be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note that x can take positive and negative values: for x ≥ 0 this corresponds to the setting in 12 We write ˜O(f(n, ϵ, W)) to denote running times of the form f(n, ϵ, W) · polylog(n, ϵ, log W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 50 Dynamic Maintenance of Monotone Dynamic Programs and Applications which flow is sent from the parent of v into Tv and for x < 0 this corresponds to the setting in which flow is sent from Tv towards the parent of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We further follow the convention that when the demands in Tv cannot be satisfied when v receives flow x from its parent, then we set DP(v, x) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that this DP has rows I = V and columns J = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will store the rows DP(v, ·) using our data structure from Section 2 using monotone piecewise constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we observe that each DP(v, ·) is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, the DP satisfies Property (1) of Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Observation 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The function DP(v, ·): R → [n + 1] ∪ {∞} is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This follows immediately from the definition of DP(v, x): Consider x, x′ ∈ R with x ≤ x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then any solution in which Tv receives flow at most x from the parent of v is also feasible when Tv receives flow at most x′ from the parent of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, DP(v, x) ≥ DP(v, x′), which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ Observe that the global solution for the simultaneous source location problem on T can be obtained by evaluating DP(r, 0), where r is the root of T: First, r has no parent and, therefore, it must be a source itself or have its demand satisfied by its children;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' this explains the choice of x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, (by definition) DP(r, 0) is the minimum number of sources that we need to satisfy all demands in Tr = T and, thus, the flow that we obtain is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We conclude that DP(r, 0) gives the global optimum solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Relationship to the approach by Andreev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, let us elaborate on the relationship of our DP and the function f used by Andreev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In [4], the function f computed by a dynamic program is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Given a vertex v and an integer i ∈ N, Andreev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' define a function f(v, i) that denotes the minimum amount of flow that v needs to receive from its parent if all demands in Tv need to be satisfied and if we can place i sources in the subtree Tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Similar to above, f(v, i) can take positive and negative values: if the demand in Tv can only be satisfied by receiving flow from the parent, then f(v, i) is positive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' if the demand in Tv is already satisfied by the sources in the subtree Tv, then it is possible that v can send flow to its parent and f(v, i) is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' It is not hard to see that the function f(v, i) is monotonically decreasing in i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='13 Now consider f(v, ·): N → R as a function and consider its “inverse”14 function f −1(v, ·): R → N, where f −1 is defined on the whole set of real numbers (including negative numbers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' That is, f −1(v, x) denotes the minimum number of sources that we need to place in Tv such that the demand that v requires from its parent is at most x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' But this was exactly the definition of DP(v, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, DP(v, x) = f −1(v, x) for all v and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Why Did We Not Use f?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In [4] it is shown how the function f can be computed in polynomial time by a bottom-up dynamic program using just a few case distinctions and a (min, +)-convolution in each DP cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, one might wonder why we picked DP(v, ·) = f −1(v, ·) and not f for our DP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Indeed, it seems quite natural to interpret the function f as a monotone piecewise constant function and to use it for our dynamic program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 13 This follows immediately from the definition of f and the fact that by adding more sources to a subtree Tv, the amount of flow that Tv needs to receive from the parent of v only decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 14 We note that, formally, f(v, ·) has no inverse since it is possible that multiple values map to the same number, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', f(v, i) = f(v, i′) for i ̸= i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, formally, we set f −1(v, x) = min{i: f(v, i) ≤ x}, where we follow the convention min{∅} = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we interpret f −1(v, ·) as a piecewise constant function from R to [n + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 51 While for the case of exact computations this is possible, we now sketch why this appears unhandy for the approximate case later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Suppose that we used the function f in our approximate computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To obtain efficient approximation algorithms, we will have to ensure that f has only few pieces and our main way to achieve this is by rounding f as per Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, this becomes tricky because the function values of f are positive and negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the following, it will be illustrative to think of positive function values for f as vertex demands that need to be satisfied and of negative values for f as available edge capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The main issue is that since the function values of f are positive and negative, it is not clear how we should perform the rounding: if we rounded positive and negative values up (towards +∞) then this would correspond to increasing the vertex demands while at the same time decreasing the edge capacities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' however, this could render some feasible solutions (in the exact computation) infeasible (in the rounded computation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' On the other hand, it is conceivable that by always rounding f down (towards −∞), we would essentially decrease the vertex demands while increasing the edge capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Potentially, this approach could work when we are allowed to violate the edge capacities by a (1 + ϵ)-factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, even if we did that, we would have another issue: to only use a small number of pieces for representing f, we would have to use different rounding mechanisms for those function values in [−1, 1] and those in [−W, W] \\ [−1, 1], where W is the largest edge capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Indeed, if we rounded the values of f to powers of (1 + δ)j then there are only O(log1+δ(W)) function values in [−W, W] \\ [−1, 1] but there are infinitely many function values in [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Similarly, if we rounded to multiples of δ then there are only O(1/δ) function values in [−1, 1] but this would lead to O(W/δ) function values in [−W, W] \\ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In both cases, our functions would have too many pieces and, thus, one would have to pick a rounding function which provides a tradeoff between these two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, we would have to find an analysis that shows that this “more involved” rounding function does not introduce too much error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that in the above discussion, all of the issues come from the fact that f(v, ·) can also take negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' On the other hand, our DP (which is f −1(v, ·)) only takes non-negative function values and, therefore, we avoid all of the above complications because we can use the standard rounding function ⌈·⌉1+δ that rounds to powers of 1 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we bypass all of the issues above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our approach also has the positive side effects that instead of getting factors of polylog(W) in our running times, we only get factors of polylog(n) because the codomain of our monotone piecewise constant functions became [n + 1] rather than some potentially large interval [−W, W].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 Computing the DP Now we describe the exact computation of our DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This will reveal the procedures Pi from Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let v ∈ V be any vertex in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We describe how to compute DP(v, ·) efficiently assuming that we have already computed the solutions for the children of v (if they exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recall that for each vertex v we also obtain as input, whether v can be used as a source or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In our following case distinctions, whenever we consider the case that v is used as a source, we will implicitly condition on the fact that it is also possible to use v as source;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' if v cannot be used as a source, we simply skip this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the construction for each DP cell DP(v, ·) for a vertex v with parent p, we will additionally ensure that we do not violate the capacity of the edge (p, v) when x is very small or very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, we will ensure that DP(v, ·) satisfies the additional property 52 Dynamic Maintenance of Monotone Dynamic Programs and Applications that DP(v, x) = ∞ for x < − cap(p, v) and DP(v, x) = DP(v, cap(p, v)) for all x > cap(p, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will denote this property as the feasible capacity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Case 1: v is a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Suppose that v is a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We initialize DP(v, ·) as the function which takes value ∞ on all of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the following, we add at most two pieces to DP(v, ·) depending on whether v can be used as a source or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, suppose v can be used as a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we can send flow up to cap(p, v) to the parent p of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, since v is a leaf, there is exactly one source in Tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we update DP(v, ·) and set DP(v, x) = 1 for all x ≥ − cap(p, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This adds one piece to DP(v, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Second, suppose v is not a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then if x ≥ d(v) and cap(p, v) ≥ d(v), v can receive all of its demand d(v) from its parent and the flow is feasible because we do not exceed the capacity of the edge (p, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, if cap(p, v) ≥ d(v), then we update DP(v, ·) again and add the piece with DP(v, x) = 0 for all x ≥ d(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If x ≥ d(v) but d(v) > cap(p, v) then we do nothing because the parent of v cannot satisfy the demand of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that DP(v, ·) is a monotonically decreasing function with at most three pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, it clearly satisfies the feasible capacity property and Property (3) of Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Case 2: v is not a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Suppose that v is not a leaf and that v has children v1 and v2, as well as a parent p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recall that we assume that we have already computed the DP entries DP(v1, ·) and DP(v2, ·) for both children of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We now show how to compute two DP solutions DPA(v, ·) and DPB(v, ·) depending on whether v is a source (in Case A) or not (in Case B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then, if v can be used as a source, we set DP(v, ·) = min{DPA(v, ·), DPB(v, ·)}, where we compute the min-operation via Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If v cannot be used as a source, we set DP(v, ·) = DPB(v, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Case A: Suppose v is used as a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We initialize DPA(v, ·) as the function which takes value ∞ on all of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now, since v can be used as a source, v can send flow cap(p, v) to its parent and flow cap(v, v1) and cap(v, v2) to its children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, for x ≥ − cap(p, v), the number of sources in DPA(v, x) is 1 (since v is a source) plus the number of sources that we require in Tv1 when v1 can receive flow cap(v, v1) from its parent v plus the same quantity for Tv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, it suffices to set DPA(v, x) = 1 + DP(v1, cap(v, v1)) + DP(v2, cap(v, v2)) for all x ≥ − cap(p, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that here we exploited that the functions DP(v1, ·) and DP(v2, ·) are monotonically decreasing and that both of them satisfy the feasible capacity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We conclude that in this case DPA(v, ·) is a monotonically decreasing function with two pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Case B: Suppose that v is not used as a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We initialize DPB(v, ·) as the function which takes value ∞ on all of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To compute the value of DPB(v, x), we need to obtain the minimum number of sources such that v receives flow at most x from its parent and such that all demands in Tv are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since v is not a source, its demand d(v) must be satisfied either by its parent p or by its children (or a combination of them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, to obtain that we have to pick the children solutions DP(v1, x1) and DP(v2, x2) such that d(v) ≤ x − x1 − x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since we did not make v a source, the number of sources in DPB(v, x) is the number of sources that we need to place in the subtrees Tv1 and Tv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we get DPB(v, x) = min x1∈R{DP(v1, x1) + DP(v2, x − x1 − d(v))}, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 53 where we used that x2 ≤ x − x1 − d(v) and by monotonicity of DP(v2, ·) we minimize the number of sources in Tv2 if we consider x2 = x − x1 − d(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Here, the flows that we computed for DPB(v, x) are set feasible because the solutions DP(vi, ·) satisfy the feasible capacity property and therefore we do not violate the edge constraints to the children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since the above equality holds for all values of x, DPB(v, ·) corresponds to a shifted (min, +)-convolution of two monotonically decreasing functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, via Lemma 6 we can first compute the shifted function DP(v2, · − d(v)) and then we can set DPB(v, ·) = DP(v1, ·) ⊕ DP(v2, · − d(v)), which we compute via Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, as a postprocessing step, we set DP(v, ·) = min{DPA(v, ·), DPB(v, ·)} if v can be used as a source and DP(v, ·) = DPB(v, ·) otherwise, as we already mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' But we also need to ensure that DP(v, ·) satisfies the feasible capacity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, we set DP(v, x) = ∞ for x < − cap(p, v) and we set DP(v, ·) = DP(v, cap(p, v)) for x > cap(p, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that these changes to DP(v, ·) can be done in time linear in the number of pieces of DP(v, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Properties of the DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that in the DP above for each vertex v we only required the DP solutions for its children v1 and v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, our dependency graph is given by our input tree T where all edges are directed towards the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This implies that every node in the dependency graph can only reach those nodes on a path to the root and thus Property (2) of Definition 8 is satisfied with h being the height of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Additionally, one can verify that above all operations also satisfy Property (3) of Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, observe that in each step we only used a constant number of operations from Lemma 6 and at most one (min, +)-convolution from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 The Approximate DP Now we explain how we solve the above DP more efficiently by computing approximate solutions ADP(v, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This will reveal the procedures ˜Pi from Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In our approximation algorithm, we do everything exactly as above except that we replace each exact solution DP(v, ·) with the approximate solution ADP(v, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we add a postprocessing step in which we round ADP(v, ·), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', we set ADP(v, ·) = ⌈ADP(v, ·)⌉1+δ (12) for a parameter δ > 0 that we will set later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that all of our operations are exact except the rounding step which loses a factor of α = 1 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, Property (4a) of Definition 8 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Additionally, observe that in each step we only used a constant number of operations from Lemma 6 and at most one (min, +)- convolution from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This implies that Property (4b) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, all functions we consider are monotone and our rounding step ensures that each row ADP(v, ·) has at most p = O(log1+δ n) pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, Property (4c) is also satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This implies that the DP is (h, 1+δ, O(log1+δ(n)))-well-behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By applying Theorem 9 with δ = ln(1 + ϵ)/(h + 1), we obtain the following proposition which shows that on binary trees, the approximation algorithm computes a bicriteria (1 + ϵ, 1)-approximate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note that for constant ϵ, the running time essentially becomes ˜O(n · h2), where h is the height of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, for trees of height ˜O(1), we obtain a near-linear running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Proposition 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The approximation algorithm computes a bicriteria (1 + ϵ, 1)- approximate solution for the simultaneous source location problem on binary trees in time O(n · (h log(n)/ϵ)2 log(h log(n)/ϵ)), where h is the height of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 54 Dynamic Maintenance of Monotone Dynamic Programs and Applications E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='3 Extension to General Graphs (Proof of Theorem 4) We prove Theorem 4 by giving reductions to the binary setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, suppose that G is a tree with potentially unbounded degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we turn G into a binary tree T using the same construction as in the proof of Lemma 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' That is, we replace each vertex u in G by a balanced binary tree τu with deg(u) leaves cu,v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , cu,vdeg(u), where the vi are the children of u in G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' the internal edges of τu have capacity ∞ and we denote the root of each τu by ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, for each edge (u, v) in G, we insert the edge (cu,v, rv) into T with capacity cap(cu,v, rv) = cap(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By the same arguments as in the proof of Lemma 29, T has O(n) vertices and height O(h log n), where h is the height of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' It is straight-forward to see that with this construction, there exists a flow from u to v in G if and only if there exists a flow from ru to rv in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now, in T we have already set the edge capacities and it remains to set the vertex demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For each vertex ru in T, we set d(ru) = d(u), and for all other vertices v in T, we set d(v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, in our instance of the simultaneous source location problem we set that each vertex ru can be picked as a source and none of the other vertices in T can be picked as a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that there exists a one-to-one correspondence between sources in G and sources in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Together with our observation for flows above, this means that solving the simultaneous source location problem on T gives a solution for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To obtain the bicriteria (1 + ϵ, 1)-approximation result for trees, we apply the approxima- tion algorithm from Proposition 33 on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, to obtain the (1 + ϵ, O(log4 n))-approximate solution for a general graph G, we proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We build the binarized Räcke tree T for G as per Lemma 29 and recall that T has quality q = O(log4 n) and height ˜O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In T, we set the bits to indicate that all leaves can be used as sources but none of the other vertices might be used as a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We apply the approximation algorithm from Proposition 33 on T to obtain a (1 + ϵ, 1)-approximate solution on T in time ˜O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now let us point out that the Räcke tree from Theorem 15 (and, therefore, also the binarized Räcke tree from Lemma 29) is also a tree flow sparsifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' That is, if there exists a feasible flow F in G, then there exists a flow of the same value between the corresponding leaves in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Additionally, for any feasible flow F with value v between leaves in T, there exists a feasible flow with value 1 qv between the corresponding vertices in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, if we are allowed to exceed the edge capacities in G by a factor of q = O(log4 n), the flow that we compute in T is feasible in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This gives that we can compute a (1 + ϵ, O(log4 n))-approximate solution in time ˜O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='4 Extension to the Dynamic Setting (Proof of Theorem 5) To prove Theorem 5, we first consider the special case of dynamic binary trees (which is not mentioned in the theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We show that for binary trees we can maintain a bicriteria (1 + ϵ, 1)-approximate solution with worst-case update time ˜O(h3/ϵ2), where h is an upper bound on the height of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we show that the results of the theorem can be derived from this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consider a dynamic binary tree on which we maintain the approximate DP from Sec- tion E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will exploit that T and the dependency tree of our DP coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, an update in T will trigger the same update in the dependency tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that the update operation SetDemand(v, d) triggers a change to ADP(v, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we can recompute the global approximate DP table using Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since the DP is well-behaved, the tree has height at most h and since we set δ = O(h/ϵ), the theorem implies that we need time ˜O(h3/ϵ2) to recompute the ADP solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Similarly, for SetCapacity((u, v), c) we can again update the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 55 rows ADP(u, ·) and ADP(v, ·) and we update the entire DP table using Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By the same arguments as above, this takes time ˜O(h3/ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For Remove(u, v), we remove the edge (u, v) from T and by the same reasoning as before we get update time ˜O(h3/ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, consider Insert((u, v), c), where we assume that v becomes the child of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we might have the issue that before the update, v is not the root of its connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To mitigate this issue, we run the same re-rooting procedure as described in Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As described in Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='5, this will only recompute the solutions of O(h) DP cells and thus we again have a total update time of ˜O(h3/ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we prove the results from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, consider the case in which all edge capacities are set to 1 and where we want to obtain a bicriteria (1 + ϵ, no(1))-approximate solution with amortized update time no(1)/ϵ2 and preprocessing time O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let G be the dynamic input graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We maintain the dynamic binarized Räcke tree T for G as per Lemma 31 and remark that the dynamic Räcke tree from Theorem 16 is also a tree flow sparsifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note that any update to G triggers an update operation on T that requires amortized update time no(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' On T, we allow the leaves to act as sources but no other vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, we set the demands of the leaves in T to the demands of the corresponding vertices in G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' all other vertices have demand 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we use the data structure for binary trees from the previous paragraph to maintain a dynamic bicriteria (1 + ϵ, 1)-approximate solution on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' That is, when a vertex demand changes in G, we update the corresponding vertex demand in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' When an edge is inserted or deleted in T due to the subroutine from Lemma 31, then we update the data structure from the previous paragraph that maintains the DP solution on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By the same argumentation as in Section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='3, we obtain that since T has quality no(1), if we can exceed the edge capacities in G by a no(1) factor then any feasible flow in T is also feasible in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This implies the result claimed in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If G is a tree of height h but (potentially) with unbounded degrees, we can maintain a bicriteria (1 + ϵ, 1)-approximate solution with worst-case update time ˜O(h3/ϵ2) and prepro- cessing time O(n2/ϵ2) similar to above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' That is, we transform G into a binary tree using the same procedure that we use in the proof of Lemma 31, where we replace each vertex u by a subtree τu with root ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Similar to what we argued in Section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='3, we only allow the vertices ru as roots in T and obtain any flow in T corresponds to a flow in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then by applying the dynamic data structure for binary trees on T, we obtain the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, let us consider the case in which we wish to obtain bicriteria approximation algorithms when we only allow the update operations SetDemand(v, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In this case, we observe that the underlying graph is static, since only the vertex demands change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, for our input graph G, we can build the Räcke tree from Theorem 15 which is also a tree flow sparsifier and we consider its binarized version T as per Lemma 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that building this tree with quality ˜O(log4 n) takes time ˜O(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Given such a static Räcke tree, we can use our dynamic data structure for binary trees from above to support the operations SetDemand(v, d) on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since T has height ˜O(1), we obtain the result with the bicriteria (1 + ϵ, O(log4 n))- approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To obtain the bicriteria (1 + ϵ, O(log2 n log log n))-approximation we do exactly the same as above, but instead of using the Räcke tree from Theorem 15, we use the Räcke tree from Harrelson, Hildrum and Rao [37] which can be used as a tree flow sparsifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As it has quality O(log2 n log log n) and can be constructed in time poly(n), we obtain the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 56 Dynamic Maintenance of Monotone Dynamic Programs and Applications F Recourse Bounds In this section discuss the recourse bounds we derive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To motivate these lower bounds, let us note that “classic” dynamic algorithms with polylogarithmic update time maintain a single explicit solution in memory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' this is desirable in many practical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, some dynamic algorithms (like our DP algorithms above) only return the value of an approximate solution in polylogarithmic time, which is sometimes referred to as implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To understand whether for our problems implicit solutions are necessary, we consider algorithms which maintain multiple explicit solutions, of which only one has to be feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We believe that this is an interesting setting to look at, as it essentially interpolates between the two scenarios above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If even algorithms with multiple solutions must have high recourse, this suggests that implicit solutions are somehow inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We show below that for fully dynamic knapsack and fully dynamic k-balanced partitioning the latter is the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Here, we consider dynamic algorithms over inputs that are undergoing insertions and deletions via an update operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The algorithms are allowed to maintain multiple explicit solutions and must ensure that after every time step, there exists a solution with certain guarantees while minimizing the recourse for updating the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, we consider algorithms which explicitly maintain s solutions S(t) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , S(t) s for each time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Here, we assume that after each time step a single update operation is performed, after which an algorithm can make changes to its solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We say that an algorithm maintains an α-approximate solution if for each time step t, there exists an index i = i(t) such that S(t) i is a feasible and α-approximate solution for the problem we study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that in this setting, the algorithm might have much lower recourse, since for each time t it may pick a different solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, it may not have to update any of the solutions significantly after the update operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Further note that this notion of ensuring that at each time step there exists a feasible solution is somewhat reminiscent of list decoding in coding theory, where the decoder can output a list of messages and only has to ensure that the correct messages is contained in that list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Measuring the recourse will be problem-specific, based on how the solutions for the problems are stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In general, given two solutions from consecutive time steps, we let d(S(t) i , S(t+1) i ) denote the (problem-specific) recourse incurred by the i’th solution at time step t (see below for how to set d(·, ·) for the problems we study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The total recourse of an algorithm is given by � t s � i=1 d(S(t) i , S(t+1) i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we will present the concrete recourse lower bounds that we derive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recourse Bounds for Knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In knapsack, the solutions S(t) i simply correspond to subsets of items which are contained in the knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To measure the recourse, we set d(S(t) i , S(t+1) i ) = ���S(t) i △S(t+1) i ���, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', we consider the cardinality of the symmetric difference of the i’th solution at time steps t and t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our main result shows that for a fixed accuracy ϵ, any dynamic (1 − ϵ)-approximation algorithm must maintain Ω(1/ϵ) solutions or it must have recourse Ω( n ϵ ), even when only a single item is inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Theorem 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ ∈ (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Assume s < 1 8ϵ(1+2ϵ) and n ∈ N is a sufficiently large multiple of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then any dynamic randomized (1 − ϵ)-approximation algorithm for knapsack with s solutions must have recourse Ω( n s ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This holds even for a single item insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 57 Recourse Bounds for k-Balanced Partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In k-balanced partitioning, each solution S(t) i consists of k clusters V (i,t) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , V (i,t) k that partition the set of vertices V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To measure the recourse, we set d(S(t) i , S(t+1) i ) = �k j=1 ���V (i,t) j △V (i,t+1) j ���, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', we consider the total number of vertices that change their set V (i,·) j from time t to t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our main result shows that for any C and fixed ϵ, any algorithm that maintains a (C, 1 + ϵ)-approximate solution must use Ω(1/ϵ) solutions or it must have amortized recourse Ω(ϵ2 n k ), even when only O(1/ϵ) edges are inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Here, the amortized recourse refers to the total recourse divided by the total number of update operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Theorem 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let C > 0 be arbitrary and ϵ ∈ (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Assume k ≥ 4 and s < 1 4ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then any dynamic randomized (C, 1 + ϵ)-approximation algorithm for k-balanced partitioning with s solutions must have amortized recourse Ω(ϵ2 n k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This holds even for O(1/ϵ) edge insertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 Proof of Theorem 34 We prove Theorem 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We use Yao’s principle [63], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', we consider a deterministic algorithm and give a distribution over inputs, showing that in expectation the algorithm will have recourse Ω( n s ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We consider an instance in which initially we have n items, and each item i has weight wi = 1 and price pi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We refer to these items as small items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We set the budget of our knapsack to B = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that in this instance, OPT = n because all small items fit into the knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we sample an integer j uniformly at random from [2s − 1] = {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , 2s − 1}, and we set k = i · n 2s + n 4s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We insert a single heavy item with p = n − k + 2ϵn and w = n − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that after inserting the heavy item, we have that OPT = n + 2ϵn since the optimal solution consists of the heavy item and k small items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We let S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , Ss denote the solutions maintained by the algorithm before the heavy item was inserted and we let S′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , S′ s denote the solutions after the heavy item was inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We write small(Si) to denote the number of small items in solution Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the following, we will show that any (1 − ϵ)-approximate solution S′ i must contain the heavy item and “almost” k small items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, we will also show that with constant probability all solutions Si had “much less” or “much more” than k small items initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This then gives that obtaining any (1 − ϵ)-approximate solution must encur high recourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We follow this proof strategy in reverse order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We start by showing that with constant probability, all Si have “much less” or ”much more” than k small items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' With probability at least 1/2, it holds that |k − small(Si)| ≥ n 4s for all i ∈ [s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Suppose that we partition the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , n} into 2s consecutive intervals, each of length n 2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that k is the middle point of one of these intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, as there are only s solutions and 2s intervals, at least half of the intervals do not contain a number from the set {small(Si): i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , s};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' we call these intervals empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, with probability at least 1/2, k is the middle point of an empty interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If the interval containing k is empty, then k has distance at least n 4s to small(Si) for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ Next, recall that S′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , S′ n are the solutions maintained by the algorithm after the insertion of the heavy item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We show that any (1 − ϵ)-approximate solution must contain the heavy item and some small items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Suppose that S′ i is a (1 − ϵ)-approximate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then S′ i contains the heavy item and a positive number of small items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 58 Dynamic Maintenance of Monotone Dynamic Programs and Applications Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, suppose that a solution S′ i only contains small items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then its total price is at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, we have that (1 − ϵ) OPT = (1 − ϵ)(1 + 2ϵ)n > n, where we used that ϵ < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, S′ i is not a (1 − ϵ)-approximate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Second, suppose that S′ i only contains the heavy item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we have that (1 − ϵ) OPT = (1 − ϵ)(p + k) = p + k − ϵ(p + k) ≥ p + n 4s − ϵ(1 + 2ϵ)n > p + 2ϵ(1 + 2ϵ)n − ϵ(1 + 2ϵ)n > p, where we used that k ≥ n 4s, p+k = n+2ϵn and s < 1 8ϵ(1+2ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, a solution containing only the heavy item is not (1 − ϵ)-approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ Next, we show that any (1−ϵ)-approximate solution must contain “almost” k small items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Suppose S′ i is a (1 − ϵ)-approximate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then k − n 8s ≤ small(S′ i) ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The upper bound follows from the fact S′ i must contain the heavy item (by Lemma 37) of weight n − k and then it can only include k small items since the budget constraint is set to B = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To prove the lower bound, note that since S′ i is a (1 − ϵ)-approximate solution, we have that its solution has value p + small(S′ i) ≥ (1 − ϵ) OPT = (1 − ϵ)(p + k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, we get that small(S′ i) ≥ k − ϵ(p + k) = k − ϵ(1 + 2ϵ)n > k − n 8s, where we used that s < 1 8ϵ(1+2ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ To finish the proof of the theorem, we condition on the event from Lemma 36, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', we have that |k − small(Si)| ≥ n 4s for all i ∈ [s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now consider any solution S′ i after the insertion of the heavy item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If S′ i is not (1 − ϵ)- approximate, we can ignore S′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If S′ i is (1 − ϵ)-approximate then it satisfies k − n 8s ≤ small(S′ i) ≤ k by Lemma 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since we are assuming the event from Lemma 36, the algorithm had to insert/delete at least n 8s small items into/from Si to obtain S′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since the event from Lemma 36 occurs with probability at least 1/2 and the above argument holds for all (1 − ϵ)-approximate solutions S′ i, we have that the expected recourse is Ω( n s ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 59 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 Proof of Theorem 35 We prove Theorem 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Again, we apply Yao’s principle [63], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', we consider a deterministic algorithm and give a distribution over inputs, showing that in expectation the algorithm will have amortized recourse Ω(ϵ2 n k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We consider a graph with n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our initial instance consists of k 2ϵ star graphs, each of which contains 2ϵ n k vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that here an optimal solution places the vertices from exactly 1 2ϵ star graphs into each partition Vj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' there are no edges between vertices from different Vj and hence the optimal cut-value is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, the solution of any (C, 1 + ϵ)-approximate solution must also have cut-value zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the update phase, we sample s edges between the central nodes of the star graphs uniformly at random and insert them into the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that after the insertion of the edges, we connected at most s star graphs and the largest connected component has size at most s·2ϵ n k ≤ 1 2 n k , where we used that s ≤ 1 4ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, the optimal solution still has cut-value zero and thus any (C, 1 + ϵ)-approximate solution must have cut-value zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, let us analyze the recourse of an algorithm which starts with initial solutions S(0) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , S(0) s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In a first step, we show that solutions which at time 0 splits one of the star graphs up “too much” must entail high recourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In a second step, we consider all other solutions and show that our insertions still trigger high recourse in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We say that a solution S(0) i = {V (i,0) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , V (i,0) k } is useful if for all star graphs H, it holds that there exists an index j such that V (i,0) j contains at least ϵ n k vertices from H and at most ϵ n k vertices are placed in � j′̸=j V (i,0) j′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Given a star graph H and a solution S(0) i , we write j(H, i) to the denote the index j such that V (i,0) j contains at least at least ϵ n k vertices from H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If a solution is not useful, we call it useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, consider solutions which are useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' There exist two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Case A: Suppose there exists a star graph H such that for all indices j it holds that � j′̸=j V (i,0) j′ contains more than ϵ n k vertices from H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that if the algorithm wants to use this solution after the edge insertions finished, it must ensure that the cut-value is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus it must move at least ϵ n k vertices to one of the V (i,0) j which requires ���� j′̸=j V (i,0) j′ ��� ≥ ϵ n k vertex moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Case B: Suppose there exists a star graph H such that for all j it holds that V (i,0) j contains less than ϵ n k vertices from H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Using that ϵ ∈ (0, 1 2), also in this case the algorithm must move at least (1 − ϵ) n k ≥ ϵ n k vertices such that eventually all of H is contained in the same set V (i,s) j when the updates finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We conclude that for useless solutions our theorem holds after amortizing over s ≤ 1 ϵ insertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Second, for the remainder of the proof consider only solutions S(0) i = {V (i,0) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , V (i,0) k } which are useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that when we insert an edge between two star graphs H1 and H2, then if j(H1, i) ̸= j(H2, i) the algorithm must move at least ϵ n k vertices to ensure that after the s insertions finished, all vertices from H1 and H2 are placed in the same set V (i,s) j for some j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We call such an insertion expensive for solution i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that if our edge insertions are such that they contain an expensive insertion for all solutions, then updating any solution S(0) i such that S(s) i is (C, 1 + ϵ)-approximate will incur recourse at least ϵ n k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The rest of our proof is devoted to showing that with constant probability this event occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This will prove the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We start by considering a fixed solution S(0) i and a single random edge insertion between randomly picked star graphs H1 and H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recall that there are k 2ϵ star graphs in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, we have that ���V (i,0) j ��� ≤ (1 + ϵ) n k for all j and thus for each j there can be at most (1+ϵ) ϵ star graphs H with j = j(H, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, for the probability that the edge insertion 60 Dynamic Maintenance of Monotone Dynamic Programs and Applications is expensive we get that Pr (j(H1, i) ̸= j(H2, i)) = 1 − Pr (j(H1, i) = j(H2, i)) ≥ 1 − (1 + ϵ)/ϵ k/(2ϵ) = 1 − 2(1 + ϵ) k ≥ 1 2, where we used that k ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we consider a fixed solution S(0) i and s edge insertions between star graphs which were picked independently and uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then with probability at least 1 − 2−s, at least one of these edge insertions is expensive for solution i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, observe that probability that for all solutions i there exists an expensive edge insertion is at least � 1 − 2−s�s = exp � s ln(1 − 2−s) � ≥ 1 + s ln(1 − 2−s) ≥ 1 − s2−s ≥ 1 4, where we used that exp(x) ≥ 1 + x for all x ∈ R, the Taylor expansion of ln(x) for x close to 1 and the fact that s2−s ≤ 3 4 for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We conclude that with constant probability, for all solutions i there exists an expensive edge insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In this case, the algorithm has total recourse at least ϵ n k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, the expected total recourse of the algorithm is Ω(ϵ n k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since we only performed s edge insertions, this gives an amortized recourse of Ω(ϵ2 n k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' G Non-Monotone Functions and ℓ∞-Necklace Alignment So far we have only considered monotone piecewise constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we will generalize some of our results to piecewise constant functions with multiple non-monotonicities and provide the details in Section G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We also derive new approximation algorithms for the ℓ∞-necklace problem in Section G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In particular, for ℓ∞-necklace we present the first approximation algorithm with near-linear running time with additive error ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We also present the first dynamic approximation algorithm for this problem which achieves additive error ϵ and has update time O((1/ϵ)2 log(1/ϵ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' the algorithm has preprocessing time O(1) when starting with empty vectors x and y and requires sublinear space O(1/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' See Theorem 44 for the details of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 Piecewise Constant Functions With Non-Monotonicities We now show that we can perform efficient operations on piecewise constant functions even when these functions contain non-monotonicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, the running times of our subprocedures will typically have some dependency on the number of non-monotonicities of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let us formalize our notion of non-monotonicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We say that a function f : [0, t) → [0, W] ∪ {−∞, ∞} has k monotone segments if there exist values 0 = x0 < x1 < · · · < xk = t M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 61 such that on each interval [xi, xi+1), f is monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Here, we require that either f is monotonically decreasing on all segments or it is monotonically increasing on all segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that a monotone function has k = 1 monotone segments (by setting x0 = 0 and x1 = t) and that the points x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , xk−1 can be viewed as the points in which f is non-monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' One crucial operations will again be rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' However, unlike previously we will mostly talk about rounding to multiples of δ instead of rounding to powers of 1 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This will be convenient for our applications to ℓ∞-necklace later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will also briefly mention how to extend our results from this subsection to the setting in which we round to powers of 1 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, let δ > 0 and consider a simple rounding function that rounds down to multiples of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, for y ∈ R we set ⌊y⌋∗ δ = max{i · δ: i · δ ≤ y, i ∈ Z} and we follow the convention that ⌊−∞⌋∗ δ = −∞ and ⌊∞⌋∗ δ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We also extend the rounding operation to functions f : [0, t) → [0, W] ∪ {−∞, ∞} by defining ⌊f⌋∗ δ : [0, t) → [0, W] ∪ {−∞, ∞} to be the function with ⌊f⌋∗ δ(x) = ⌊f(x)⌋∗ δ for all x ∈ [0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we show that the function ⌊f⌋∗ δ can be computed efficiently and that it has only few pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let δ > 0 and let f : [0, t) → [0, W] ∪ {−∞, ∞} be a piecewise constant function with p pieces and k monotone segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we can compute the function ⌊f⌋∗ δ in time O(p log p) and ⌊f⌋∗ δ has O(k · W/δ) pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , (xp, yp) denote the list representation of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We construct the list representation (x′ 1, y′ 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , (x′ p, y′ p) of ⌊f⌋∗ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , p, we set x′ i = xi and y′ i = ⌊yi⌋∗ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' After that, we merge all consecutive pieces that have the same y′ i-values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' this can be done exactly as in the pruning step described in the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since f takes values in [0, W] ∪ {−∞, ∞}, there are O(W/δ) choices for multiples of δ in [0, W].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In particular, on each monotone segment of f, ⌊f⌋∗ δ has O(W/δ) pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since f has k monotone segments this implies that ⌊f⌋∗ δ has O(k · W/δ) pieces in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that all operations from above can be performed in linear time and the running time bound stems from the fact that we also need to store the pieces in a binary search tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ Next, we show that we can compute the (min, +)-convolution of two piecewise constant functions in time that is quadratic in the number of their pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The lemma generalizes the result from Lemma 7 because we drop the assumption that one of the functions needs to be monotone (but this comes at the cost of a more complicated proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We prove the lemma in Section G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let f1, f2 : [0, t) → [0, W] ∪ {−∞, ∞} be piecewise constant functions which have at most p pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we can compute f = f1 ⊕ f2 in time O(p2 log p) and f has O(p2) pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By combining the two lemmas above, we can show that we can efficiently compute additive approximations of (min, +)-convolutions even in the case of non-monotonicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, we say that f : [0, t) → [0, W] ∪ {−∞, ∞} is an additive ϵ-approximation of g: [0, t) → [0, W] ∪ {−∞, ∞} if g(x) − ϵ ≤ f(x) ≤ g(x) for all x ∈ [0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we obtain the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Theorem 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let f, g: [0, t) → [0, W] ∪ {−∞, ∞} be two functions with k monotone segments and suppose we have already computed ⌊f⌋∗ δ and ⌊g⌋∗ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then the function (⌊f⌋∗ δ) ⊕ (⌊g⌋∗ δ) is an additive 2δ-approximation of f ⊕ g, has at most O((k · W/δ)2) pieces and can be computed in time O((k · W/δ)2 log((k · W/δ)2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The approximation ratio follows from the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The claims about the number of pieces and the running time follow from combining Lemma 39 and Lemma 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ 62 Dynamic Maintenance of Monotone Dynamic Programs and Applications We note that by stating Lemma 39 for the rounding operation ⌈·⌉1+δ that rounds to powers of 1 + δ (see Lemma 6), we can obtain the following version of Theorem 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Theorem 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let f, g: [0, t) → [0, W] ∪ {−∞, ∞} be two functions with k monotone segments and suppose we have already computed ⌈f⌉1+δ and ⌈g⌉1+δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then (⌈f⌉1+δ)⊕(⌈g⌉1+δ) is a (1+δ)-approximation of f ⊕g, has at most O((k·log1+δ(W)2) pieces and can be computed in time O((k · log1+δ(W))2 log((k · log1+δ(W))2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This result generalizes our previous method of first rounding a monotone function via Lemma 6 and then applying the efficient convolution from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, observe that monotone functions have one monotone segment and, thus, after rounding both functions, our algorithm from Lemma 7 computes the (min, +)-convolution in time O(log2 1+δ(W) log log1+δ(W)) which is the same running time that we obtain by combining the two lemmas above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, the algorithm from Theorem 42 matches this result for k = 1 and it generalizes it when we apply it for k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 Proof of Lemma 40 We assume that fi for i = 1, 2 is given as a doubly linked list (xi 1, yi 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , (xi p, yi p) such that xi j < xi j+1 for all 1 ≤ j < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will output f in the same representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To compute f we will make use of the following non-overlapping interval data structure (NOI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let [a, b] and [a′, b′] be two subsets of the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We call each of them an interval and say that they overlap if [a, b] ∩ [a′, b′] ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We say that an interval [a, b] is empty if a ≥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The NOI data structure stores a set S of non-overlapping, non-empty intervals I = [a, b] and supports the following operations: ClosestLargerInterval(z), which given a number z returns the interval [a, b] together with a Boolean value bool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If bool is true, then z ≤ b and there is no interval [a′, b′] in S with z ≤ b′ < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that it is possible that z belongs to [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If bool is false, then there exists no interval [a, b] with z ≤ b and the returned values for a and b are undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' InsertInterval(a, b), which inserts the interval [a, b] into S, merging it with any interval that it overlaps with and updating S accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' There exists an efficient implementation of such a data structure as stated in the next claim, which we prove at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▷ Claim 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' There exists an implementation of the non-overlapping interval data structure such that any sequence of q operations takes time O(q log q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We compute f as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that the function values of f1 and of f2 are constant over each 2-dimensional rectangle whose corners are (x1 s, x2 t), (x1 s, x2 t+1), (x1 s+1, x2 t), and (x1 s+1, x2 t+1) for any 1 ≤ s ≤ p and 1 ≤ t ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We call this rectangle Rst and denote by [x1 s + x2 t, x1 s+1 + x2 t+1] the range of the rectangle Rst and by y1 s + y2 t the function value of the rectangle, where we assume that ∞ + y with y ∈ W∞ equals ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' There are K2 such rectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now note that for any value x with x1 s +x2 t ≤ x ≤ x1 s+1 +x2 t+1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', x is in the range of the rectangle Rst, the function value y1 s + y2 t is one of the sums that occurs in the computation of f(x) = min¯x{f1(¯x) + f2(x − ¯x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will compute f(x) (for all values x “simultaneously”) by comparing the function values of all rectangles Rst to whose range x belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The main observation that we exploit is the following: As we will consider the rectangles by decreasing function values, the first rectangle (in this order) to whose range a value x belongs is the rectangle whose function value equals f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 63 Thus, when processing a rectangle, we need to determine all ranges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e, subintervals of [0, t], to which no function value has yet been assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To do so, we use the NOI data structure to store the intervals of all values x for which we have already assigned a function value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore we use a balanced binary search tree B that stores at its leaves every interval to which a function value has already been assigned, together with its (constant) function value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Specifically, we will store these ranges in the leaves of B, ordered by their smaller boundary value x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The difference between the two is that the NOI data structures merges overlapping intervals, no matter what their function value is, while every interval stored as a leaf of B has the same function value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', has a constant f-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To be precise we proceed as follows: We first generate all rectangles Rst by iterating over the lists of f1 and f2 and sort them by non-decreasing order of their function value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This takes time O(p2 log p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we process the rectangles in this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To do so, we first initialize an empty NOI data structure as well as an empty balanced binary search tree B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next we describe how to process the rectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let Rst be the next rectangle to be processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We execute the following steps for Rst: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' z = x1 s + x2 t 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (a, b, bool) = ClosestLargerInterval(z) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' while bool is true and b < x1 s+1 + x2 t+1 do a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' if z ̸∈ [a, b] then insert the interval [z, a] together with the function value of Rst into B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' z = b c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (a, b, bool) = ClosestLargerInterval(z) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If bool is true then insert the interval [z, a] together with the function value of Rst into B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' else insert the interval [z, x1 s+1 + x2 t+1] together with the function value of Rst into B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' InsertInterval(x1 s + x2 t, x1 s+1 + x2 t+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Once all rectangles have been processed, we traverse the leaves of B in order and connect them by a doubly linked list to create an (ordered) list representation of the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As we process the rectangles in increasing order of function value this guarantees that for each value x the smallest function value of any rectangle Rst is returned as f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that each insertion into B takes time O(log p) and the number of calls to the NOI data structure is proportional to the number of rectangles plus the number of intervals merged in the NOI data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As processing a rectangle creates at most one new interval, and merged intervals are never separated again, the number of interval merges is at most the number of rectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, there are at most p interval merges and at most 2p2 insertions into B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, the total running time for the above algorithm is O(p2 log p) plus the time for the NOI data structure, which, by Claim 43, is also O(p2 log p) as q = O(p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We still have to prove Claim 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof of Claim 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We implement the NOI data structure with a balanced binary search tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The leaves store the non-overlapping intervals, ordered by their upper endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The ClosestLargerInterval(z) operation searches for the interval [a, b] such that b is the smallest upper endpoint of an interval that is at least z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If no such interval exists, bool is set to false, otherwise it is set to true and [a, b] is returned as interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that finding [a, b] takes time O(log q), as q is the maximum number of intervals stored in the balanced binary tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The InsertInterval(a, b) operation first executes a ClosestLargerInterval(a) operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let (a′, b′, bool) be the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If bool is false, then the interval [a, b] is inserted as new interval and the procedure terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Otherwise the interval [a′, b′] is the interval with smallest upper endpoint such that a ≤ b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that [a′, b′] might overlap with [a, b] and we test for this next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If b < a′ then a leaf with range [a, b] is inserted into the balanced search tree and 64 Dynamic Maintenance of Monotone Dynamic Programs and Applications InsertInterval(a, b) terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Otherwise (b ≥ a′), let L be the leaf of the balanced search tree that stores [a′, b′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If b ≤ b′, the two intervals are merged by updating L to store the interval [min(a, a′), b′] and InsertInterval(a, b) terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If, however, b > b′, it is possible that the new interval [a, b] overlaps with even more intervals in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we execute the following steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' z = b′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' (a′′, b′′, bool) = ClosestLargerInterval(z) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' while bool is true do a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If b < a′′ then the leaf L is updated to store the interval [min(a, a′), b] and InsertInterval terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The leaf storing the interval [a′′, b′′] is removed from the balanced search tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If b ≤ b′′ then the leaf L is updated to store the interval [min(a, a′), b′′] and InsertIn- terval terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Otherwise, z = b′′ and (a′′, b′′, bool) = ClosestLargerInterval(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' L is updated to store the interval [min(a, a′), b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that this algorithm merges all intervals that overlap with [a, b] into one interval and updates the balanced search tree accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let t be the number of iterations executed by InsertInterval(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The running time is O((t + 1) log q) as each iteration executes one call to ClosestLargerInterval, one deletion of a leaf in the balanced binary tree, and at most one modification of a label at a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Every such iteration decreases the number of leaves in the balanced binary tree by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, each call to InsertInterval that does not execute any iterations of the above while-loop increases the number of leaves by at most 1 and there is no other operation that modifies the number of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As there are at most q calls to InsertInterval, the while-loop can be executed at most q times over all calls to InsertInterval, each taking time O(log q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, the total runnning time for q calls to InsertInterval is O(q log q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 ℓ∞-Necklace Alignment Using our techniques from above, we present a novel approximation algorithm for the ℓ∞-necklace alignment problem [14, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In this problem, the input consists of two neck- laces represented as two sorted vectors of n real numbers, x = ⟨x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , xn−1⟩ and y = ⟨y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , yn−1⟩, where the xi, yi ∈ [0, 1) represent points on the unit-circumference circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will sometimes refer to the elements xi and yj as beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We define the distance between two beads xi and yj by the minimum of the clockwise and counterclockwise distances along the circumference of the unit-perimeter circular necklaces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', we set d◦(xi, yj) = min{|xi − yj| , 1 − |xi − yj|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the ℓ∞-necklace alignment problem, we need to find an offset c ∈ [0, 1) and a shift s ∈ [n + 1] that minimize n−1 max i=0 (d◦((xi + c) mod 1, y(i+s) mod n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the above definition, the offset c encodes how much we rotate the first necklace clockwise relative to the second necklace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Additionally, the shift s defines a perfect matching between the beads such that bead i of the first necklace is matched with bead (i + s) mod n of the second necklace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 65 Bremner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [14] showed that the ℓ∞-necklace alignment problem can be solved exactly in time ˜O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We complement this by showing that we can compute a solution with additive error ϵ in time ˜O(n + ϵ−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We also consider the dynamic version of the problem in which beads are inserted and deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, we assume that initially x and y are empty and we offer the following update operations: Insert(i, α, β) which inserts α ∈ [0, 1) into x at the i’th position and it further inserts β ∈ [0, 1) into y at the i’th position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We require that after the insertion, x and y are still ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Delete(i) which deletes xi from x and yi from y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that both of these operations change the number of entries in x and y but they ensure that x and y always have the same length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We show that we can maintain a solution with additive error ϵ using update time O(1/ϵ2 log(1/ϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The preprocessing time is O(1) and the space usage is only O(1/ϵ) which is sublinear in the size of the vectors x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Theorem 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' There exists a static algorithm for the ℓ∞-necklace alignment problem that computes a solution with additive error ϵ in time O(n + (1/ϵ)2 log(1/ϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Fur- thermore, there exists a fully dynamic algorithm for the ℓ∞-necklace alignment problem that maintains a solution with additive error ϵ with update time O(1/ϵ2 log(1/ϵ)) and preprocessing time O(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' the space usage of the algorithm is O(1/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To obtain the result for the dynamic algorithm, we show that for vectors A, B ∈ Rn that are undergoing element insertions and deletions, we can dynamically maintain an approximation of the (min, +)-convolution A ⊕ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We expect that this result will have further applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The proof of the theorem follows from Propositions 45 and 48 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 The Static Algorithm Now we consider our static algorithm and prove the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Proposition 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' There exists a static algorithm for the ℓ∞-necklace alignment problem that computes a solution with additive error ϵ in time O(n + (1/ϵ)2 log(1/ϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We devote the rest of this subsection to the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Our algorithm is rather simple and (up to the part in which we perform the rounding) it is the same as the one used by Bremner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consider the input ϵ (as error parameter), x = ⟨x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , xn−1⟩ and y = ⟨y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , yn−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we set δ = ϵ/2 and perform a single pass over x and y and apply the rounding function ⌊·⌋∗ δ to each of the entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' While doing so, we compute the list representations of x and y (where we interpret x and y as functions from [0, n) to [0, 1)) which have at most O(1/δ) pieces (by applying Lemma 39 with W = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we compute the vectors x′ = ⟨x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , xn−1, ∞, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , ∞ � �� � n times ⟩, x′′ = ⟨x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , xn−1, −∞, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , −∞ � �� � n times ⟩, y′ = ⟨yn−1, yn−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , y0, yn−1, yn−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , y0⟩, but we do not store them explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Instead, we only store their list representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note that x′ is a monotonically increasing vector, x′′ has two monotonically increasing segments and y′ has two monotonically decreasing segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 66 Dynamic Maintenance of Monotone Dynamic Programs and Applications Next, we set a to the (min, −)-convolution of x′ and y′ and we set b to the (max, −)- convolution of x′′ and y′ (we show below in Lemma 46 that we can compute these functions efficiently).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, we set v = 1 2(b − a) and return min{vs : s ∈ [n]} as the solution for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We note that v can be efficiently computed via the list representations of a and b and we can also quickly find the minimum over the vs by iterating over the list representation of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we turn to the analysis of the algorithm above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We adapt the proof of Theorem 6 in Bremner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [14] for approximate solutions and argue how to implement it using piecewise constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We start by showing that we can compute (min, −)-convolution and (max, −)-convolution as efficiently as the classic (min, +)-convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let f and g be two piecewise constant functions with p pieces and suppose that g has k monotonically decreasing segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Suppose that we can compute the (min, +)- convolution of f ′ and g′ in time t(p, k) if f ′ and g′ have k monotonically decreasing segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then in time O(t(p, k) + p log p) we can compute: The (max, −)-convolution of f and g if f has k monotonically increasing segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The (min, −)-convolution of f and g if f has k monotonically increasing segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, suppose that we wish to compute the (max, −)-convolution of two functions f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We show that we can compute the (max, −)-convolution of f and g via the (min, +)-convolution of −f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Indeed, for all x it holds that: max ¯x∈[0,x]{f(¯x) − g(x − ¯x)} = max ¯x∈[0,x]{−(−f(¯x) + g(x − ¯x))} = − min ¯x∈[0,x]{−f(¯x) + g(x − ¯x)} = −(((−f) ⊕ g)(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To see that the running time is correct, note that we can compute the list representation of −f in time O(p) and it takes takes O(p log p) to update the binary search tree in which we store the pieces of −f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, −f has k monotonically decreasing segments since f has k monotonically increasing segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we can apply the efficient algorithm for (min, +)-convolution in time t(p, k) on −f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We can prove the result for (min, −)-convolution similarly by computing a (min, +)- convolution of g and −f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, for all x it holds that min ¯x∈[0,x]{f(¯x) − g(x − ¯x)} = min ¯x∈[0,x]{−f(¯x) + g(x − ¯x)} = ((−f) ⊕ g)(x), where in the first step we used the symmetry of (min, −)-convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The running time analysis is exactly as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ In the proof of Proposition 45 we need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We will use the lemma to find the optimal offset c for a given shift s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Lemma 47 (Fact 5 in [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let z = ⟨z0, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , zn−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then min c∈R n−1 max i=0 |zi + c| = 1 2 � n−1 max i=0 zi − n−1 min i=0 zi � and the minimizer for this quantity is given by c = − 1 2(minn−1 i=0 zi + maxn−1 i=0 zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we can prove Theorem 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 67 Proof of Theorem 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We prove the theorem in three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In Step 1, we will prove that we compute the correct result in the exact case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', when we perform no rounding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This first part is essentially the same proof as in in Bremner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [14] but with more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In Step 2, we argue about approximation guarantee of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In Step 3, we prove its running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Step 1: The Exact Case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, we use Theorem 2 of Bremner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' [14] which states that if ˜y = ⟨y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , yn−1, y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , yn−1⟩ then min c,s n−1 max i=0 d◦((xi + c) mod 1, y(i+s) mod n) = min c,s n−1 max i=0 d−(xi + c, ˜yi+s), where d−(a, b) = |a − b| for all a, b ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, instead of directly optimizing the origi- nal objective function minc,s maxn−1 i=0 d◦((xi + c) mod 1, y(i+s) mod n), we will consider the more convenient objective function minc,s maxn−1 i=0 d−(xi + c, ˜yi+s) which involves no modulo operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Indeed, consider the new objective function and for all s ∈ [n] we define the vector z(s) ∈ Rn such that z(s)i = xi − y(i+s) mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we obtain that for the new objective function it holds that: min c,s n−1 max i=0 d−(xi + c, ˜yi+s) = min c,s n−1 max i=0 ��xi + c − y(i+s) mod n) �� = min s min c max i |z(s)i + c| = min s 1 2 � max i {z(s)i} − min i {z(s)i} � , where in the first step we used the definition of d−(·, ·) and that ˜yk = yk mod n for all k ∈ [2n], in the second step we substituted the definition of z(s)i and in the third step we applied Lemma 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The above implies that we need to compute the quantities maxi{z(s)i} and mini{z(s)i} effi- ciently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Even more, consider the vector v ∈ Rn with entries vs = 1 2 (maxi{z(s)i} − mini{z(s)i}) and observe that the calculation above shows that the optimal objective function value is the same as the smallest entry in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, in the following we show that we can compute v efficiently using the vectors a and b that we computed in our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recall the definitions of the two vectors x′ and y′: x′ = ⟨x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , xn−1, ∞, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , ∞ � �� � n times ⟩, y′ = ⟨yn−1, yn−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , y0, yn−1, yn−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , y0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we let a ∈ R2n be the vector resulting from the (min, −)-convolution of x′ and y′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', ak = mini{x′ i −yk−i} for all k ∈ [2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we observe that for each entry an+s′ with s′ ∈ [n], it holds that an+s′ = n+s′ min i=0 {x′ i − y′ n+s′−i} = n−1 min i=0 {xi − y(i−s′−1) mod n}, where in the second step we used that x′ i = ∞ for i ≥ n and that y′ n+s′−i = y((n−1)−(n+s′−i)) mod n = y(i−s′−1) mod n since in y′ we concatenated the entries of y twice but in reverse order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now observe that if s′ = n − 1 − s then a2n−s−1 = an+s′ = n−1 min i=0 {xi − y(i−s′−1) mod n} = n−1 min i=0 {xi − y(i+s) mod n} = n−1 min i=0 {z(s)i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 68 Dynamic Maintenance of Monotone Dynamic Programs and Applications Next, we define the vector x′′ such that: x′′ = ⟨x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , xn−1, −∞, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , −∞ � �� � n times ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We let b denote the vector resulting from the (max, −)-convolution of x′′ and y′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', bk = maxi{x′′ i − y′ k−i} for all k ∈ [2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now a similar argument as above shows that b2n−s−1 = maxn−1 i=0 {z(s)i} for all s ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, for each entry bn+s′ with s′ ∈ [n] it holds that bn+s′ = n+s′ max i=0 {x′ i − y′ n+s′−i} = n−1 max i=0 {xi − y(i−s′−1) mod n}, where we used that x′′ i = −∞ for i ≥ n and the same argument relating the entries of y′ and y as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, if s′ = n − 1 − s then b2n−s−1 = bn+s′ = n−1 max i=0 {xi − y(i−s′−1) mod n} = n−1 max i=0 {xi − y(i+s) mod n} = n−1 max i=0 {z(s)i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Combining the results above we get that vs = 1 2(b2n−s−1 −a2n−s−1) for all s ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' There- fore, we get that the optimal objective function value is given by mins vs = mins 1 2(b2n−s−1 − a2n−s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In other words, to compute the optimal objective function value it suffices to compute the difference 1 2(b−a) and then to return the smallest entry in v with index between n and 2n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Step 2: Approximation Guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We argue that the algorithm returns an additive ϵ-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, observe that in the algorithm all computations are performed exactly except for the rounding at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the rounding process, we decrease each entry by at most δ = ϵ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, the triangle inequality implies that when we match bead xi to bead yi+s, the error that was introduced by the approximation is at most 2δ = ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since in the objective function we are only interested in the maximum error over all matched beads, this implies that we obtain an additive ϵ-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Step 3: Running Time Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' It is left to analyze the running time of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Iterating over the input vectors x and y, rounding the entries and computing the list representation of x and y can be done in time O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recall that x and y have O(1/δ) pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, we can also compute the vectors x′, x′′ and y in time O(1/δ log 1/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then Lemmas 46 and 40 imply that we can compute the (min, −)-convolution and the (min, +)-convolutions in time O(1/δ2 log(1/δ)) and the resulting vectors have O(1/δ2) pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, the vector v can be computed in time O(1/δ2 log(1/δ)) and the minimum that we return can be found by simply iterating over the pieces of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since previously we have set δ = ϵ/2, this finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 The Dynamic Algorithm We now give our extension to the dynamic setting of the ℓ∞-necklace alignment problem in which there are insertions and deletions from x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ▶ Proposition 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' There exists a fully dynamic algorithm for the ℓ∞-necklace align- ment problem that maintains a solution with additive error ϵ with update time O(1/ϵ2 log(1/ϵ)) and preprocessing time O(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' the space usage of the algorithm is O(1/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the preprocessing, we initialize x and y as empty vectors and store them as piecewise constant functions (as per Section 2) and we do not store them explicitly as vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, we set δ = ϵ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' These operations can be done in time O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 69 Next, consider an operation Insert(i, α, β) which asks to insert α into x at the i’th position and to insert β into y at the i’th position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since we are in the approximate setting, instead of inserting the exact values of α and β, we insert ⌊α⌋∗ δ into the i’th position of x and ⌊β⌋∗ δ into the i’th position of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We perform these insertions by manipulating the list representations of x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We only describe how to perform the manipulations for x, as for y they are essentially the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Denote the list representation of x as (X0, Y0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , (Xp, Yp) where p is the number of pieces of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we iterate over all pieces of x and check whether there exists a piece with value Yj = ⌊α⌋∗ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If no such piece exists, we insert (i, ⌊α⌋∗ δ) into the list representation at the appropriate position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we find the smallest integer j such that Yj > ⌊α⌋∗ δ and for all k ≥ j, we increment Xk by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Intuitively, we are moving all pieces that are larger than ⌊α⌋∗ δ one unit to the right in order to make space for the element that was just inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Once we have updated x and y as described above, we simply run the static algorithm without the step in which we initialize x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that, since we assume that after each insertion x and y are still ordered and since we only insert rounded entries into x and y, we get that x and y never have have more than O(1/δ) pieces by Lemma 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now, since above we have set δ = ϵ/2, the proof of Proposition 45 implies that we obtain a solution with additive error ϵ in time O(1/ϵ2 log(1/ϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, note that since we do not store x and y explicitly (we only store their rounded version represented by their list representations), the space usage is O(1/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, we note that the operation Delete(i) can be implemented similar to above by first manipulating the list representations of x and y to remove the i’th entries from x and y and then running the static algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ◀ We remark that by storing two dynamic vectors x and y that are undergoing element insertions and deletions as described in the proof of Proposition 48, we can also efficiently maintain an approximation of their (min, +)-convolution x ⊕ y via Lemma 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' H Omitted Proofs H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='1 Proof of Lemma 6 Denote the list representations of g and h as (xg 1, yg 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , (xg pg, yg pg) and (xh 1, yh 1 ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , (xh ph, yh ph), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recall that both list representation are stored in doubly linked lists and that the pieces of g and h are stored in a binary search tree such that for all x ∈ [0, t] we can evaluate g(x) and f(x) in time O(log pg) and O(log ph), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We show how to construct each of the functions fmin, fshift, fadd and fround by showing how to construct their list representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, let us consider fmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We construct the list representation (xmin 1 , ymin 1 ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' of fmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The intuition of our approach is that each piece of fmin must start and end at one of the start or end points of the pieces of g and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we will evaluate the function min{g, h} at all points xg i and xh j and set fmin accordingly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' then if fmin contains multiple pieces with the same ymin i value, we will remove these duplicate pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, we consider the set X = {xg 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , xg pg, xh 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , xh ph} and order it from small to large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we set xmin i to the i’th smallest element in X for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , pg +ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that on the interval [xmin i−1, xmin i ), fmin must take the value min{g(xmin i−1), h(xmin i−1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, we set ymin i = min{g(xmin i−1), h(xmin i−1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This gives an initial list representation of f min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we “prune” the list representation of fmin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', we iterate over all pairs (xmin i , ymin i ) in increasing order of i and if ymin i−1 = ymin i then we remove the pair (xmin i−1, ymin i−1) from the list representation of fmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that at 70 Dynamic Maintenance of Monotone Dynamic Programs and Applications the end of this process, all values of ymin i are pairwise disjoint (since the functions g and h are monotone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To see that fmin(x) = min{g(x), h(x)} for all x ∈ [0, t], we observe that for all x ∈ X (where X is as in the paragraph above) we have set fmin(x) correctly by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, on all contiguous intervals in [0, t] \\ X, g and h are constant and thus fmin is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, for all x ∈ [0, t] \\ X, fmin(x) is also set correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, we observe that fmin has at most pg+ph pieces because X consisted of at most pg+ph elements and after that we only removed pieces from fmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, ordering the elements in X can be done in time O((pg + ph) log(pg + ph)) and evaluating min{g(xmin i ), h(xmin i )} can be done in time O(log(pg) + log(ph)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' After that we only performed a single pass over the list representations of fmin in time O(|X|) = O(pg + ph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, it took time O((pg + ph) log(pg + ph)) to create the list representation of fmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, note that to store the elements xmin i in the binary search tree, we need additional time O((pg +ph) log(pg +ph)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now we observe that fadd can be computed similarly to fmin: the function fadd only changes its functions values at the points in X (where X is as above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, we let xadd i be the i’th smallest element in X and set yadd i = g(xadd i−1) + h(xadd i−1), followed by the same pruning step as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The rest of the proof goes through as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, consider fshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We construct the list representation (xshift 1 , yshift 1 ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , (xshift pg , xshift pg ) of fshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , pg, we set xshift i = xg i + c and yshift i = yg i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The correctness is straightforward and from the construction it is evident that there are only pg pieces and that everything can be done in time O(pg log(pg)) (since we still need to construct the binary tree for the pieces of fshift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, us consider fround.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' As before, we construct the list representation of fround, (xround 1 , yround 1 ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , (xround pg , xround pg ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , pg, we set xround i = xg i and yround i = ⌈yg i ⌉1+δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' After that, we perform the same pruning step as in the construction of fmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since g takes values in W∞ = {0} ∪ [1, W] ∪ {+∞} and g is monotone, fround can take at most 2 + ⌈log1+δ(W)⌉ different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Again, the running time bound stems from the fact that we have to construct the binary search tree for the pieces of fround.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='2 Proof of Lemma 7 Let (xs 1, ys 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , (xs ps, ys ps) be the list representation of fs for s = 1, 2, where ps ≤ p is the number of pieces of fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We create pairs (y1 i , y2 j ) for all (i, j) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , p1} × {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , p2}, and order them such that y1 i + y2 j becomes monotonically increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We iterate over all pairs in this order, and in each iteration we set the function value f(x) for some x-values to y := y1 i + y2 j , where (y1 i , y2 j ) is the pair considered during the iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Here, we start with large x-values (at which f takes the smallest values) and keep on decreasing x (and the function values increase);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' in other words, we construct f on its domain [0, t] from right to left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' More concretely, let xmax denote the highest x-value for which we did not yet set a function value (or −∞ if all function values have been set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let x′ = x1 i−1 + x2 j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We set the function values for all x ∈ [x′, xmax) to y and then set xmax = x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For each such new piece of f, we store that we combined the indices i and j of the pieces that we used from f1 and from f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we proceed with next iteration until all function values have been set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The following two statements show that this procedure is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Each x is assigned a function value that is at most the correct value f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To see this let x ∈ [0, t] and recall that f(x) = min ¯x∈[0,x] f1(¯x) + f2(x − ¯x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 71 Let ¯x∗ denote the value of ¯x that attains the minimum in the above expression, and let i∗ and j∗ denote the indices of the pieces that ¯x∗ and x − ¯x∗ fall into, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' the list representations of f1 and f2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This means ¯x∗ ∈ [x1 i∗−1, x1 i∗) and x − ¯x∗ ∈ [x2 j∗−1, x2 j∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, x ≥ x′ = x1 i∗−1 +x2 j∗−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, either in the iteration for the pair (y1 i∗, y2 j∗) or before, the procedure assigns a function value to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Because the procedure assigns function-values in monotonically increasing fashion we are guaranteed that the function value that is assigned is at most the correct value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The function value y that is assigned is at least the correct value f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Suppose that during some iteration we assign the function value y = y1 i + y2 j to x ∈ [x′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , xmax), where x′ = x1 i−1 + x2 j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We have f(x) = min ¯x∈[0,x] f1(¯x) + f2(x − ¯x) (definition) ≤ f1(x1 i−1) + f2(x − x1 i−1) (consider ¯x = x1 i−1) ≤ f1(x1 i−1) + f2(x′ − x1 i−1) (x′ ≤ x, f2 monotonically decreasing) = f1(x1 i−1) + f2(x2 j−1) = y1 i + y2 j (y1 i = f1(x1 i−1) and y2 j = f(x2 j−1)) = y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, the assigned value is at least f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Observe that we can implement the above procedure in time O(p2 log p): We first sort the at most p2 pairs in time O(p2 log p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then every iteration can be executed in constant time because setting the function values for x ∈ [x′, xmax) to y can be performed by adding the pair (xmax, y) to the list-representation of f and updating xmax to x′ takes time O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Finally, suppose we already computed f and, given x ∈ [0, t], we shall return a value ¯x∗ ∈ [0, t] such that f(x) = f1(¯x∗) + f2(x − ¯x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' First, let (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , (xp, yp) denote the list representation of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then we can determine the piece ℓ of f such that x ∈ [xℓ, xℓ+1) in time O(log p) since we store the pairs (xi, yi) of f in a binary search tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Recall that for each piece of f, we stored the indices i and j of the pieces from f1 and f2 that we combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now observe that we have ¯x∗ ∈ [x1 i−1, x1 i ) and x − ¯x∗ ∈ [x2 j−1, x2 j), where i and j are such that these pieces from f1 and f2 form the corresponding piece of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, to find ¯x∗ we can first try to set ¯x∗ = x1 i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If x − ¯x∗ = x − x1 i−1 ∈ [x2 j−1, x2 j) then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Otherwise, we must have that x − x1 i−1 ≥ x2 j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we have to increase the value of ¯x∗ from x1 i−1 until it is large enough such that x − ¯x∗ ∈ [x2 j−1, x2 j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This can be achieved by setting ∆ = (x − x1 x−1) − x2 j and ¯x∗ = x1 i−1 + ∆ + 1 2 min{x1 i − (x1 i−1 + ∆), x2 j − x2 j−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that this value of ¯x∗ can be computed in time O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, the total time to return ¯x∗ is O(log p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='3 Proof of Theorem 9 Recall that the dependency graph is a DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We call a vertex without any incoming edges a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The level of a vertex u is the length of the longest path from a leaf to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that since each node can only reach h other nodes, every vertex has level at most h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We compute the DP bottom-up, starting at the leaves of the DAG and then recursively computing the solutions for rows i for which the solutions of In(i) have already been computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We store the approximate solutions ADP(i, ·) using monotone piecewise constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We prove the theorem by induction over the level of i in the dependency graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We show the stronger statement that for every DP row i of level ℓ, ADP(i, ·) is an αℓ+1-approximation of DP(i, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 72 Dynamic Maintenance of Monotone Dynamic Programs and Applications We start with leaf vertices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', vertices of level 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For a leaf i, we use Properties 4(b) and 4(c) to obtain that ˜Pi returns ADP(i, ·) which is a monotonone piecewiese constant function with at most p pieces and which is an α-approximation of DP(i, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Next, consider a row i of level ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We use ˜Pi to compute ADP(i, ·) = ˜Pi({ADP(i′, ·): i′ ∈ In(i)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By induction hypothesis, all solutions ADP(i′, ·), i′ ∈ In(i), are stored as monotone piecewise constant functions and each of them has at most p pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since we apply the operations from Lemma 6 only O(1) times, the number of pieces only grows by a factor O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since we only apply the (min, +)-convolution from Lemma 7 at most a single time, the number of pieces after the convolution is bounded by O(p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Thus, we will never operate on functions with more than O(p2) pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The bounds from Lemmas 6 and 7 imply that all operations to compute ˜Pi can be performed in time at most O(p2 log(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, by induction hypothesis and since each i′ is at level ℓ′ ≤ ℓ − 1, we know that ADP(i′, ·) is an αℓ-approximation of DP(i′, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Using Properties (3) and 4(a), we get that ADP(i, ·) is an αℓ+1-approximation of DP(i, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The theorem’s approximation guarantee follows from Property (2) which implies that ℓ ≤ h for all DP rows i in the dependency graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Furthermore, above we argued that each solution ADP(i, ·) can be computed in time O(p2 log(p)) which gives a total running time of O(|I| · p2 log(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='4 Proof of Theorem 10 Consider a row i for which DP(i, ·) changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Note that we only have to compute DP solutions for rows i′ which are reachable from i in the dependency graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since we assume that the dependency graph is a DAG and Reach(i) ≤ h for all rows i, there can be at most h such rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In the proof of Theorem 9 we argued that each solution ADP(i, ·) can be computed in time O(p2 log(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This gives the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='5 Property of the Räcke Tree Let G = (VG, EG) be an undirected graph and let T = (VT , ET ) be a Räcke tree for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We prove that mincutT (A, B) ≥ mincutG(A, B) by showing that for any set of vertices ST ⊆ VT , it holds that capT (ST ) ≥ capG(S) where S ⊆ VG is the set of leaf vertices in VT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let ST ⊆ VT and consider the cut (ST , ¯ST ) in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We use S to denote the restriction of ST to the leaf vertices and observe that (S, ¯S) forms a cut in G as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then: capT (ST ) = � (xt,yt)∈ST × ¯ST capT (xt, yt) (definition of capT (ST )) = � (xt,yt)∈ST × ¯ST capG(Vxt ∩ Vyt) (definition of tree edge capacity) = � (xt,yt)∈ST × ¯ST � (x,y)∈Vxt× ¯Vxt capG(x, y) (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' assume Vxt ⊆ Vyt) = � {x,y}∈EG capG(x, y) � (xt,yt)∈ST × ¯ST 1{x ∈ Vxt ∧ y ∈ ¯Vxt} (change order of summation) ≥ � (x,y)∈S× ¯S capG(x, y) = capG(S) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Here the inequality follows because a pair (x, y) ∈ S × ¯S whose capacity is counted on the right hand side corresponds to a graph edge {x, y} ∈ EG (between x ∈ S and y ∈ ¯S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' This M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Henzinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Räcke 73 graph edge contributes to the capacity on every edge of the x-y path in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' One of these edges must be cut by ST , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', 1{x ∈ Vxt ∧y ∈ ¯Vxt} = 1 for this tree edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Hence, its capacity is also counted on the left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='6 Proof of Lemma 18 The lower bound is immediate since ˜Pi is an α-approximation of Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For the upper bound we use induction over ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' For ℓ = 0 observe that ADP(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ·) = ˜Pi({ADP(i′): i′ ∈ In(i)}) (definition) ≤ αPi({ADP(i′): i′ ∈ In(i)}) ( ˜Pi is α-approximate) = αPi(∅) (vi is a leaf) = αDP(i) (the DP is okay-behaved) For ℓ > 0 we have ADP(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ·) = ˜Pi({ADP(i′): i′ ∈ In(i)}) (definition) ≤ αPi({ADP(i′): i′ ∈ In(i)}) ( ˜Pi is α-approximate) = αPi({αℓDP(i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ·): i′ ∈ In(i)}) (induction hypothesis) = αℓ+1DP(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' ·) (the DP is okay-behaved) Here the induction hypothesis exploits the fact that all i′ ∈ In(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' have level strictly less than ℓ in the dependency graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='7 Proof of Lemma 19 The claim about the approximation guarantee follows immediately from Lemma 18 and the fact that the root has level at most h (since the longest leaf-root path in the depen- dency tree has length h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' To obtain running time O(|VT | · t), we compute the solutions ADP(v1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' , ADP(vn) in this order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', based on the topological ordering of the dependency DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Then by assumption on the ordering of the rows i and since all ˜Pi can be computed in time t, the lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='8 Proof of Lemma 20 Suppose the inserted or deleted edge is incident upon a vertex i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Since the DPs we consider are well-behaved, we only need to recompute DP solutions for those vertices j such that there exists a directed path from i to j, j ≥ i, in the dependency graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By construction of the dependency graph, there can be at most h such vertices (since the longest leaf-root path in the dependency graph has length h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Therefore, we can recompute all of these solutions in time O(h · t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' After we finished the recomputation, the guarantees on the approximation ratio are implied by Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='9 Proof of Lemma 21 We only prove the case if all functions fi are monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The case for monotonically increasing functions is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let P denote the set of all pieces in functions fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Consider a piece p ∈ P that starts at t1 ends at t2 and has value α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We construction a piece-wise constant function fp : [0, t] → W∞ with two pieces that has value ∞ on [0, t1), and value α on [t1, t] (this means we extend the piece from t2 to t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' 74 Dynamic Maintenance of Monotone Dynamic Programs and Applications Because the functions are monotonically decreasing we can rewrite fmin as a minimum of the piece-functions fp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', fmin(x) = min p∈P fp(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' We now sort all pieces in P by there start-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' By processing the pieces in sorting order we can build the result function step-by-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Let fr−1 denote the piece-wise constant function encoding the minimum over the first r −1 pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' In order to compute fr we have to compare the last piece of fr−1 to the r-th piece pr in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' If the value of pr is higher than fr−1(∞) (the value of the last piece in fr−1) we ignore the piece pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Otherwise, we end the current last piece of fr−1 at the start time tr of piece pr and add the piece pr with its start time, its value, and an end time of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' The running time is dominated by sorting the pieces and inserting them into a binary search tree when adding them to the result function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='10 Proof of Lemma 22 We can assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' that f2 is monotonically decreasing (this follows from the symmetry of (min, +)-convolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=' Now the lemma is implied by the following computation, where in the third step we use the monotonicity of f2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} +page_content=', we use that f2(x′) ≤ f2(x) for all x′ ≥ x: f(x′) = min ¯x∈[0,x′] f1(¯x) + f2(x′ − ¯x) ≤ min ¯x∈[0,x] f1(¯x) + f2(x′ − ¯x) ≤ min ¯x∈[0,x] f1(¯x) + f2(x − ¯x) = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf'} diff --git a/KdE2T4oBgHgl3EQfAgZ0/content/tmp_files/2301.03592v1.pdf.txt b/KdE2T4oBgHgl3EQfAgZ0/content/tmp_files/2301.03592v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e039f9585be5a1a1457dbe0a04d757cc56371a45 --- /dev/null +++ b/KdE2T4oBgHgl3EQfAgZ0/content/tmp_files/2301.03592v1.pdf.txt @@ -0,0 +1,1427 @@ +EFFICIENT INTRA-RACK RESOURCE DISAGGREGATION FOR +HPC USING CO-PACKAGED DWDM PHOTONICS +A PREPRINT +George Michelogiannakis +Lawrence Berkeley National Laboratory +mihelog@lbl.gov +Yehia Arafa +Qualcomm Technologies, Inc +yarafa@nmsu.edu +Brandon Cook +Lawrence Berkeley National Laboratory +BGCook@lbl.gov +Liang Yuan Dai +Columbia University +ld2719@columbia.edu +Abdel-Hameed Badawy +New Mexico State University +badawy@nmsu.edu +Madeleine Glick +Columbia University +msg144@columbia.edu +Yuyang Wang +Columbia University +yw3831@columbia.edu +Keren Bergman +Columbia University +bergman@ee.columbia.edu +John Shalf +Lawrence Berkeley National Laboratory +jshalf@lbl.gov +January 11, 2023 +ABSTRACT +The diversity of workload requirements and increasing hardware heterogeneity in emerging high +performance computing (HPC) systems motivate resource disaggregation. Disaggregation separates +servers into their constituent compute and memory resources so that they can be allocated as required +to each workload. Previous work has shown the potential of intra-rack resource disaggregation, +but it is not clear how to realize these gains and cost-effectively meet the stringent bandwidth +and latency requirements of HPC applications. To that end, we describe how modern photonic +components can be co-designed with modern HPC racks to implement flexible intra-rack resource +disaggregation and fully meet the high escape bandwidth of contemporary multi chip module (MCM) +packages and all chip types in modern HPC racks with negligible power overhead. We show how to +use distributed indirect routing to meet these demands without the need for significant complexity +for reconfiguration that spatial optical switches require. We then show that intra-rack resource +disaggregation implemented using emerging photonics and parallel optical wavelength-selective +switches satisfies bit error rate (BER) and bandwidth constraints and provides an average application +speedup of 23.9% for 31 out-of-order CPU and 61.5% for 27 GPU benchmarks compared to a similar +system that instead uses modern electronic switches for disaggregation, due to their higher latency. +Using observed resource usage from a production system, we estimate that an iso-performance +intra-rack disaggregated HPC system using photonics would require 4× fewer memory modules and +2× fewer NICs than a non-disaggregated baseline. +Keywords Photonics · disaggregation · AWGR · spatial +arXiv:2301.03592v1 [cs.DC] 9 Jan 2023 + +arXiv Template +A PREPRINT +1 +Introduction +Leading high performance computing (HPC) systems are steadily embracing heterogeneity of compute and memory +resources as a means to preserve performance scaling and reduce system power Liu et al. [2012], Top [2018], Ujaldón +[2016]. This trend is already apparent with the integration of GPUs Mittal and Vetter [2015], Tiwari et al. [2015], +Gao and Zhang [2016] and is expected to continue with fixed-function or reconfigurable accelerators such as field +programmable gate arrays (FPGAs), Milojicic [2020], Asaadi and Chapman [2017], Segal et al. [2014], Hogervorst +et al. [2021], Lant et al. [2020], Dimond et al. [2011], Ramirez-Gargallo et al. [2019], emerging customized accelerators, +and heterogeneous memory Venkata et al. [2017]. In addition, key HPC workloads show considerable diversity in +computational and memory access patterns Michelogiannakis et al. [2022], Rodrigo et al. [2016]. +This expectation of resource heterogeneity, workload diversity, and today’s method of allocating resources to applications +in units of statically-configured nodes where every node is identical and unused resources are left to idle, raises the +concern of resource underutilization (referred to as “marooned resources”). These marooned resources increase both +capital and operational costs without improving performance. This has led to the emergence of resource disaggregation. +Disaggregation refers to decomposing servers into their constituent compute and memory resources so that these can be +allocated as required according to the needs of each workload. Hyperscale datacenters have readily embraced resource +disaggregation and have demonstrated that it significantly improves utilization of GPUs and memory Guleria et al. +[2019a], Peng et al. [2020], Taylor [2015], Li et al. [2022], Koh et al. [2019], Papaioannou et al. [2016], Gonzalez et al. +[2020], Cheng et al. [2019a], Guleria et al. [2019b]. +Although file storage is routinely disaggregated in modern systems Per, Michelogiannakis et al. [2022], Petersen and +Bent [2017], HPC has been slow to embrace disaggregation of compute and memory resources Glick et al. [2020], Guo +et al. [2021] due to the sensitivity of HPC workloads to bandwidth and latency that cannot be met by current PCIe/CXL +or Ethernet link technologies used in contemporary disaggregated architectures. Studies showed that disaggregation only +among resources in the same rack (i.e., intra-rack resource disaggregation) in HPC could reduce resources by 5.36% to +69.01% while avoiding the overhead of full-system disaggregation Michelogiannakis et al. [2022], but the impact of +increased memory latency and specific architectural trade offs have not been explored. Thus, although disaggregation +using electronic networks have been demonstrated in hyperscale datacenters Lin et al. [2020], Papaioannou et al. [2016], +Call et al. [2020], minimizing adverse effects to and addressing the stringent bandwidth density and latency demands of +HPC workloads requires a thorough investigation. +In this work, our contributions are as follows. Firstly, we describe how to use emerging photonic links and switches to +design modern and practical resource-disaggregated HPC racks based on an existing GPU-accelerated HPE/Cray EX +supercomputer Per. Secondly, we show how state of the art commercially available photonics and advanced packaging +multi chip modules (MCMs) meet bit error rate (BER) requirements, impose a negligible power overhead, and deliver +sufficient bandwidth to satisfy the escape bandwidth of all chips in modern HPC racks. Thirdly, we show how to use +distributed indirect routing and arrayed waveguide grating routers (AWGRs) Liu et al. [2020], Zhang et al. [2019] to +satisfy all bandwidth requirements without the overhead and latency for reconfiguration that spatial Seok et al. [2019a], +Ding et al. [2016] and wave-selective Huang et al. [2020] switches require. Moreover, having demonstrated negligible +adverse impact to all other metrics, we show that intra-rack disaggregation using emerging photonics provides an +average application speedup of 23.9% for 31 out-of-order (OOO) CPU and 61.5% for 27 GPU benchmarks compared to +a similar system that instead uses state of the art electronic switches, which also increase power overhead by three orders +of magnitude. Finally, based on observed resource usage, we estimate that a system based on racks using photonics +for resource disaggregation can have 43% fewer overall chips compared to a non-disaggregated system with the same +computational throughput. +2 +Related Work +Hyperscale datacenters predominantly focus on full-system resource disaggregation where applications can allocate +fine-grain resources of different types, today typically graphical processing units (GPUs) Guleria et al. [2019a] and +memory Peng et al. [2020], Lim et al. [2009], Koh et al. [2019], Gonzalez et al. [2020], which are located anywhere in +the system or within a group of racks Lin et al. [2020], Papaioannou et al. [2016], Call et al. [2020]. In such a system, +resources of the same type are typically placed in the same rack. +However, full-system, flexible, and fine-grain disaggregation introduces significant overhead because of the higher +latency and lower bandwidth density of contemporary hardware that is used to implement resource disaggregation – +typically PCIe, 100Gig Ethernet, and eventually compute express link (CXL) Van Doren [2019] over electronic links. +This overhead does not simply increase power and procurement cost, but rather adds potentially substantial latency +between key resources such as central processing units (CPUs) and memory that traditionally exhibit latency-sensitive +2 + +arXiv Template +A PREPRINT +communication. The aforementioned studies quote a several orders of magnitude increase in network and memory +latency due to full-system resource disaggregation to improve resource utilization by 35% at most Zervas et al. [2018]. +Another study found that application performance degradation depends on both network bandwidth and latency, but +can still reach up to 40% even with high bandwidth, low-latency networks Gao et al. [2016]. Work on SPEC and +commercial benchmarks also found an up to 27% application slowdown due to the additional memory latency Abali +et al. [2015]. A study on Microsoft’s Azure found a range of performance slowdowns up to 30% from an extra 65 ns to +access main memory Li et al. [2022]. Software defined networks (SDNs) based on electrical networks fare no better in +terms of overhead Gao et al. [2016], Han et al. [2013], Call et al. [2020]. +Hybrid full-system photonic–electronic approaches have also been proposed that rely on circuit switching Zervas et al. +[2018] for reconfiguration. As a result, a few studies call for intra-rack disaggregation in datacenters Taylor [2015], +Lim et al. [2009], Guleria et al. [2019b]. Even the low latency and high bandwidth density of modern photonics cannot +fully satisfy the bandwidth, energy, and latency requirements of full system disaggregation. This makes system-wide +disaggregation impractical in many cases Lin et al. [2020], Zervas et al. [2018], Cheng et al. [2019a], Cheng et al. +[2018]. +Recent full system approaches in high performance computing (HPC) rely on optics to connect CPUs and memory, and +electronic switches for hard disk drives (HDDs) to increase resource CPU utilization by 36.6% and memory 21.5% Guo +et al. [2021]. In contrast, another study confirms that production HPC systems can reduce resources from 5.36% +to 69.01% with intra-rack disaggregation and still satisfy the worst-case average rack utilization Michelogiannakis +et al. [2022]. Similar to datacenters, intra-rack disaggregation in HPC promises the lowest overhead and impact to +applications Glick et al. [2020], Taylor [2015], Guleria et al. [2019b]. +Related work has research other aspects necessary to make resource disaggregation practical in a system, such as +job scheduling Fan et al. [2019], Agosta et al. [2018], Amaral et al. [2021], Domeniconi et al. [2019], how the +operating system (OS) and runtime should adapt Maccabe [2017], Hwu et al. [2015], Shan et al. [2018], programming +of code portability in heterogeneous systems Gioiosa et al. [2020], Agosta et al. [2018], partitioning of application +data Khaleghzadeh et al. [2020], fault tolerance Hussain [2020], how to fairly compare the performance of different het- +erogeneous systems Jamieson et al. [2018], and the impact of heterogeneous resources to application performance Tang +et al. [2017], Lastovetsky [2015], Venkata et al. [2017]. These are important but out of scope topics for our study. +2.1 +Under-utilization in Production Systems +We use NERSC’s Cori system as an exemplar production HPC system, while recognizing workload requirements +on other systems may differ. In NERSC’s Cori, before Perlmutter came online and thus Cori was runnign the full +NERSC workload, three quarters of the time Haswell nodes use less than 17.4% of memory capacity (50.1% for +KNL nodes) and less than 0.46 GB/s of memory bandwidth Michelogiannakis et al. [2022]. These observations are +similar to observations collected on LANL clusters Peng et al. [2020] and Alibaba machines that execute batch jobs. +Likewise, half of the time Cori nodes use no more than half of their compute cores and three quarters of the time 1.25% +of available network interface controller (NIC) bandwidth. Similarly, in Lawrence Livermore National Laboratory +(LANL) clusters, approximately 75% of the time, no more than 20% of memory capacity is used Peng et al. [2020]. +Alibaba’s published data Guo et al. [2019] show that memory is underutilized similarly to Cori for machines that +execute batch jobs. Data from Google systems shows that memory and disk capacity of tasks is spread over three orders +of magnitude and typically underutilized Han et al. [2013]. Azure reports 25% of memory under-utilization Li et al. +[2022]. Datacenters have also reported 28% to 55% CPU idle in the case of Google trace data Patel et al. [2015] and +20% to 50% most of the time in Alibaba Guo et al. [2019]. Early studies also suggest GPU under-utilization Li et al. +[2015], Jeon et al. [2019], Li et al. [2011]. +3 +Photonics for Resource Disaggregation +Here we walk through available optical link and switch technologies and argue that photonics today meet the strict +performance and error rate requirements to efficiently implement intra-rack resource disaggregation in HPC. +3.1 +Memory Technologies and Requirements +IO systems in HPC are already largely disaggregated over conventional system-scale interconnects since the underlying +technologies (disk or SSD) are relatively high latency and lower bandwidth Michelogiannakis et al. [2022], Terzenidis +et al. [2018]. By contrast, memory technologies (particularly high bandwidth memorys (HBMs) needed by GPUs) are +much higher bandwidth and much less tolerant of latency and require much lower bit error rates (BERs). Given that +memory disaggregation imposes the most challenging constraints among other resources in today’s compute nodes, we +3 + +arXiv Template +A PREPRINT +Figure 1: Logical schematic of a DWDM link using ring resonator technology and a comb-laser source. Each ring is +tuned to a different frequency of light and can be used to modulate that specific wavelength of light (a channel). Comb +laser sources provide a comb of frequencies of light to provide those wavelengths for encoding. All of the encoded +optical channels share the same optical fiber and are decoded using the rings on the receiving side to route channels to +the photodetectors. +Aggregated comb laser sources +Active photonic interposer +(a) Active Optical MCM +(b) Blade +Optical Circuit Switches +Optical Fiber +(c) Rack/Pod +Figure 2: Overall physical structure of Rack/Pod scale resource disaggregation from photonically connected MCMs +to the Rack/Pod scale pooling of disaggregated resources. The conversion from CXL-over-fiber to HBM or NVM +electrical protocol is implemented in the active interposer for the photonics MCM. +will use DDR and HBM memory technology to set our performance target. A typical DDR4 memory has a response +latency of approximately 90 ns and for HBM the average response latency is 90-140 ns Wang et al. [2020]. Still, any +added latency between the CPU and memory from resource disaggregation may penalize application performance as +we quantify later. Server-class memories typically require BERs of less than 10−18 to achieve tolerable failures in +time (FIT) rates with conventional single-error-correct/double-error-detect (SEC-DED) protection Meza et al. [2015], +Sridharan et al. [2015]. Forward error correction (FEC) can reduce the BER, but with additional latency Luyi et al. +[2012]. +3.2 +Optical Link Technologies +We consider a range of photonic link technologies that include conventional 100 Gbps Ethernet physical interfaces that +represent the current baseline link technology for memory disaggregation. We also introduce a range of cutting-edge +dense wavelength division multiplexing (DWDM) link technologies that are either demonstrated as research prototypes +or are commercially available. The photonic components all come from existing commercial technologies (100 Gbps, +400 Gbps, Ayar TeraPhy) and some research prototypes from DARPA PIPES (the 1-2 Tb link technologies). These +higher performance link technologies must be co-packaged in order to achieve their bandwidth density. These link +technologies are summarized in Table 1. The technology for the optical links is depicted in Figure 1. Delivering +multiple channels of laser light to the package has been challenging to scale cost-effectively if each "color" of light +were to require a separate laser source. This concern was alleviated by the emergence of quantum dot and soliton comb +laser sources shown in Figure 4 that can produce hundreds of usable light frequencies with wall-plug efficiencies of up +to 41% Kim et al. [2019a]. +4 + +7NNN00NNN:.MMCustomAccelGPuCPUDisaggregatedresourceRack/PodarXiv Template +A PREPRINT +Figure 3: Copackaged optics are required for DWDM link technologies to achieve the kind of bandwidth density +required to operate at native memory bandwidths. +3.3 +Active Photonic MCMs +Many CPUs and GPUs do not have the necessary off-chip bandwidth for full utilization of their compute resources +because operating their I/O pins at higher bandwidth incurs a power cost Chen et al. [2017], Jouppi et al. [2017]. +Using emerging high-speed optical links directly to the MCM, illustrated in Figure 3 provides to the order of 10× +gains in escape bandwidth Glick et al. [2020], Wade [2019], Bergman et al. [2018], Maniotis et al. [2021]. This is +a necessary property to enable efficient resource disaggregation as well as handle changing bandwidth requirements +of key applications such as machine learning that drastically shifts bandwidth between inter-GPUs and off-chip from +inference to training. +MCMs with integrated photonics have been demonstrated in both 2.5D and 3D interposer platforms Glick et al. [2020], +Minkenberg et al. [2021], Sutono et al. [1998], Abrams et al. [2020]. They can use different die-to-die link standards +such as UCIe. Active interposer platforms combine the photonic integrated circuit (PIC) and interposer into a single +integrated substrate. The active interposer allows photonic components to be fabricated and directly integrated with +through silicon vias (TSVs) and additional metal redistribution layers. Electronic circuits are flip-chipped on top of +active interposers using copper pillars Dittrich et al. [2017]. Further work has embedded photonic switch fabrics within +MCM platforms with a crosstalk suppression and extinction ratio of >50dB and on-chip loss as low <1.8dB Glick et al. +5 + +2.5DIntegratedTransceiver +ComputeNode +Optical +Fibers +FCBumps +PIC +FC Bumps +EIC +Interposer +ASIC/FPGA +PCBEIO +PIC +25umpitchpad +array to be bumpec +25umpitchpad +arraytobeb +MCM +C3 +ElectricalloonPcBto +be wirebondedto PiC +EIC +PICarXiv Template +A PREPRINT +Figure 4: Comb laser sources provide the many discrete optical frequencies for the DWDM link with up to 41% +experimentally measured conversion efficiency. +[2020]. This was further scaled up to support more than 100 ports with microring resonators using a scalable switch +fabric that combined switching in the space domain with wavelength-selectivity to define fine-grained connectivity for +node disaggregation Huang et al. [2020], Glick et al. [2020]. +3.3.1 +Link Protocol +We adopt CXL as our link protocol Van Doren [2019]. CXL is an overlay on the PCIe-Gen6 physical layer, it includes +guaranteed ordering of events and is a broadly adopted industry standard with published specifications. However, our +study does not rely on any features of any particular protocol so alternatives such as UCIe also apply. +3.3.2 +Link Propagation and Encoding/Decoding Latency +The target reach for an intra-rack disaggregation solution is approximately 1-4 meters. Given the speed of light c +and light propagating through optical material that has an index of refraction that is near r1.5, the effective latency +of propagating through an optical fiber at nominally 0.75c is approximately 5 ns per meter. Therefore, rack-scale +disaggregation adds 5-15 ns to our latency budget, less than 20% of the typical DRAM latency. The link latency for +SERDES and photonic ring modulation is negligible. Intra-rack fiber lengths up to 4 meters require no intervening +Electrical Optical (OEO) conversions. +3.3.3 +Bit Error Rates and FEC +To achieve 10−18 BER required for memory technologies, FEC Luyi et al. [2012] will likely be required. Using the +lightweight FEC scheme that is proposed for CXL Van Doren [2019] and PCIe Gen6 Sharma [2020] as an example, +the all-inclusive latency for FEC can be as low as 2 ns. Therefore, for 200 Gbps, serialization delay is 10 ns and the +FEC calculations add 2-3 ns. At 400 Gbps and above, the net latency for FEC would be 5 ns plus 2-3 ns. Of note, this +approach to achieving these BER targets is achievable with less than a 0.1% bandwidth loss. +6 + +100um-6 dBm +10 +-10 dBm +S-band +C-band +L-band +0 +Bm) +-10 +NO +-20 +-30 +-40 +1,520 +1,540 +1,560 +1,580 +1,600 +Wavelength (nm)arXiv Template +A PREPRINT +BW +(Gbps) +Energy +(pJ/bit) +Link +Gbps × +Chan- +nels +#Links +(2 +TB/s +es- +cape) +Agg. +Ws (2 +TB/s +es- +cape) +Ref. +100 +30 +25 × 4 +160 +480 +Fathololoumi +et +al. +[2021], +Agrell +et +al. +[2016] +400 +30 +100 × 4 +40 +197 +Wei +et +al. +[2015] +768 +< 1 +32 × 24 +21 +14.4 +Wade +[2019] +1,024 +0.45 +16 × 64 +16 +7.2 +Kim +et +al. +[2019b] +2,048 +0.3 +16 × 128 +8 +4.8 +Kim +et +al. +[2019b] +Table 1: A range of WDM photonic link technologies. +In terms of impact on BER, this PCIe/CXL-like correction scheme corrects all single bursts of up to 16 bits. Double +bursts will likely be mis-corrected, but the chance of a bad flit decreases quadratically (e.g., a flit BER of 10−6 becomes +10−12 as you need two error bursts per flit to fail). Each flit is protected with a strong 64-flit CRC such that the flit FIT +rate (CRC escapes) is significantly less than one part per billion. Lastly, the FEC escapes become link retransmissions +and the ASIC-to-ASIC connection sees close to zero errors. As a result, emerging memory fabric protocols such as +CXL, which could be run over our evaluated physical links, are capable of achieving a BER rate that meets the stringent +memory system requirements and minimizes performance loss due to retransmission. +3.4 +Optical Circuit Switch Technologies +Motivated by minimizing latency, our vision for a disaggregated rack is to have photonically-enabled MCMs that are +connected via an optical circuit switch, as shown in Figure 2. Compute and memory chips would be in the center of the +MCM and the edge of the MCM would contain co-packaged optical silicon in-package photonics (SiPs). Switches with +all-optical paths include spatial- and wave-selective approaches, shown in Table 2. +3.4.1 +Spatial Optical Switches +In recent years, the primary switching cells investigated are microelectromechanical systems (MEMS) actuated +couplers, Mach-Zehnder interferometers (MZIs), and microring resonators (MRRs). Taking after their free-space +counterpart, photonic MEMS-actuated switches are broadband spatial switches that have demonstrated radix scaling up +to 240×240 Seok et al. [2019b]. However, MEMS switching cells generally require high driving voltages (greater than +20 V), which make them less attractive for co-integration with electronic drivers, but typically offer low inter-channel +cross-talk and low optical losses. Spatial switches can also use mirrors Cal, photonic integrated circuits Ding et al. +[2016], or tiled planar silicon photonics Seok et al. [2019a]. MZI switches are more co-integration friendly compared +to MEMS but have only been shown to scale up to 32×32 Ikeda et al. [2020]. This limit can be seen as a consequence +of the higher insertion-loss scaling resulting from cascaded MZI cells, as well as the susceptibility of popular MZI +topologies to first-order crosstalk. +The challenge for scaling-up the spatial approach is the quantization of package and MCM escape bandwidth and +reduced configuration options. For example, at 768 Gbps (the Ayar TeraPhy Wade [2019]), the number of fibers escaping +the package is 21 fibers, which means the package can be connected only up to 21 different potential destinations using +a spatial switch. +7 + +arXiv Template +A PREPRINT +Switch +Type +Radix +Wave- +lengths +per +port +B/W +per +channel +(wave- +length) +Insertion +Loss +Crosstalk +Mach- +Zehnder +based Ikeda +et +al. +[2020] +32×32 +1 +439 +Gbps +12.8 +dB +-26.6 +dB +MEMS- +actuated Seok +et +al. +[2019b] +240×240 1 +– +9.8 dB +-70 dB +Microring +res- +onator Khope +et +al. +[2017], +Cheng +et +al. +[2019b] +8×8 +(128×128) +8 +(128) +100 +Gbps +(42 +Gbps) +5dB +(10dB) +(-35 +dB) +Casc. +AW- +GRs Sato +[2018] +370×370 370 +25 +Gbps +15 dB +-35 dB +Table 2: High-radix CMOS-compatible photonic switches. +3.4.2 +Wavelength Selective Optical Switches and AWGRs +The inherent wavelength-selectivity of MRR switching cells allows for the straightforward implementation of +wavelength-selective switching (WSS) topologies. This enables one to establish all-to-all networks by leveraging +wavelength-division multiplexing (WDM). Currently, MRR-based switches with the largest radix include the 8×8 +crossbar Khope et al. [2017] and switch-and-select Nikolova et al. [2017], but have been experimentally emulated to +include a 16×16 Clos Dai et al. [2020]. The metrics in Dai et al. [2020] can be seen to correlate very closely with the +scaling proposed in Cheng et al. [2019b], making a practical case for the 128×128 shown in Table 2. +All-to-all networks via WDM signals can also be achieved by arrayed waveguide grating routers (AWGRs) Liu et al. +[2020], Zhang et al. [2019], Proietti et al. [2013], Lea [2015], Terzenidis et al. [2018]. As AWGRs are passive optical +elements, no reconfiguration is possible within the routing fabric itself. Instead, fast wavelength-tunable lasers must +be leveraged at the transmitter of every node if it wishes to address a different destination since AWGRs shuffle the +light frequencies such that one lambda goes to each endpoint from each source. AWGRs enable us to implement an +N×N all-to-all topology using just O(N) fibers (each carrying N frequencies of light) whereas an implementation +using copper would require N2 wires. Although the cost of fast wavelength-tunable lasers is still an ongoing research +topic Dhoore, Sören and Roelkens, Günther and Morthier, Geert [2019], AWGRs are mature, commercially available, +and well established in literature FSp. +In AWGRs, only a limited number of ports can be practically supported due to the walk-off of passband center +frequencies from the carrier wavelength grid and the worse crosstalk associated with a larger number of ports (N). +A feasible implementation of AWGR-based optical switches with large N has been demonstrated utilizing cascaded +small-size AWGRs Sato [2018]. Specifically, N M × M AWGRs (front-AWGRs) are interconnected with M N × N +AWGRs (rear-AWGRs) to effectively act as an MN × MN AWGR. Each output port of a front-AWGR is connected +to an input port of a rear-AWGR, where the interconnection pattern can be optimized with knowledge of port-specific +insertion losses to minimize the worst-case insertion loss of the aggregated AWGR. Further up-scaling of the switch +radix can be achieved by interconnecting small K × K delivery-coupling switchs (DC-switchs) with multiple copies +of the MN × MN AWGRs, yielding a KMN × KMN switching capability. This architecture has been verified by +hardware prototypes of 270 × 270 and 1440 × 1440 Sato et al. [2013], Ueda et al. [2016], showing ∼15 dB insertion +loss and below −35 dB crosstalk suppression. In order to accommodate the 350 MCMs of our rack, a reasonable +8 + +arXiv Template +A PREPRINT +1 +5 +3 +2 +8 +4 +6 +7 +Figure 5: With an AWGR, endpoint 1 has one wavelength directly connecting it to endpoint 3. If it desires more +bandwidth, it can route through another intermediate endpoint (indirect routing) chosen in a Valiant fashion Liu et al. +[2020], Teh et al. [2020]. Here, the link from 1 to 7 is available (green) but the link from 7 to 3 is not (red). The chosen +path is from 1 to 6 to 3 because both links are available. +configuration is KMN = 3 × 12 × 11 = 396. This results in 370 ports and 370 wavelengths per port (Table 2). Since +AWGRs typically have a 25 GHz optical bandwidth if the wavelength grid is 50 GHz, with PAM4 we assume 25 Gbps +per wavelength Dai et al. [2020], Bhoja [2017]. +Wave-selective switches Huang et al. [2020], Marom et al. [2017] can steer any subset of wavelengths to a given +destination, not just all (spatial) or one (AWGR). Dynamic programming methods can avoid sending the same frequency +of light from two different sources to the same destination. Since this is a relatively new technology, we constructed a +model shown in Table 2 that projects the performance of a larger radix switch that is comprised of smaller demonstrated +building blocks. +3.4.3 +Reconfiguration Time +Spatial and wave-selective switches typically require centralized scheduling Teh et al. [2020] to reach a steady globally +optimal solution. The reconfiguration time can range from tens of nanoseconds to tens of milliseconds. In production +HPC systems, multi-node jobs start every few seconds and last from minutes to hours Michelogiannakis et al. [2019, +2022]. Also, job resource usage and communication becomes predictable early, does not change fast, and typically +remains predictable throughout a job’s execution time Michelogiannakis et al. [2022, 2019], Shalf et al. [2005], Vetter +and Mueller [2002]. Therefore, even milliseconds of reconfiguration time is ample. +4 +Control Logic +Here we describe how we can perform indirect routing to increase point-to-point bandwidth using only per-source logic. +4.1 +Indirect routing in AWGRs +AWGRs dedicate exactly one wavelength between any source–destination pair. If a source–destination pair requests +more bandwidth than what a single wavelength can satisfy, sources can use indirect routing an example of which is +shown in Figure 5. Sources can split traffic to N intermediate destinations in parallel in order to use the bandwidth of +9 + +arXiv Template +A PREPRINT +N wavelengths. This does not consume additional power in the photonic components assuming lasers are constantly +powered. Sources consider indirect paths only if the direct (single-hop) bandwidth to their desired destination does +not suffice. A source considers indirect destinations for which the direct bandwidth from the source is available and +whose wavelengths from the intermediate hop to the desired final destination is available. Among potentially multiple +candidates, sources choose one in a Valiant fashion Liu et al. [2020], Teh et al. [2020], Domke et al. [2019]. This +is done on a per-flow basis in order to avoid out of order packet delivery. This routing logic can be modelled as an +allocator problem and implemented with a low latency and area penalty Ma et al. [2014], Becker and Dally [2009]. +Indirect routing relies on sources knowing which other sources attached to the same AWGR are utilizing their local +wavelengths in order to identify a productive intermediate destination. For instance, in Figure 5 endpoint 1 should +know whether the wavelengths from 7 to 3 and 6 to 7 are occupied. For that, we rely on piggybacking where traffic +between a source and a destination periodically includes the state of the sources’s wavelengths as a way to broadcast +local state to the rest of the endpoints attached to the same AWGR Jiang et al. [2009]. In the case of a N×N AWGR, +each source uses N bits to encode which of its N local wavelengths it is using with one-hot encoding. Even if we +piggyback this information multiple times a second, the bandwidth impact is negligible. For instance, if we multiplex +multiple flows into a wavelength and therefore denote 8 bits per wavelength, the status vector per source becomes +256 × 8 = 2048bits = 256bytes. If, due to stale information, sources pick an intermediate destination whose +wavelength direct to the final destination is not available, the intermediate destination performs indirect routing through +a second intermediate destination, and so on. If no data is exchanged between a pair, thus presenting no opportunity for +piggybacking, that pair can exchange a separate control message with the same information. +4.2 +Spatial and Wave-Selective Switches +Spatial and wave-selective switches can use indirect routing in tandem with reconfiguration. Indirect routing reduces +the need for reconfiguration, but intermediate hops should be chosen among hops that already have a direct connection +with the final destination; otherwise, the intermediate hop itself may trigger a reconfiguration. The synergy between +indirect routing and switch reconfiguration was explored in Teh et al. [2020]. +5 +Disaggregated Rack Design +For the rest of our study, we will model an HPC rack based on a GPU-accelerated HPE/Cray EX Supercomputer Per +where a rack contains 128 GPU-accelerated nodes. Each node of our model system contains an AMD Milan CPU +that has eight memory controllers each supporting a 3200MHz DDR4 module. Therefore, each CPU has 256 GB of +memory with a maximum bandwidth of 204.8 GBps. A compute node also has four NVIDIA Ampere A100 GPUs. +Each GPU supports 12 third generation NVLink links each supporting 25 GBps per direction. Each GPU also has 40 +GB of co-located HBM with a bandwidth of 1555.2 GBps. Each node also has four 31.5 GBps PCI Gen4 links to +connect each GPU to the CPU. The CPU also connects to four Slingshot 11 NICs with 200 Gbps per direction De Sensi +et al. [2020a]. Note that our photonic disaggregation hardware is orthogonal to and thus does not impair past work +related to disaggregation such as runtimes, OS support, endpoint sharing management, and security. +5.1 +MCMs and Escape Bandwidth +We organize chips within each rack into an MCMs package. For simplicity, we restrict all MCMs to have the same +escape bandwidth and we place chips of only the same type in MCMs. We then make conservative assumptions for next +generation photonics that are entering the market today based on our analysis of Section 3. In particular, each MCM has +32 optical fibers attached to it, a conservative assumption compared to the five arrays of 24 fibers demonstrated Hosseini +et al. [2021]. Each fiber supports 64 wavelengths (channels) of 25 Gbps each for a 6400 GBps escape bandwidth per +MCM. We vary the number of chips per MCM such that each chip enjoys the same escape bandwidth as in our baseline +rack Per. Therefore, our photonic architecture does not restrict chip escape bandwidth. Table 3 shows the number of +chips per MCM and the total number of MCMs containing chips of that type to satisfy chip escape bandwidth. Each +MCM contains a controller chip that interfaces the native protocol of the disaggregated resource to the CXL protocol +over the photonic links. CXL’s overhead and its associated FEC is included in our model of the overall architecture. +5.2 +Optical Switches +The radix and wavelengths per port of optical switches dictate number of MCMs we can fully connect optically with +a single switch as well as the amount of direct (single-hop) bandwidth. From Section 3.4, we pick state-of-the-art +representatives of wave-selective, cascaded AWGRs, and spatial optical switches. Their parameters are shown in +Table 4. Even though spatial Seok et al. [2019b] and wave-selective switches Huang et al. [2020] are capable of 100 +10 + +arXiv Template +A PREPRINT +Chip type +Chips per MCM +# MCMs per rack +CPU +14 +10 +GPU +3 +171 +NIC +203 +3 +HBM +4 +128 +DDR4 +27 +38 +Total +350 +Table 3: The number of chips ((CPU, GPU, NIC, HBM, or DDR4 module) per MCM and MCMs in a rack assuming +32 fibers per MCM, 64 wavelengths of 25 Gbps per fiber. The target BER to and from memory is 10−18 (Section 3.1). +Switch type +State of the art +Switch radix +Cascaded AWGRs Sato [2018] +370 +Spatial Seok et al. [2019b] +240 +Wave-Selective Huang et al. [2020] +256 +Gbps per wavelength +All switches +25 +Wavelengths per port +Cascaded AWGRs Sato [2018] +370 +Spatial Seok et al. [2019b] +240 +Wave-Selective Huang et al. [2020] +256 +Table 4: Switch configuration for our study. +Gbps per wavelength, most links available widely today do not support that (Table 1). In addition, we show that we can +still satisfy bandwidth demands with the conservative assumption of 25 Gbps per wavelength. +To connect our 350 MCMs using 370×370 AWGRs, we can combine MCM fibers in five groups of six and connect each +group to one port of five parallel AWGRs. However, this would require each AWGR port to handle 384 wavelengths. +To respect the per port 370 wavelength limitation of our AWGR configuration but still satisfy the full escape bandwidth +of MCMs, we combine the remaining 14 wavelengths along with the remaining two fibers per MCM (128 + 14 = 142 +wavelengths total) that were left unconnected into an extra parallel AWGR, for a total of six parallel AWGRs. We then +connect MCM fibers to AWGRs in a staggered manner such that each MCM connects to each other MCM using at least +five 25 Gbps direct-path wavelengths, for a direct MCM–MCM bandwidth of 25 × 5 = 125 Gbps. +For simplicity, because of their relative small difference and because wave-selective switches can also achieve configu- +rations that spatial switches can, we treat both wave-selective and spatial switches as 256 ports with 256 wavelengths +per port. Each MCM can connect to 2048 +256 = 8 parallel switches. However, because the radix of optical switches is lower +than the number of MCMs, we instantiate 11 optical switches and connect MCMs in a staggered manner such that +optical switch with an index I connects to MCMs that have an index starting from (32 × I) mod 350 until (I + 255) +mod 350. This way, a small number of optical switch ports are left unconnected in order to not exceed the 32 fibers +per MCM. Similar to AWGRs, these ports can support future larger racks. If the switches configure appropriately, +each MCM has at least three direct paths to any other MCM. Each path has 256 wavelengths, thus the direct MCM +bandwidth is 256 × 3 × 25 = 2304 Gbps. +6 +Evaluation +Having previously evaluated in Section 3.3.3 that photonic switches satisfy BER requirements, in this Section we +analyze the impact of photonic-based intra-rack resource disaggregation to bandwidth, latency, and power. We then +compare against electronic switches and estimate system-wide savings. +6.1 +Bandwidth Evaluation +We distinguish two test cases based on Section 5.2: (A) Six parallel AWGRs and (B) 11 parallel wave-selective switches. +6.1.1 +Available Bandwidth +Using indirect routing and switch reconfiguration, any one particular MCM can use its full escape bandwidth to reach a +single destination MCM. In test case (A), all wavelengths escaping an MCM can reach the same destination MCM +using indirect routing. In test case (B), 768 wavelengths can be configured to route directly to a destination MCM +11 + +arXiv Template +A PREPRINT +and the other 2048 − 768 = 1280 wavelengths can be configured to route indirectly through intermediate MCMs. +This assumes that other MCMs will not contend for bandwidth that may disrupt indirect routing or complicate switch +reconfiguration. While the direct (single-hop) bandwidth between cases (A) and (B) has a large difference, case (A) +always provides that direct bandwidth between MCMs whereas a spatial or wave-selective switch requires a scheduler +and leaves the majority of input–output combinations unconnected at any one time, thus also has to use indirect routing +to compensate. +Based on system profiling data of a production open-science HPC system Michelogiannakis et al. [2022], the 125 +Gbps direct bandwidth between MCMs in test case (A) suffices over 99.5% of the time between CPUs and main +memory (DDR4) and virtually all the time between memory and NICs. In addition, the bandwidth of a single AWGR +wavelength of 25 Gbps suffices 97% of the time between CPUs and memory as well as between memory and NICs. +This means that with a 97% probability, four of the five wavelengths between a memory and CPUs or NICs and +memory pair are available to use for indirect routing in case the direct 125 Gbps bandwidth does not suffice between +another memory–CPU or NIC–memory pair. Therefore, the probability at any one time that the direct bandwidth does +not suffice for a number of CPU–memory and NIC–memory pairs large enough such that they cannot find unused +bandwidth in other pairs to use for indirect routing is multiple orders of magnitude less than 0.1% and thus negligible. +To further reduce the probability, congested pairs can use direct paths from CPUs to CPUs that communicate minimally +and NICs to other NICs that do not communicate at all Michelogiannakis et al. [2022]. Therefore, test case (A) satisfies +bandwidth between CPUs, NICs, and main memory (DDR4). +Figure 6: Average and maximum slowdown for each suite and input set size. The slowdown is for an additional 35ns of +latency between the LLC and main memory from the additional photonic components. Left: in-order pipeline compute +cores. Right: Out of order (OOO) compute cores. +For GPUs, in test case (A) with indirect routing a single GPU can use a total of 125 × 512 = 8000 GBps to access any +one HBM or more in case a GPU is allocated more than one HBMs. This well satisfies the 1555.2 GBps that NVIDIA +Ampere A100 GPUs in our model rack Per access HBMs with today and leaves 8000 − 1555.2 = 6444.8 GBps unused +per GPU. In addition, in the worst case, an MCM containing three GPUs will communicate at full bandwidth (12 +NVLink links of 25 GBps per each of the three GPU equals 900 GBps) to other MCMs containing GPUs. Here, if +all GPUs in the rack acts similarly, we cannot rely on indirect routing from a GPU through an intermediate GPU to +reach a destination GPU. The direct 125 Gbps bandwidth between GPU MCMs do not suffice. Therefore, each GPU +can use the 6444.8 GBps of unused bandwidth to and from HBMs for indirect routing to well cover the 900 GBps +bandwidth that would otherwise use NVLink GPU–GPU links. This leaves 6444.8 − 900 = 5544.8 GBps per GPU that +can support direct HBM–HBM communication such as due to GPUDirect RDMA, indirect routing for other MCMs, or +simply increase available bandwidth to memory. Of note, our analysis does not use direct optical paths from GPUs to +main memory (DDR4). Future protocols may use for these paths or they can be used to provide even more indirect +routing bandwidth. +Our analysis shows that test case (A) with AWGRs more than satisfies bandwidth demands and avoids the need for a +scheduler to reconfigure spatial and wave-selective switches that would otherwise add overhead and reduce reaction +time. +6.2 +Latency Evaluation +Intra-rack resource disaggregation based on modern photonics increases the latency significantly less than full system +disaggregation. For intra-rack disaggregation we assume an additional latency between MCMs of 35 ns. That additional +latency covers 15 ns for electrical–optical–electrical conversion and 4 meters of photonic propagation at 5 ns per meter, +which covers round-trip distance of typical two-meter tall racks (Section 3.3.2). The small impact of distance to latency +12 + +Average +Maximum +100 +1 +(%) +75 +umopmos +50 +Percentage +25 +0 +Parsec +Parsec +Parsec +NAS A +NAS B +NAS C +Rodinia +small +medium +largeAverage +Maximum +125 +100 +(%) +slowdown +75 +Percentage s +50 +25 +0 +Parsec +Parsec +Parsec +NAS A +NAS B +NAS C +Rodinia +small +medium +largearXiv Template +A PREPRINT +with photonics practically makes MCMs in a rack equi-distant, thus mitigating a traditional queuing delay versus +locality tradeoff in job scheduling Jeon et al. [2019]. Indirect routing would increase latency by a few extra ns, but the +probability of routing indirectly is low. Because 35 ns is orders of magnitude lower than system-wide network latency, +we do not consider the effect of the additional 35 ns to inter-rack communication through NICs. +6.2.1 +CPU Evaluation +We experimentally quantify the impact to application performance with in-order pipeline and out-of-order (OOO) +compute cores. In-order cores provide insight of the impact of memory latency when the compute core does not mask +latency, whereas OOO cores are representative of modern cores. We use full system simulation in Gem5 Binkert et al. +[2011] of x86 compute cores running an Ubuntu 18.4 guest OS. We configure the cache hierarchy to match the CPUs +of our model HPC rack Per. We calculate the slowdown of application execution time when we add 35 ns of latency +between the LLC and main memory, compared to a baseline system with no additional latency to memory. Latency is +the only potential source of application slowdown since our architecture satisfies the full escape bandwidth for each +chip. +We evaluate the impact in three benchmark suites: PARSEC 3.1 Bienia et al. [2008], NAS parallel benchmarks +3.4.1 Bailey et al. [1992], and Rodinia Che et al. [2009]. For PARSEC we evaluate small, medium, and large input sets. +For NAS, we evaluate input sizes “A”, “B”, and “C”. For Rodinia we use the single default input set. These benchmark +suites have been widely used and contain a large variety of computation kernels that are representative of key HPC +applications such as stencils, graph processing, linear algebra, computational mathematics, grid, sorting, and many +others that have been observed to be important workloads in NERSC’s systems ?. Overall, we use 58 benchmarks to +provide a wide representation. We use a single compute core to better focus on the effect of the additional latency to +memory. +Figure 6 shows slowdown percentages for benchmarks across our three suites for an in-order core on the left and OOO +core on the right. As shown, NAS benchmarks are negligibly affected by the increased latency. Rodinia benchmarks +have an average slowdown of 15% with in-order cores and 13% for OOO cores. However, a single benchmark (NW) +has a slowdown of approximately 76% for in-order cores. The largest slowdown for the rest of Rodinia benchmarks +across in-order and OOO cores is 12%. Finally, PARSEC benchmarks are impacted the most, but the average slowdown +remains below 25% except for large inputs using OOO cores. OOO cores typically tolerate memory access latency +better, but they also produce more memory accesses per unit of time compared to in-order cores. +Figure 7 shows slowdown for individual PARSEC benchmarks for large inputs and in-order cores. As shown, only +three benchmarks exceed a 25% slowdown, while eight benchmarks have a slowdown of no more approximately 3.5%. +Therefore, our experiments show that while some benchmarks (three in PARSEC and one in Rodinia) experience +important slowdowns, the majority of benchmarks are impacted minimally even without mitigation strategies. This is +the case with all of the NAS benchmarks we used, eight PARSEC, and all but one Rodinia benchmarks. For benchmarks +that are more affected, there is a range of hardware and software techniques Mutlu et al. [2006], Parcerisa and Gonzalez +[2001], Mowry et al. [1998], Nekkalapu et al. [2008] to increase memory tolerance that we can apply to further reduce +application slowdown. +6.2.2 +Recovering Performance +To gauge the effectiveness of strategies to recover application performance, we test the impact of the following remedies +applied one at a time: (i) 256 miss status handling registers (MSHRs) in the LLC, (ii) doubling the LLC size with the +default number of 16 MSHRs, and (iii) default LLC configuration but a strided prefetcher with a larger stride than the +default four. Figure 8 shows the slowdown percentage that we were able to recover for PARSEC benchmarks through +these three techniques at the best, average, and worst case. This figure is the only one that includes these remedies in +this results of this section. As shown, about a 20% performance loss for small and large inputs is recovered by average. +The most effective remedy is doubling the LLC size. The reason for the smaller speedup for medium is that due the +particular LLC size, memory access patterns, and input sizes in PARSEC, medium experienced a smaller benefit from a +larger LLC. These findings motivate future work to mitigate the latency impact of the disaggregation hardware, similar +to mitigating the increased latency to access emerging memory technologies Mittal and Vetter [2016]. +6.2.3 +Sensitivity to Latency +To show the sensitivity of application performance to the amount of additional latency, Figure 9 shows application +slowdown for 25 ns, 30 ns, and 35 ns for in-order cores (OOO cores show comparable trends). As shown, reducing +the additional latency to 25 ns from 35 ns reduces application slowdown by as much as half. This motivates latency +improvements in photonic components or shorter rack distances. +13 + +arXiv Template +A PREPRINT +Figure 7: Results for individual PARSEC benchmarks with large inputs. +Figure 8: Percentage of slowdown that we can recover with LLC modifications. +14 + +Parsec slowdown: Large inputs. Single in-order core +100 +75 +(%) easad +50 +25Best case +Average +Worst case +(%) +50 +recovered +40 +30 +slowdown +20 +Percentage +10 +0 +Parsec small +Parsec medium +Parsec largearXiv Template +A PREPRINT +Figure 9: Percentage slowdown for 25ns, 30ns, and 35ns of additional LLC–memory latency for in order cores. +The overall average slowdown across all benchmarks is approximately 13% for both in-order cores and OOO cores +without architectural remedies, for large PARSEC inputs, and “B” size NAS inputs. This considerably less than +slowdowns quoted in past work for full-system disaggregation, furthering the case for intra-rack disaggregation. +6.2.4 +GPU Evaluation +To evaluate the impact of the additional latency between GPUs and HBMs or DDR4 main memory, we extend the +publicly available version of PPT-GPU Arafa et al. [2021] toolkit to account for the additional latency between the +main memory of the GPU and the LLC. In our evaluation, we modeled one NVIDIA A100 GPU Choquette and Gandhi +[2020] running a total of 27 applications that have a total of 2133 kernels from different benchmark suites. We run 13 +applications from Rodinia Che et al. [2009] and 10 applications from Polybench Grauer-Gray et al. [2012]. Polybench +applications are linear algebra applications that stress the GPU cache and main memory. Furthermore, we run AlexNet, +CifarNet, GRU, and LSTM from the Tango deep network Karki et al. [2019] benchmark suite. We use the default input +sizes and configuration that came with the benchmarks, detailed in Arafa et al. [2021]. We run applications using the +“SASS” model, where we extract memory and instruction traces for each application. +Figure 10 shows the effect of different latencies on the performance of our GPU benchmarks. We compare performance +in terms of the total predicted cycles. As shown, the highest average slowdown is 24% for Polybench. The overall +average slowdowns across the 27 applications is only 8%, 10%, and 12% for the 25 ns, 30 ns, and 35 ns additional +latency, respectively. For these benchmarks, doubling the LLC size recovers an average of 8% of the performance loss. +6.2.5 +CPU–GPU Comparison +We illustrate the difference in memory latency tolerance of in-order CPUs, OOO CPUs, and GPUs in Figure 11 for the +intersection of Rodinia benchmarks that correctly ran on both CPU and GPU with their default input sets. As shown, +GPUs tolerate the additional 35 ns latency significantly better with a maximum slowdown of 3.3%. This is promising +for resource disaggregation given the steady growth of GPU presence in HPC systems. +6.3 +Power Overhead +We calculate the per-rack power overhead of our photonic solution for 350 MCMs with 2048 escape wavelengths from +each MCM and 25 Gbps per wavelength. If we use a DFB laser array demonstrated in Rahimi et al. [2022] with a 11% +wall plug efficiency (WPE) at 10 dDm, a total of 256 × 256 such lasers consumes 64.5 kW. For the components and +distances in our study, the required optical power per wavelength is 10 dBm. Furthermore, 350×2048 of the modulators +15 + +25ns +30ns +35ns +100 +75 +50 +25 +0 +Parsec +par +ROarXiv Template +A PREPRINT +Figure 10: Percentage slowdown for 25ns, 30ns, and 35ns of additional LLC–memory latency for different GPU +benchmark suites. +Figure 11: Percentage slowdown for CPU and GPU Rodinia benchmarks. +16 + +25ns +30ns +35ns +80.00% +% +60.00% +S +40.00% +20.00% +0.00% +Rodinia-avg Rodinia-max PBench-avg PBench-max Tango-avg +Tango-max35ns in-order CPU +35nsO0OCPU +35ns GPU +80 +(%) +60 +slowdown ( +40 +ercentage +20 +ParXiv Template +A PREPRINT +and receivers of Sun et al. [2020] that consume 0.8 and 2.12 pJ/bit at 25 Gbps respectively result in a total additional +power of 52.5 kW. Finally, the switches of Table 2 consume no more than 1 kW at the worst case. In summary, the total +power overhead taking into account parallel switches is no more than 150 kW. Our analysis assumes the components +are constantly on. Considering that the maximum power consumption of a single A100 GPU is a few hundreds of Ws +and our modelled rack contains 512 such GPUs, the power overhead for our photonic solution is negligible. +6.4 +Comparison With Electronic Switches +Electronic SERDES signalling rate per wire is only 112 Gbps for a short reach. Also, typical CXL or PCIe signaling +rates top out at 35 GHz/wire. In fact, as SERDES rates increase, the distance that those signals can reach reduces down +to even a few millimeters due to the resistance and capacitance of copper wires. Photonics break the reach limitations +of copper and with co-packaging can achieve 4 Tbps per mm of shoreline on the chip die. +Focusing on electronic switches, Rosetta De Sensi et al. [2020b] and Infiniband Katebzadeh et al. [2020] have a +measured per hop latency of no less than approximately 200 ns. Emerging PCIe Gen5 switches add just 10 ns per +hop Vasa et al. [2020], but only support 100 lanes per switch. To fully connect our disaggregated rack, we consider a +two-level tree network with four hops (the top level is composed of an internal two-hop subnetwork). These four hops +will be in addition to the 35 ns we previously evaluated for FEC and propagation (propagation delay is comparable +between copper and photonic for rack distances), since our photonic solution uses switches with negligible traversal +latency. Therefore, the additional latency for disaggregation in the PCIe case becomes 85 ns compared to 35 ns for +our photonic architecture. Finally, we also consider the latency through one hop of an Anton 3 network, which is +approximately 90 ns by average Shim et al. [2022], though scaling up to match our rack size would require multiple +hops. These latencies represent the best case for electronic packet switches because scheduler decisions or congestion +can cause higher worst-case (tail) latencies that may further penalize application performance. This assumes that we +connect only one lane per endpoint which carries 32 Gbps for PCIe Gen5 and 29 Gbps for Anton 3. This is multiple +times less than the per-chip bandwidth of photonics our photonic architecture. +Figure 12 shows the speedup of a system that implements intra-rack disaggregation with emerging photonics with an +additional 35 ns latency to and from DDR4 and HBM memory compared to a similar system that uses modern electronic +switches instead. 85 ns is the lowest case for electronic switches and corresponds to a four-hop PCIe Gen5 network or a +single-hop Anton 3 network. As shown, for CPU benchmarks if we only take into account “medium” from PARSEC to +avoid counting PARSEC benchmarks three times, the average speedup for in-order CPUs is 12.7% and the maximum +76%. For OOO compute cores, the average is 23.9% and maximum 78.3%. For GPUs, the average and maximum are +both 61.5%. This analysis clearly shows the adverse impact of the additional latency of electronic switches and further +motivates the use of photonics for intra-rack resource disaggregation. Furthermore, the four electronic switches of this +analysis consume at least many tens of Watts of power, which is multiple orders of magnitude higher than our photonic +solution. +6.5 +Iso-Performance Comparison +Based on our performance evaluations, in order to preserve system-wide average computational throughput as our +baseline GPU-accelerated HPE/Cray EX system Per, our photonically-disaggregated system requires 13% more CPUs +and 8% more GPUs. However, intra-rack resource disaggregation allows our rack to have an average 4× fewer memory +modules and 2× fewer NICs Michelogiannakis et al. [2022]. Combining the two effects, our disaggregated rack has +1082 total modules compared to 1920 in the baseline system, a 43% reduction. Alternatively, we can preserve all +rack resources and instead add 128 of a combination of CPUs and GPUs (with their HBMs), which is only a 7% chip +increase across the rack. Doing so doubles computational throughput. +7 +Conclusion +We have designed a resource disaggregated HPC rack that uses modern photonic links and switches to meet BER and +bandwidth requirements of HPC applications, has a negligible power impact, uses distributed indirect routing instead of +complex switch reconfiguration, and provides a 23.9% for CPUs or 61.5% for GPUs speedup compared to a similar +disaggregated rack implemented with modern electronic switches. Our architecture enables a disaggregated system to +preserve its performance but use 43% fewer overall chips. +17 + +arXiv Template +A PREPRINT +Figure 12: Speedup of a system that uses emerging photonics to implement intra-rack resource disaggregation that adds +35 ns of additional latency to and from memory compared to a similar system that uses modern electronic switches and +adds 85 ns of memory latency instead. +References +B. Liu, D. Zydek, H. Selvaraj, and L. Gewali. Accelerating high performance computing applications: Using cpus, +gpus, hybrid CPU/GPU, and fpgas. In 2012 13th International Conference on Parallel and Distributed Computing, +Applications and Technologies, pages 337–342, 2012. +The top500 HPC list, November 2018. URL https://www.top500.org/green500/lists/2018/11/. +M. Ujaldón. Hpc accelerators with 3d memory. In 2016 IEEE Intl Conference on Computational Science and +Engineering (CSE) and IEEE Intl Conference on Embedded and Ubiquitous Computing (EUC) and 15th Intl +Symposium on Distributed Computing and Applications for Business Engineering (DCABES), pages 320–328, 2016. +doi:10.1109/CSE-EUC-DCABES.2016.203. +Sparsh Mittal and Jeffrey S. Vetter. A survey of cpu-gpu heterogeneous computing techniques. ACM Comput. Surv., 47 +(4), July 2015. ISSN 0360-0300. doi:10.1145/2788396. +D. Tiwari, S. Gupta, J. Rogers, D. Maxwell, P. Rech, S. Vazhkudai, D. Oliveira, D. Londo, N. DeBardeleben, P. Navaux, +L. Carro, and A. Bland. Understanding gpu errors on large-scale hpc systems and the implications for system design +and operation. In 2015 IEEE 21st International Symposium on High Performance Computer Architecture (HPCA), +pages 331–342, 2015. doi:10.1109/HPCA.2015.7056044. +Y. Gao and P. Zhang. A survey of homogeneous and heterogeneous system architectures in high performance +computing. +In 2016 IEEE International Conference on Smart Cloud (SmartCloud), pages 170–175, 2016. +doi:10.1109/SmartCloud.2016.36. +D. Milojicic. Accelerators for artificial intelligence and high-performance computing. Computer, 53(2):14–22, 2020. +H. Asaadi and B. Chapman. Comparative study of deep learning framework in hpc environments. In 2017 New York +Scientific Data Summit (NYSDS), pages 1–7, 2017. doi:10.1109/NYSDS.2017.8085040. +O. Segal, N. Nasiri, M. Margala, and W. Vanderbauwhede. High level programming of FPGAs for HPC and data +centric applications. In 2014 IEEE High Performance Extreme Computing Conference (HPEC), pages 1–3, 2014. +doi:10.1109/HPEC.2014.7040979. +18 + +CPUtiming +CPUOOO +GPU +80 +Photonic speedup over electronic (%) +60 +40 +20 +xew +avg +avg +small +avg +max +max +avg +avg +large +Prsc +Prsc +Prsc +PrscarXiv Template +A PREPRINT +Tom Hogervorst, Tong Dong Qiu, Giacomo Marchiori, Alf Birger, Markus Blatt, and Razvan Nane. Hardware +acceleration of HPC computational flow dynamics using HBM-enabled FPGAs, 2021. +J. Lant, J. Navaridas, M. Luján, and J. Goodacre. Toward fpga-based HPC: Advancing interconnect technologies. IEEE +Micro, 40(1):25–34, 2020. doi:10.1109/MM.2019.2950655. +Rob Dimond, Sebastien Racanière, and Oliver Pell. Accelerating large-scale hpc applications using fpgas. In 2011 +IEEE 20th Symposium on Computer Arithmetic, pages 191–192, 2011. doi:10.1109/ARITH.2011.34. +G. Ramirez-Gargallo, M. Garcia-Gasulla, and F. Mantovani. Tensorflow on state-of-the-art HPC clusters: A machine +learning use case. In 2019 19th IEEE/ACM International Symposium on Cluster, Cloud and Grid Computing +(CCGRID), pages 526–533, 2019. doi:10.1109/CCGRID.2019.00067. +M. G. Venkata, F. Aderholdt, and Z. Parchman. SharP: Towards programming extreme-scale systems with hierarchical +heterogeneous memory. In 2017 46th International Conference on Parallel Processing Workshops (ICPPW), pages +145–154, 2017. doi:10.1109/ICPPW.2017.32. +George Michelogiannakis, Benjamin Klenk, Brandon Cook, Min Yee Teh, Madeleine Glick, Larry Dennison, Keren +Bergman, and John Shalf. A case for intra-rack resource disaggregation in HPC. ACM Trans. Archit. Code Optim., +jan 2022. ISSN 1544-3566. doi:10.1145/3514245. Just Accepted. +G. P. Rodrigo, P. Östberg, E. Elmroth, K. Antypas, R. Gerber, and L. Ramakrishnan. Towards understanding job +heterogeneity in hpc: A nersc case study. In 2016 16th IEEE/ACM International Symposium on Cluster, Cloud and +Grid Computing (CCGrid), pages 521–526, 2016. +A. Guleria, J. Lakshmi, and C. Padala. +Quadd: +Quantifying accelerator disaggregated datacenter effi- +ciency. In 2019 IEEE 12th International Conference on Cloud Computing (CLOUD), pages 349–357, 2019a. +doi:10.1109/CLOUD.2019.00064. +I. Peng, R. Pearce, and M. Gokhale. On the memory underutilization: Exploring disaggregated memory on HPC +systems. In 2020 IEEE 32nd International Symposium on Computer Architecture and High Performance Computing +(SBAC-PAD), pages 183–190, 2020. +J. Taylor. Facebook’s data center infrastructure: Open compute, disaggregated rack, and beyond. In 2015 Optical Fiber +Communications Conference and Exhibition (OFC), pages 1–1, 2015. +Huaicheng Li, Daniel S. Berger, Stanko Novakovic, Lisa Hsu, Dan Ernst, Pantea Zardoshti, Monish Shah, Ishwar +Agarwal, Mark D. Hill, Marcus Fontoura, and Ricardo Bianchini. First-generation memory disaggregation for cloud +platforms. arXiv, 2022. doi:10.48550/ARXIV.2203.00241. +K. Koh, K. Kim, S. Jeon, and J. Huh. Disaggregated cloud memory with elastic block management. IEEE Transactions +on Computers, 68(1):39–52, 2019. +A. D. Papaioannou, R. Nejabati, and D. Simeonidou. The benefits of a disaggregated data centre: A resource +allocation approach. +In 2016 IEEE Global Communications Conference (GLOBECOM), pages 1–7, 2016. +doi:10.1109/GLOCOM.2016.7842314. +J. Gonzalez, A. Gazman, M. Hattink, M. G. Palma, M. Bahadori, R. Rubio-Noriega, L. Orosa, M. Glick, O. Mutlu, +K. Bergman, and R. Azevedo. Optically connected memory for disaggregated data centers. In 2020 IEEE 32nd +International Symposium on Computer Architecture and High Performance Computing (SBAC-PAD), pages 43–50, +2020. doi:10.1109/SBAC-PAD49847.2020.00017. +Qixiang Cheng, Keren Bergman, Yishen Huang, Hao Yang, Meisam Bahadori, Nathan Abrams, Xiang Meng, +Madeleine Glick, Yang Liu, and Michael Hochberg. Silicon photonic switch topologies and routing strategies +for disaggregated data centers. IEEE Journal of Selected Topics in Quantum Electronics, PP:1–1, 12 2019a. +doi:10.1109/JSTQE.2019.2960950. +A. Guleria, J. Lakshmi, and C. Padala. Emf: Disaggregated gpus in datacenters for efficiency, modularity and flexibility. +In 2019 IEEE International Conference on Cloud Computing in Emerging Markets (CCEM), pages 1–8, 2019b. +doi:10.1109/CCEM48484.2019.000-5. +NERSC’s Perlmutter configuration. URL https://docs.nersc.gov/systems/perlmutter/. +Torben Kling Petersen and John Bent. +Hybrid flash arrays for hpc storage systems: +An alternative to +burst buffers. +In 2017 IEEE High Performance Extreme Computing Conference (HPEC), pages 1–7, 2017. +doi:10.1109/HPEC.2017.8091092. +M. Glick, N. C. Abrams, Q. Cheng, M. Y. Teh, Y. H. Hung, O. Jimenez, S. Liu, Y. Okawachi, X. Meng, L. Johansson, +M. Ghobadi, L. Dennison, G. Michelogiannakis, J. Shalf, A. Liu, J. Bowers, A. Gaeta, M. Lipson, and K. Bergman. +PINE: Photonic integrated networked energy efficient datacenters (enlitened program) [invited]. IEEE/OSA Journal +of Optical Communications and Networking, 12(12):443–456, 2020. doi:10.1364/JOCN.402788. +19 + +arXiv Template +A PREPRINT +Xiaotao Guo, Xuwei Xue, Bitao Pan, Fulong Yan, Georgios Exarchakos, and Nicola Calabretta. Fosda: A hybrid +disaggregated hpc architecture based on distributed nanoseconds optical switches. In Future computing, 2021. ISBN +978-1-61208-846-4. +R. Lin, Y. Cheng, M. D. Andrade, L. Wosinska, and J. Chen. Disaggregated data centers: Challenges and trade-offs. +IEEE Communications Magazine, 58(2), 2020. +Aaron Call, Jordà Polo, David Carrera, Francesc Guim, and Sujoy Sen. Disaggregating non-volatile memory for +throughput-oriented genomics workloads. CoRR, abs/2007.02813, 2020. +Gengchen Liu, Roberto Proietti, Marjan Fariborz, Pouya Fotouhi, Xian Xiao, and S. J. Ben Yoo. Architecture +and performance studies of 3d-hyper-flex-lion for reconfigurable all-to-all hpc networks. In Proceedings of the +International Conference for High Performance Computing, Networking, Storage and Analysis, SC ’20. IEEE Press, +2020. ISBN 9781728199986. +Y. Zhang, X. Xiao, K. Zhang, S. Li, A. Samanta, Y. Zhang, K. Shang, R. Proietti, K. Okamoto, and S. J. B. Yoo. +Foundry-enabled scalable all-to-all optical interconnects using silicon nitride arrayed waveguide router interposers +and silicon photonic transceivers. IEEE Journal of Selected Topics in Quantum Electronics, 25(5):1–9, 2019. +doi:10.1109/JSTQE.2019.2910415. +Tae Joon Seok, Kyungmok Kwon, Johannes Henriksson, Jianheng Luo, and Ming C. Wu. Wafer-scale silicon +photonic switches beyond die size limit. Optica, 6(4):490–494, Apr 2019a. doi:10.1364/OPTICA.6.000490. URL +http://opg.optica.org/optica/abstract.cfm?URI=optica-6-4-490. +Yunhong Ding, Valerija Kamchevska, Kjeld Dalgaard, Feihong Ye, Rameez Asif, Simon Gross, Michael Withford, +Michael Galili, Toshio Morioka, and L.K. Oxenlowe. Reconfigurable sdm switching using novel silicon photonic +integrated circuit. Scientific Reports, 6, 08 2016. doi:10.1038/srep39058. +Yishen Huang, Qixiang Cheng, Anthony Rizzo, and Keren Bergman. Push—pull microring-assisted space-and- +wavelength selective switch. Opt. Lett., 45(10):2696–2699, May 2020. doi:10.1364/OL.392482. +Kevin Lim, Jichuan Chang, Trevor Mudge, Parthasarathy Ranganathan, Steven K. Reinhardt, and Thomas F. Wenisch. +Disaggregated memory for expansion and sharing in blade servers. SIGARCH Comput. Archit. News, 37(3):267–278, +jun 2009. ISSN 0163-5964. doi:10.1145/1555815.1555789. +S. Van Doren. Abstract - hoti 2019: Compute express link. In 2019 IEEE Symposium on High-Performance Interconnects +(HOTI), pages 18–18, 2019. doi:10.1109/HOTI.2019.00017. +G. Zervas, H. Yuan, A. Saljoghei, Q. Chen, and V. Mishra. Optically disaggregated data centers with minimal remote +memory latency: Technologies, architectures, and resource allocation [invited]. IEEE/OSA Journal of Optical +Communications and Networking, 10(2):A270–A285, 2018. doi:10.1364/JOCN.10.00A270. +Peter X. Gao, Akshay Narayan, Sagar Karandikar, Joao Carreira, Sangjin Han, Rachit Agarwal, Sylvia Ratnasamy, and +Scott Shenker. Network requirements for resource disaggregation. In Proceedings of the 12th USENIX Conference +on Operating Systems Design and Implementation, OSDI’16. USENIX Association, 2016. ISBN 9781931971331. +Bulent Abali, Richard J. Eickemeyer, Hubertus Franke, Chung-Sheng Li, and Marc A. Taubenblatt. Disaggregated and +optically interconnected memory: when will it be cost effective?, 2015. +Sangjin Han, Norbert Egi, Aurojit Panda, Sylvia Ratnasamy, Guangyu Shi, and Scott Shenker. Network support for +resource disaggregation in next-generation datacenters. In Proceedings of the Twelfth ACM Workshop on Hot Topics +in Networks, HotNets-XII, 2013. ISBN 9781450325967. doi:10.1145/2535771.2535778. +Y. Cheng, M. De Andrade, L. Wosinska, and J. Chen. Resource disaggregation versus integrated servers in data centers: +Impact of internal transmission capacity limitation. In 2018 European Conference on Optical Communication +(ECOC), pages 1–3, 2018. doi:10.1109/ECOC.2018.8535214. +Yuping Fan, Zhiling Lan, Paul Rich, William E. Allcock, Michael E. Papka, Brian Austin, and David Paul. Scheduling +beyond CPUs for HPC. Proceedings of the 28th International Symposium on High-Performance Parallel and +Distributed Computing, Jun 2019. +Giovanni Agosta, William Fornaciari, Giuseppe Massari, Anna Pupykina, Federico Reghenzani, and Michele Zanella. +Managing heterogeneous resources in HPC systems. PARMA-DITAM ’18, page 7–12, New York, NY, USA, 2018. +Association for Computing Machinery. ISBN 9781450364447. +Marcelo Amaral, Jorda Polo, David Carrera, Nelson Gonzalez, Chih-Chieh Yang, Alessandro Morari, Bruce D’Amora, +Alaa Youssef, and Malgorzata Steinder. DRMaestro: orchestrating disaggregated resources on virtualized data-centers. +Journal of Cloud Computing, 10, 03 2021. doi:10.1186/s13677-021-00238-6. +G. Domeniconi, E. Lee, Vanamala Venkataswamy, and Swaroopa Dola. CuSH: Cognitive scheduler for heterogeneous +high performance computing system. 2019. +20 + +arXiv Template +A PREPRINT +Arthur B. Maccabe. Operating and runtime systems challenges for hpc systems. In Proceedings of the 7th Inter- +national Workshop on Runtime and Operating Systems for Supercomputers ROSS 2017, ROSS ’17, 2017. ISBN +9781450350860. doi:10.1145/3095770.3095771. +W. Hwu, L. Chang, H. Kim, A. Dakkak, and I. El Hajj. +Transitioning hpc software to exascale heteroge- +neous computing. In 2015 Computational Electromagnetics International Workshop (CEM), pages 1–2, 2015. +doi:10.1109/CEM.2015.7237412. +Yizhou Shan, Yutong Huang, Yilun Chen, and Yiying Zhang. Legoos: A disseminated, distributed os for hardware +resource disaggregation. +In Proceedings of the 13th USENIX Conference on Operating Systems Design and +Implementation, OSDI’18, page 69–87. USENIX Association, 2018. ISBN 9781931971478. +Roberto Gioiosa, Ozcelik Burcu Mutlu, Seyong Lee, S. Jeffrey Vetter, Giulio Picierro, and Marco Cesati. The minos +computing library - efficient parallel programming for extremely heterogeneous systems. GPGPU@PPoPP, pages +1–10, 2020. +H. Khaleghzadeh, R. R. Manumachu, and A. Lastovetsky. A hierarchical data-partitioning algorithm for performance +optimization of data-parallel applications on heterogeneous multi-accelerator numa nodes. IEEE Access, 8:7861– +7876, 2020. +Zaeem Hussain. Heterogeneity aware fault tolerance for extreme scale computing. August 2020. URL http: +//d-scholarship.pitt.edu/39456/. +P. Jamieson, A. Sanaullah, and M. Herbordt. Benchmarking heterogeneous hpc systems including reconfigurable fabrics: +Community aspirations for ideal comparisons. In 2018 IEEE High Performance extreme Computing Conference +(HPEC), pages 1–6, 2018. +K. Tang, D. Tiwari, S. Gupta, S. S. Vazhkudai, and X. He. Effective running of end-to-end hpc workflows on emerging +heterogeneous architectures. In 2017 IEEE International Conference on Cluster Computing (CLUSTER), pages +344–348, 2017. doi:10.1109/CLUSTER.2017.22. +Alexey Lastovetsky. Heterogeneous parallel computing: from clusters of workstations to hierarchical hybrid platforms. +Supercomputing Frontiers and Innovations, 1(3), 2015. ISSN 2313-8734. +Jing Guo, Zihao Chang, Sa Wang, Haiyang Ding, Yihui Feng, Liang Mao, and Yungang Bao. Who limits the resource +efficiency of my datacenter: An analysis of alibaba datacenter traces. In Proceedings of the International Symposium +on Quality of Service, IWQoS ’19, 2019. ISBN 9781450367783. doi:10.1145/3326285.3329074. +J. Patel, V. Jindal, I. Yen, F. Bastani, J. Xu, and P. Garraghan. Workload estimation for improving resource management +decisions in the cloud. In 2015 IEEE Twelfth International Symposium on Autonomous Decentralized Systems, pages +25–32, 2015. doi:10.1109/ISADS.2015.17. +Teng Li, Vikram K. Narayana, and Tarek A. El-Ghazawi. Efficient resource sharing through GPU virtualization on +accelerated high performance computing systems. CoRR, abs/1511.07658, 2015. +Myeongjae Jeon, Shivaram Venkataraman, Amar Phanishayee, unjie Qian, Wencong Xiao, and Fan Yang. Analysis of +large-scale multi-tenant gpu clusters for dnn training workloads. In Proceedings of the 2019 USENIX Conference on +Usenix Annual Technical Conference, USENIX ATC ’19, page 947–960, USA, 2019. USENIX Association. ISBN +9781939133038. +Teng Li, Vikram K. Narayana, Esam El-Araby, and Tarek El-Ghazawi. Gpu resource sharing and virtualization on high +performance computing systems. In 2011 International Conference on Parallel Processing, pages 733–742, 2011. +doi:10.1109/ICPP.2011.88. +N. Terzenidis, M. Moralis-Pegios, G. Mourgias-Alexandris, T. Alexoudi, K. Vyrsokinos, and N. Pleros. High-port and +low-latency optical switches for disaggregated data centers: The hipolaos switch architecture. IEEE/OSA Journal of +Optical Communications and Networking, 10(7):102–116, 2018. doi:10.1364/JOCN.10.00B102. +Zeke Wang, Hongjing Huang, Jie Zhang, and Gustavo Alonso. Benchmarking high bandwidth memory on FPGAs, +2020. +Justin Meza, Qiang Wu, Sanjeev Kumar, and Onur Mutlu. Revisiting memory errors in large-scale production data +centers: Analysis and modeling of new trends from the field. In 45th Annual IEEE/IFIP International Conference +on Dependable Systems and Networks, DSN 2015, Rio de Janeiro, Brazil, June 22-25, 2015, pages 415–426. IEEE +Computer Society, 2015. doi:10.1109/DSN.2015.57. +Vilas Sridharan, Nathan DeBardeleben, Sean Blanchard, Kurt B. Ferreira, Jon Stearley, John Shalf, and Sudhanva +Gurumurthi. Memory errors in modern systems: The good, the bad, and the ugly. In Özcan Özturk, Kemal Ebcioglu, +and Sandhya Dwarkadas, editors, Proceedings of the Twentieth International Conference on Architectural Support +for Programming Languages and Operating Systems, ASPLOS ’15, Istanbul, Turkey, March 14-18, 2015, pages +297–310. ACM, 2015. doi:10.1145/2694344.2694348. +21 + +arXiv Template +A PREPRINT +S. Luyi, F. Jinyi, and Y. Xiaohua. Forward error correction. In 2012 Fourth International Conference on Computational +and Information Sciences, pages 37–40, 2012. doi:10.1109/ICCIS.2012.158. +Bok Young Kim, Yoshitomo Okawachi, Jae K. Jang, Mengjie Yu, Xingchen Ji, Yun Zhao, Chaitanya Joshi, Michal +Lipson, and Alexander L. Gaeta. Turn-key, high-efficiency kerr comb source. Opt. Lett., 44(18):4475–4478, Sep +2019a. doi:10.1364/OL.44.004475. +Shaoming Chen, Samuel Irving, Lu Peng, Yue Hu, Ying Zhang, and Ashok Srivastava. Using switchable pins to +increase off-chip bandwidth in chip-multiprocessors. IEEE Trans. Parallel Distrib. Syst., 28(1):274–289, jan 2017. +ISSN 1045-9219. doi:10.1109/TPDS.2016.2546246. +Norman P. Jouppi, Cliff Young, Nishant Patil, David Patterson, Gaurav Agrawal, Raminder Bajwa, Sarah Bates, Suresh +Bhatia, Nan Boden, Al Borchers, Rick Boyle, Pierre-luc Cantin, Clifford Chao, Chris Clark, Jeremy Coriell, Mike +Daley, Matt Dau, Jeffrey Dean, Ben Gelb, Tara Vazir Ghaemmaghami, Rajendra Gottipati, William Gulland, Robert +Hagmann, C. Richard Ho, Doug Hogberg, John Hu, Robert Hundt, Dan Hurt, Julian Ibarz, Aaron Jaffey, Alek +Jaworski, Alexander Kaplan, Harshit Khaitan, Daniel Killebrew, Andy Koch, Naveen Kumar, Steve Lacy, James +Laudon, James Law, Diemthu Le, Chris Leary, Zhuyuan Liu, Kyle Lucke, Alan Lundin, Gordon MacKean, Adriana +Maggiore, Maire Mahony, Kieran Miller, Rahul Nagarajan, Ravi Narayanaswami, Ray Ni, Kathy Nix, Thomas Norrie, +Mark Omernick, Narayana Penukonda, Andy Phelps, Jonathan Ross, Matt Ross, Amir Salek, Emad Samadiani, +Chris Severn, Gregory Sizikov, Matthew Snelham, Jed Souter, Dan Steinberg, Andy Swing, Mercedes Tan, Gregory +Thorson, Bo Tian, Horia Toma, Erick Tuttle, Vijay Vasudevan, Richard Walter, Walter Wang, Eric Wilcox, and +Doe Hyun Yoon. In-datacenter performance analysis of a tensor processing unit. In Proceedings of the 44th Annual +International Symposium on Computer Architecture, ISCA ’17, pages 1–12, New York, NY, USA, 2017. ACM. ISBN +978-1-4503-4892-8. doi:10.1145/3079856.3080246. +M. Wade. TeraPHY: A chiplet technology for low-power, high-bandwidth in-package optical i/o. In 2019 IEEE Hot +Chips 31 Symposium (HCS), pages i–xlviii, 2019. doi:10.1109/HOTCHIPS.2019.8875658. +Keren Bergman, John Shalf, George Michelogiannakis, Sebastien Rumley, Larry Dennison, and Monia Ghobadi. Pine: +An energy efficient flexibly interconnected photonic data center architecture for extreme scalability. In 2018 IEEE +Optical Interconnects Conference (OI), pages 25–26, 2018. doi:10.1109/OIC.2018.8422036. +Pavlos Maniotis, Laurent Schares, Marc A. Taubenblatt, and Daniel M. Kuchta. Co-packaged optics for HPC and data +center networks. In Henning Schröder and Ray T. Chen, editors, Optical Interconnects XXI, volume 11692, pages 17 +– 22. International Society for Optics and Photonics, SPIE, 2021. doi:10.1117/12.2579066. +Cyriel Minkenberg, Rajagopal Krishnaswamy, Aaron Zilkie, and David Nelson. Co-packaged datacenter optics: +Opportunities and challenges. IET Optoelectronics, 15(2):77–91, 2021. doi:https://doi.org/10.1049/ote2.12020. +A. Sutono, A. Pham, J. Laskar, and W.R. Smith. Investigations of multi-layer ceramic-based mcm technology. In IEEE +7th Topical Meeting on Electrical Performance of Electronic Packaging (Cat. No.98TH8370), pages 83–86, 1998. +doi:10.1109/EPEP.1998.733854. +Nathan C. Abrams, Qixiang Cheng, Madeleine Glick, Moises Jezzini, Padraic Morrissey, Peter O’Brien, and Keren +Bergman. Silicon photonic 2.5d multi-chip module transceiver for high-performance data centers. Journal of +Lightwave Technology, 38(13):3346–3357, 2020. doi:10.1109/JLT.2020.2967235. +Michael Dittrich, Andy Heinig, Fabian Hopsch, and Robert Trieb. Heterogeneous interposer based integration of chips +with copper pillars and c4 balls to achieve high speed interfaces for adc application. In 2017 IEEE 67th Electronic +Components and Technology Conference (ECTC), pages 643–648, 2017. doi:10.1109/ECTC.2017.221. +S. Fathololoumi, D. Hui, S. Jadhav, J. Chen, K. Nguyen, M. N. Sakib, Z. Li, H. Mahalingam, S. Amiralizadeh, +N. N. Tang, H. Potluri, M. Montazeri, H. Frish, R. A. Defrees, C. Seibert, A. Krichevsky, J. K. Doylend, J. Heck, +R. Venables, A. Dahal, A. Awujoola, A. Vardapetyan, G. Kaur, M. Cen, V. Kulkarni, S. S. Islam, R. L. Spreitzer, +S. Garag, A. C. Alduino, R. Chiou, L. Kamyab, S. Gupta, B. Xie, R. S. Appleton, S. Hollingsworth, S. McCargar, +Y. Akulova, K. M. Brown, R. Jones, D. Zhu, T. Liljeberg, and L. Liao. 1.6 tbps silicon photonics integrated circuit +and 800 gbps photonic engine for switch co-packaging demonstration. Journal of Lightwave Technology, 39(4): +1155–1161, 2021. doi:10.1109/JLT.2020.3039218. +Erik Agrell, Magnus Karlsson, A R Chraplyvy, David J Richardson, Peter M Krummrich, Peter Winzer, Kim Roberts, +Johannes Karl Fischer, Seb J Savory, Benjamin J Eggleton, Marco Secondini, Frank R Kschischang, Andrew Lord, +Josep Prat, Ioannis Tomkos, John E Bowers, Sudha Srinivasan, Mae Brandt-Pearce, and Nicolas Gisin. Roadmap of +optical communications. Journal of Optics, 18(6):063002, may 2016. doi:10.1088/2040-8978/18/6/063002. +J. Wei, Q. Cheng, R. V. Penty, I. H. White, and D. G. Cunningham. 400 gigabit ethernet using advanced modulation +formats: Performance, complexity, and power dissipation. IEEE Communications Magazine, 53(2):182–189, 2015. +doi:10.1109/MCOM.2015.7045407. +22 + +arXiv Template +A PREPRINT +Bok Young Kim, Yoshitomo Okawachi, Jae K. Jang, Mengjie Yu, Xingchen Ji, Yun Zhao, Chaitanya Joshi, Michal +Lipson, and Alexander L. Gaeta. Turn-key, high-efficiency kerr comb source. Opt. Lett., 44(18):4475–4478, Sep +2019b. doi:10.1364/OL.44.004475. +D. D. Sharma. Pci express® 6.0 specification at 64.0 gt/s with pam-4 signaling: a low latency, high bandwidth, high +reliability and cost-effective interconnect. In 2020 IEEE Symposium on High-Performance Interconnects (HOTI), +pages 1–8, 2020. doi:10.1109/HOTI51249.2020.00016. +Kazuhiro Ikeda, Keijiro Suzuki, Ryotaro Konoike, Shu Namiki, and Hitoshi Kawashima. Large-scale silicon pho- +tonics switch based on 45-nm cmos technology. Optics Communications, 466:125677, 2020. ISSN 0030-4018. +doi:https://doi.org/10.1016/j.optcom.2020.125677. +Tae Joon Seok, Kyungmok Kwon, Johannes Henriksson, Jianheng Luo, and Ming C. Wu. 240×240 wafer-scale silicon +photonic switches. In 2019 Optical Fiber Communications Conference and Exhibition (OFC), pages 1–3, 2019b. +Akhilesh S. P. Khope, Andrew M. Netherton, Takako Hirokawa, Nicolas Volet, Eric J. Stanton, Clint Schow, Roger +Helkey, Adel A. M. Saleh, John E. Bowers, and Rod C. Alferness. Elastic wdm optoelectronic crossbar switch with +on-chip wavelength control. In Advanced Photonics 2017 (IPR, NOMA, Sensors, Networks, SPPCom, PS), page +PTh1D.3. Optica Publishing Group, 2017. doi:10.1364/PS.2017.PTh1D.3. +Qixiang Cheng, Meisam Bahadori, Yu-Han Hung, Yishen Huang, Nathan Abrams, and Keren Bergman. Scalable +microring-based silicon clos switch fabric with switch-and-select stages. IEEE Journal of Selected Topics in Quantum +Electronics, 25(5):1–11, 2019b. doi:10.1109/JSTQE.2019.2911421. +Ken-ichi Sato. Realization and Application of Large-Scale Fast Optical Circuit Switch for Data Center Networking. +Journal of Lightwave Technology, 36(7):1411–1419, April 2018. ISSN 1558-2213. doi:10.1109/JLT.2018.2801308. +S series optical circuit switch. URL https://www.calient.net/products/s-series-photonic-switch/. +Dessislava Nikolova, David M Calhoun, Yang Liu, Sébastien Rumley, Ari Novack, Tom Baehr-Jones, Michael Hochberg, +and Keren Bergman. Modular architecture for fully non-blocking silicon photonic switch fabric. Microsystems & +nanoengineering, 3(1):1–9, 2017. +Liang Yuan Dai, Yu-Han Hung, Qixiang Cheng, and Keren Bergman. Experimental demonstration of pam-4 transmission +through microring silicon photonic clos switch fabric. In 2020 Optical Fiber Communications Conference and +Exhibition (OFC), pages 1–3, 2020. +R. Proietti, Y. Yin, R. Yu, C. J. Nitta, V. Akella, C. Mineo, and S. J. B. Yoo. Scalable optical interconnect architecture +using awgr-based tonak lion switch with limited number of wavelengths. Journal of Lightwave Technology, 31(24): +4087–4097, 2013. doi:10.1109/JLT.2013.2285883. +C. Lea. A scalable awgr-based optical switch. Journal of Lightwave Technology, 33(22):4612–4621, Nov 2015. ISSN +0733-8724. doi:10.1109/JLT.2015.2479296. +Dhoore, Sören and Roelkens, Günther and Morthier, Geert. Fast wavelength-tunable lasers on silicon. IEEE JOURNAL +OF SELECTED TOPICS IN QUANTUM ELECTRONICS, 25(6):1500908:1–1500908:8, 2019. ISSN 1077-260X. +Optical circuit switch product availability. https://www.fs.com/products/66601.html. +Ken-ichi Sato, Hiroshi Hasegawa, Tomonobu Niwa, and Toshio Watanabe. A large-scale wavelength routing optical +switch for data center networks. IEEE Communications Magazine, 51(9):46–52, September 2013. ISSN 1558-1896. +doi:10.1109/MCOM.2013.6588649. +Koh Ueda, Yojiro Mori, Hiroshi Hasegawa, Hiroyuki Matsuura, Kiyo Ishii, Haruhiko Kuwatsuka, Shu Namiki, Toshio +Watanabe, and Ken-ichi Sato. Demonstration of 1,440×1,440 fast optical circuit switch for datacenter networking. +In 2016 21st OptoElectronics and Communications Conference (OECC) Held Jointly with 2016 International +Conference on Photonics in Switching (PS), pages 1–3, July 2016. +Sudeep Bhoja. Pam4 signaling for intra-data center and data center to data center connectivity (dci). In 2017 Optical +Fiber Communications Conference and Exhibition (OFC), pages 1–54, 2017. +D. M. Marom, P. D. Colbourne, A. D’errico, N. K. Fontaine, Y. Ikuma, R. Proietti, L. Zong, J. M. Rivas-Moscoso, +and I. Tomkos. Survey of photonic switching architectures and technologies in support of spatially and spectrally +flexible optical networking [invited]. IEEE/OSA Journal of Optical Communications and Networking, 9(1):1–26, +2017. doi:10.1364/JOCN.9.000001. +Min Yee Teh, Yu-Han Hung, George Michelogiannakis, Shijia Yan, Madeleine Glick, John Shalf, and Keren +Bergman. +Tago: Rethinking routing design in high performance reconfigurable networks. +In SC20: Inter- +national Conference for High Performance Computing, Networking, Storage and Analysis, pages 1–16, 2020. +doi:10.1109/SC41405.2020.00029. +23 + +arXiv Template +A PREPRINT +George Michelogiannakis, Yiwen Shen, Min Yee Teh, Xiang Meng, Benjamin Aivazi, Taylor Groves, John Shalf, +Madeleine Glick, Manya Ghobadi, Larry Dennison, and Keren Bergman. Bandwidth steering in HPC using silicon +nanophotonics. In Proceedings of the International Conference for High Performance Computing, Networking, +Storage and Analysis, SC ’19, 2019. ISBN 9781450362290. doi:10.1145/3295500.3356145. +John Shalf, Shoaib Kamil, Leonid Oliker, and David Skinner. Analyzing ultra-scale application communication +requirements for a reconfigurable hybrid interconnect. In Proceedings of the ACM/IEEE SC2005 Conference on +High Performance Networking and Computing, November 12-18, 2005, Seattle, WA, USA, CD-Rom, page 17, 2005. +doi:10.1109/SC.2005.12. +Jeffrey S. Vetter and Frank Mueller. Communication characteristics of large-scale scientific applications for contempo- +rary cluster architectures. In 16th International Parallel and Distributed Processing Symposium (IPDPS 2002), 15-19 +April 2002, Fort Lauderdale, FL, USA, CD-ROM/Abstracts Proceedings, 2002. doi:10.1109/IPDPS.2002.1015504. +Jens Domke, Satoshi Matsuoka, Ivan R. Ivanov, Yuki Tsushima, Tomoya Yuki, Akihiro Nomura, Shin’ichi Miura, Nie +McDonald, Dennis L. Floyd, and Nicolas Dubé. Hyperx topology: First at-scale implementation and comparison to +the fat-tree. In Proceedings of the International Conference for High Performance Computing, Networking, Storage +and Analysis, SC ’19, 2019. ISBN 9781450362290. doi:10.1145/3295500.3356140. +Sheng Ma, Zhiying Wang, Natalie Enright Jerger, Li Shen, and Nong Xiao. Novel flow control for fully adaptive +routing in cache-coherent nocs. IEEE Transactions on Parallel and Distributed Systems, 25(9):2397–2407, 2014. +doi:10.1109/TPDS.2013.166. +Daniel U. Becker and William J. Dally. +Allocator implementations for network-on-chip routers. +In Proceed- +ings of the Conference on High Performance Computing Networking, Storage and Analysis, pages 1–12, 2009. +doi:10.1145/1654059.1654112. +Nan Jiang, John Kim, and William J. Dally. Indirect adaptive routing on large scale interconnection networks. In +Proceedings of the 36th Annual International Symposium on Computer Architecture, ISCA ’09, pages 220–231, 2009. +ISBN 978-1-60558-526-0. doi:10.1145/1555754.1555783. +Daniele De Sensi, Salvatore Di Girolamo, Kim H. McMahon, Duncan Roweth, and Torsten Hoefler. An in-depth analysis +of the slingshot interconnect. In SC20: International Conference for High Performance Computing, Networking, +Storage and Analysis, pages 1–14, 2020a. doi:10.1109/SC41405.2020.00039. +Kaveh Hosseini, Edwin Kok, Sergey Y. Shumarayev, Chia-Pin Chiu, Arnab Sarkar, Asako Toda, Yanjing Ke, Allen +Chan, Daniel Jeong, Mason Zhang, Sangeeta Raman, Thungoc Tran, Kumar Abhishek Singh, Pavan Bhargava, +Chong Zhang, Haiwei Lu, Ravi Mahajan, Xiaoqian Li, Nitin Deshpande, Conor O’Keeffe, Tim Tri Hoang, Uma +Krishnamoorthy, Chen Sun, Roy Meade, Vladimir Stojanovic, and Mark Wade. 8 tbps co-packaged FPGA and +silicon photonics optical IO. In 2021 Optical Fiber Communications Conference and Exhibition (OFC), pages 1–3, +2021. +Nathan Binkert, Bradford Beckmann, Gabriel Black, Steven K. Reinhardt, Ali Saidi, Arkaprava Basu, Joel Hestness, +Derek R. Hower, Tushar Krishna, Somayeh Sardashti, Rathijit Sen, Korey Sewell, Muhammad Shoaib, Nilay Vaish, +Mark D. Hill, and David A. Wood. The gem5 simulator. SIGARCH Comput. Archit. News, 39(2):1–7, aug 2011. +ISSN 0163-5964. doi:10.1145/2024716.2024718. +Christian Bienia, Sanjeev Kumar, Jaswinder Pal Singh, and Kai Li. The parsec benchmark suite: Characterization and +architectural implications. In 2008 International Conference on Parallel Architectures and Compilation Techniques +(PACT), pages 72–81, 2008. +D.H. Bailey, E. Barszcz, L. Dagum, and H.D. Simon. +Nas parallel benchmark results. +In Supercom- +puting ’92:Proceedings of the 1992 ACM/IEEE Conference on Supercomputing, pages 386–393, 1992. +doi:10.1109/SUPERC.1992.236665. +Shuai Che, Michael Boyer, Jiayuan Meng, David Tarjan, Jeremy W. Sheaffer, Sang-Ha Lee, and Kevin Skadron. +Rodinia: A benchmark suite for heterogeneous computing. In 2009 IEEE International Symposium on Workload +Characterization (IISWC), pages 44–54, 2009. doi:10.1109/IISWC.2009.5306797. +O. Mutlu, Hyesoon Kim, and Y.N. Patt. Efficient runahead execution: Power-efficient memory latency tolerance. IEEE +Micro, 26(1):10–20, 2006. doi:10.1109/MM.2006.10. +J.-M. Parcerisa and A. Gonzalez. Improving latency tolerance of multithreading through decoupling. IEEE Transactions +on Computers, 50(10):1084–1094, 2001. doi:10.1109/12.956093. +T.C. Mowry, C.Q.C. Chan, and A.K.W. Lo. Comparative evaluation of latency tolerance techniques for software +distributed shared memory. In Proceedings 1998 Fourth International Symposium on High-Performance Computer +Architecture, pages 300–311, 1998. doi:10.1109/HPCA.1998.650569. +24 + +arXiv Template +A PREPRINT +Satyanarayana Nekkalapu, Haitham Akkary, Komal Jothi, Renjith Retnamma, and Xiaoyu Song. +A simple la- +tency tolerant processor. In 2008 IEEE International Conference on Computer Design, pages 384–389, 2008. +doi:10.1109/ICCD.2008.4751889. +Sparsh Mittal and Jeffrey S. Vetter. A survey of software techniques for using non-volatile memories for storage +and main memory systems. IEEE Trans. Parallel Distrib. Syst., 27(5):1537–1550, may 2016. ISSN 1045-9219. +doi:10.1109/TPDS.2015.2442980. +Yehia Arafa, Abdel-Hameed Badawy, Ammar ElWazir, Atanu Barai, Ali Eker, Gopinath Chennupati, Nandakishore +Santhi, and Stephan Eidenbenz. Hybrid, scalable, trace-driven performance modeling of GPGPUs. In Proceedings of +the International Conference for High Performance Computing, Networking, Storage and Analysis, SC ’21, 2021. +ISBN 9781450384421. doi:10.1145/3458817.3476221. +Jack Choquette and Wish Gandhi. NVIDIA A100 GPU: Performance & innovation for GPU computing. In 2020 IEEE +Hot Chips 32 Symposium (HCS), pages 1–43, 2020. doi:10.1109/HCS49909.2020.9220622. +S. Grauer-Gray, L. Xu, R. Searles, S. Ayalasomayajula, and J. Cavazos. Auto-tuning a high-level language targeted to +gpu codes. In Innovative Parallel Computing, InPar ’12, page 1–10, 2012. doi:10.1109/InPar.2012.6339595. +A. Karki, C. P. Keshava, S. M. Shivakumar, J. Skow, G. M. Hegde, and H. Jeon. Detailed characterization of deep +neural networks on gpus and fpgas. In Proceedings of the 12th Workshop on General Purpose Processing Using +GPUs, GPGPU ’19, page 12–21, 2019. doi:https://doi.org/10.1145/3300053.3319418. +Javad Rahimi, Joris Van Kerrebrouck, Bahawal Haq, Johan Bauwelinck, Gunther Roelkens, and Geert Morthier. +Demonstration of a high-efficiency short-cavity iii-v-on-si c-band dfb laser diode. IEEE Journal of Selected Topics in +Quantum Electronics, 28(3):1–6, 2022. doi:10.1109/JSTQE.2021.3122552. +Chen Sun, Daniel Jeong, Mason Zhang, Woorham Bae, Chong Zhang, Pavan Bhargava, Derek Van Orden, Shahab +Ardalan, Chandarasekaran Ramamurthy, Erik Anderson, Austin Katzin, Haiwei Lu, Sidney Buchbinder, Behrooz +Beheshtian, Anatoly Khilo, Michael Rust, Chen Li, Forrest Sedgwick, John Fini, Roy Meade, Vladimir Stojanovi´c, +and Mark Wade. TeraPHY: An o-band wdm electro-optic platform for low power, terabit/s optical i/o. In 2020 IEEE +Symposium on VLSI Technology, pages 1–2, 2020. doi:10.1109/VLSITechnology18217.2020.9265012. +Daniele De Sensi, Salvatore Di Girolamo, Kim H. McMahon, Duncan Roweth, and Torsten Hoefler. An in-depth analysis +of the slingshot interconnect. In Proceedings of the International Conference for High Performance Computing, +Networking, Storage and Analysis, SC ’20. IEEE Press, 2020b. ISBN 9781728199986. +M. R. Siavash Katebzadeh, Paolo Costa, and Boris Grot. Evaluation of an infiniband switch: Choose latency or +bandwidth, but not both. In 2020 IEEE International Symposium on Performance Analysis of Systems and Software +(ISPASS), pages 180–191, 2020. doi:10.1109/ISPASS48437.2020.00033. +Mallikarjun Vasa, Chun-Lin Liao, Sanjay Kumar, Ching-Huei Chen, and Bhyrav Mutnury. Pcie gen-5 design challenges +of high-speed servers. In 2020 IEEE 29th Conference on Electrical Performance of Electronic Packaging and Systems +(EPEPS), pages 1–3, 2020. doi:10.1109/EPEPS48591.2020.9231458. +Keun Sup Shim, Brian Greskamp, Brian Towles, Bruce Edwards, J. P. Grossman, and David E. Shaw. The specialized +high-performance network on anton 3, 2022. +25 + diff --git a/KdE2T4oBgHgl3EQfAgZ0/content/tmp_files/load_file.txt b/KdE2T4oBgHgl3EQfAgZ0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..34684a058663d0e09434258e4963a41bcefcc97a --- /dev/null +++ b/KdE2T4oBgHgl3EQfAgZ0/content/tmp_files/load_file.txt @@ -0,0 +1,1810 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf,len=1809 +page_content='EFFICIENT INTRA-RACK RESOURCE DISAGGREGATION FOR HPC USING CO-PACKAGED DWDM PHOTONICS A PREPRINT George Michelogiannakis Lawrence Berkeley National Laboratory mihelog@lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='gov Yehia Arafa Qualcomm Technologies, Inc yarafa@nmsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='edu Brandon Cook Lawrence Berkeley National Laboratory BGCook@lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='gov Liang Yuan Dai Columbia University ld2719@columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='edu Abdel-Hameed Badawy New Mexico State University badawy@nmsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='edu Madeleine Glick Columbia University msg144@columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='edu Yuyang Wang Columbia University yw3831@columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='edu Keren Bergman Columbia University bergman@ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='edu John Shalf Lawrence Berkeley National Laboratory jshalf@lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='gov January 11, 2023 ABSTRACT The diversity of workload requirements and increasing hardware heterogeneity in emerging high performance computing (HPC) systems motivate resource disaggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Disaggregation separates servers into their constituent compute and memory resources so that they can be allocated as required to each workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Previous work has shown the potential of intra-rack resource disaggregation, but it is not clear how to realize these gains and cost-effectively meet the stringent bandwidth and latency requirements of HPC applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' To that end, we describe how modern photonic components can be co-designed with modern HPC racks to implement flexible intra-rack resource disaggregation and fully meet the high escape bandwidth of contemporary multi chip module (MCM) packages and all chip types in modern HPC racks with negligible power overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' We show how to use distributed indirect routing to meet these demands without the need for significant complexity for reconfiguration that spatial optical switches require.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' We then show that intra-rack resource disaggregation implemented using emerging photonics and parallel optical wavelength-selective switches satisfies bit error rate (BER) and bandwidth constraints and provides an average application speedup of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='9% for 31 out-of-order CPU and 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='5% for 27 GPU benchmarks compared to a similar system that instead uses modern electronic switches for disaggregation, due to their higher latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Using observed resource usage from a production system, we estimate that an iso-performance intra-rack disaggregated HPC system using photonics would require 4× fewer memory modules and 2× fewer NICs than a non-disaggregated baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Keywords Photonics · disaggregation · AWGR · spatial arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='03592v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='DC] 9 Jan 2023 arXiv Template A PREPRINT 1 Introduction Leading high performance computing (HPC) systems are steadily embracing heterogeneity of compute and memory resources as a means to preserve performance scaling and reduce system power Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2012], Top [2018], Ujaldón [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This trend is already apparent with the integration of GPUs Mittal and Vetter [2015], Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2015], Gao and Zhang [2016] and is expected to continue with fixed-function or reconfigurable accelerators such as field programmable gate arrays (FPGAs), Milojicic [2020], Asaadi and Chapman [2017], Segal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2014], Hogervorst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2021], Lant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Dimond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2011], Ramirez-Gargallo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019], emerging customized accelerators, and heterogeneous memory Venkata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In addition, key HPC workloads show considerable diversity in computational and memory access patterns Michelogiannakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2022], Rodrigo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This expectation of resource heterogeneity, workload diversity, and today’s method of allocating resources to applications in units of statically-configured nodes where every node is identical and unused resources are left to idle, raises the concern of resource underutilization (referred to as “marooned resources”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' These marooned resources increase both capital and operational costs without improving performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This has led to the emergence of resource disaggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Disaggregation refers to decomposing servers into their constituent compute and memory resources so that these can be allocated as required according to the needs of each workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Hyperscale datacenters have readily embraced resource disaggregation and have demonstrated that it significantly improves utilization of GPUs and memory Guleria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019a], Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Taylor [2015], Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2022], Koh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019], Papaioannou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2016], Gonzalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019a], Guleria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Although file storage is routinely disaggregated in modern systems Per, Michelogiannakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2022], Petersen and Bent [2017], HPC has been slow to embrace disaggregation of compute and memory resources Glick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2021] due to the sensitivity of HPC workloads to bandwidth and latency that cannot be met by current PCIe/CXL or Ethernet link technologies used in contemporary disaggregated architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Studies showed that disaggregation only among resources in the same rack (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=', intra-rack resource disaggregation) in HPC could reduce resources by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='36% to 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='01% while avoiding the overhead of full-system disaggregation Michelogiannakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2022], but the impact of increased memory latency and specific architectural trade offs have not been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Thus, although disaggregation using electronic networks have been demonstrated in hyperscale datacenters Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Papaioannou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2016], Call et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], minimizing adverse effects to and addressing the stringent bandwidth density and latency demands of HPC workloads requires a thorough investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In this work, our contributions are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Firstly, we describe how to use emerging photonic links and switches to design modern and practical resource-disaggregated HPC racks based on an existing GPU-accelerated HPE/Cray EX supercomputer Per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Secondly, we show how state of the art commercially available photonics and advanced packaging multi chip modules (MCMs) meet bit error rate (BER) requirements, impose a negligible power overhead, and deliver sufficient bandwidth to satisfy the escape bandwidth of all chips in modern HPC racks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Thirdly, we show how to use distributed indirect routing and arrayed waveguide grating routers (AWGRs) Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019] to satisfy all bandwidth requirements without the overhead and latency for reconfiguration that spatial Seok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019a], Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2016] and wave-selective Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020] switches require.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Moreover, having demonstrated negligible adverse impact to all other metrics, we show that intra-rack disaggregation using emerging photonics provides an average application speedup of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='9% for 31 out-of-order (OOO) CPU and 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='5% for 27 GPU benchmarks compared to a similar system that instead uses state of the art electronic switches, which also increase power overhead by three orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Finally, based on observed resource usage, we estimate that a system based on racks using photonics for resource disaggregation can have 43% fewer overall chips compared to a non-disaggregated system with the same computational throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 2 Related Work Hyperscale datacenters predominantly focus on full-system resource disaggregation where applications can allocate fine-grain resources of different types, today typically graphical processing units (GPUs) Guleria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019a] and memory Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Lim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2009], Koh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019], Gonzalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], which are located anywhere in the system or within a group of racks Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Papaioannou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2016], Call et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In such a system, resources of the same type are typically placed in the same rack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' However, full-system, flexible, and fine-grain disaggregation introduces significant overhead because of the higher latency and lower bandwidth density of contemporary hardware that is used to implement resource disaggregation – typically PCIe, 100Gig Ethernet, and eventually compute express link (CXL) Van Doren [2019] over electronic links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This overhead does not simply increase power and procurement cost, but rather adds potentially substantial latency between key resources such as central processing units (CPUs) and memory that traditionally exhibit latency-sensitive 2 arXiv Template A PREPRINT communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The aforementioned studies quote a several orders of magnitude increase in network and memory latency due to full-system resource disaggregation to improve resource utilization by 35% at most Zervas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Another study found that application performance degradation depends on both network bandwidth and latency, but can still reach up to 40% even with high bandwidth, low-latency networks Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Work on SPEC and commercial benchmarks also found an up to 27% application slowdown due to the additional memory latency Abali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' A study on Microsoft’s Azure found a range of performance slowdowns up to 30% from an extra 65 ns to access main memory Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Software defined networks (SDNs) based on electrical networks fare no better in terms of overhead Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2016], Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2013], Call et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Hybrid full-system photonic–electronic approaches have also been proposed that rely on circuit switching Zervas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2018] for reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' As a result, a few studies call for intra-rack disaggregation in datacenters Taylor [2015], Lim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2009], Guleria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Even the low latency and high bandwidth density of modern photonics cannot fully satisfy the bandwidth, energy, and latency requirements of full system disaggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This makes system-wide disaggregation impractical in many cases Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Zervas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2018], Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019a], Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Recent full system approaches in high performance computing (HPC) rely on optics to connect CPUs and memory, and electronic switches for hard disk drives (HDDs) to increase resource CPU utilization by 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='6% and memory 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='5% Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In contrast, another study confirms that production HPC systems can reduce resources from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='36% to 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='01% with intra-rack disaggregation and still satisfy the worst-case average rack utilization Michelogiannakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Similar to datacenters, intra-rack disaggregation in HPC promises the lowest overhead and impact to applications Glick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Taylor [2015], Guleria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Related work has research other aspects necessary to make resource disaggregation practical in a system, such as job scheduling Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019], Agosta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2018], Amaral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2021], Domeniconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019], how the operating system (OS) and runtime should adapt Maccabe [2017], Hwu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2015], Shan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2018], programming of code portability in heterogeneous systems Gioiosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Agosta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2018], partitioning of application data Khaleghzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], fault tolerance Hussain [2020], how to fairly compare the performance of different het- erogeneous systems Jamieson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2018], and the impact of heterogeneous resources to application performance Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2017], Lastovetsky [2015], Venkata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' These are important but out of scope topics for our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1 Under-utilization in Production Systems We use NERSC’s Cori system as an exemplar production HPC system, while recognizing workload requirements on other systems may differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In NERSC’s Cori, before Perlmutter came online and thus Cori was runnign the full NERSC workload, three quarters of the time Haswell nodes use less than 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='4% of memory capacity (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1% for KNL nodes) and less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='46 GB/s of memory bandwidth Michelogiannakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' These observations are similar to observations collected on LANL clusters Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020] and Alibaba machines that execute batch jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Likewise, half of the time Cori nodes use no more than half of their compute cores and three quarters of the time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='25% of available network interface controller (NIC) bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Similarly, in Lawrence Livermore National Laboratory (LANL) clusters, approximately 75% of the time, no more than 20% of memory capacity is used Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Alibaba’s published data Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019] show that memory is underutilized similarly to Cori for machines that execute batch jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Data from Google systems shows that memory and disk capacity of tasks is spread over three orders of magnitude and typically underutilized Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Azure reports 25% of memory under-utilization Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Datacenters have also reported 28% to 55% CPU idle in the case of Google trace data Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2015] and 20% to 50% most of the time in Alibaba Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Early studies also suggest GPU under-utilization Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2015], Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019], Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 3 Photonics for Resource Disaggregation Here we walk through available optical link and switch technologies and argue that photonics today meet the strict performance and error rate requirements to efficiently implement intra-rack resource disaggregation in HPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1 Memory Technologies and Requirements IO systems in HPC are already largely disaggregated over conventional system-scale interconnects since the underlying technologies (disk or SSD) are relatively high latency and lower bandwidth Michelogiannakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2022], Terzenidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' By contrast, memory technologies (particularly high bandwidth memorys (HBMs) needed by GPUs) are much higher bandwidth and much less tolerant of latency and require much lower bit error rates (BERs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Given that memory disaggregation imposes the most challenging constraints among other resources in today’s compute nodes, we 3 arXiv Template A PREPRINT Figure 1: Logical schematic of a DWDM link using ring resonator technology and a comb-laser source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Each ring is tuned to a different frequency of light and can be used to modulate that specific wavelength of light (a channel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Comb laser sources provide a comb of frequencies of light to provide those wavelengths for encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' All of the encoded optical channels share the same optical fiber and are decoded using the rings on the receiving side to route channels to the photodetectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Aggregated comb laser sources Active photonic interposer (a) Active Optical MCM (b) Blade Optical Circuit Switches Optical Fiber (c) Rack/Pod Figure 2: Overall physical structure of Rack/Pod scale resource disaggregation from photonically connected MCMs to the Rack/Pod scale pooling of disaggregated resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The conversion from CXL-over-fiber to HBM or NVM electrical protocol is implemented in the active interposer for the photonics MCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' will use DDR and HBM memory technology to set our performance target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' A typical DDR4 memory has a response latency of approximately 90 ns and for HBM the average response latency is 90-140 ns Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Still, any added latency between the CPU and memory from resource disaggregation may penalize application performance as we quantify later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Server-class memories typically require BERs of less than 10−18 to achieve tolerable failures in time (FIT) rates with conventional single-error-correct/double-error-detect (SEC-DED) protection Meza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2015], Sridharan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Forward error correction (FEC) can reduce the BER, but with additional latency Luyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2 Optical Link Technologies We consider a range of photonic link technologies that include conventional 100 Gbps Ethernet physical interfaces that represent the current baseline link technology for memory disaggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' We also introduce a range of cutting-edge dense wavelength division multiplexing (DWDM) link technologies that are either demonstrated as research prototypes or are commercially available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The photonic components all come from existing commercial technologies (100 Gbps, 400 Gbps, Ayar TeraPhy) and some research prototypes from DARPA PIPES (the 1-2 Tb link technologies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' These higher performance link technologies must be co-packaged in order to achieve their bandwidth density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' These link technologies are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The technology for the optical links is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Delivering multiple channels of laser light to the package has been challenging to scale cost-effectively if each "color" of light were to require a separate laser source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This concern was alleviated by the emergence of quantum dot and soliton comb laser sources shown in Figure 4 that can produce hundreds of usable light frequencies with wall-plug efficiencies of up to 41% Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 4 7NNN00NNN:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='MMCustomAccelGPuCPUDisaggregatedresourceRack/PodarXiv Template A PREPRINT Figure 3: Copackaged optics are required for DWDM link technologies to achieve the kind of bandwidth density required to operate at native memory bandwidths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3 Active Photonic MCMs Many CPUs and GPUs do not have the necessary off-chip bandwidth for full utilization of their compute resources because operating their I/O pins at higher bandwidth incurs a power cost Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2017], Jouppi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Using emerging high-speed optical links directly to the MCM, illustrated in Figure 3 provides to the order of 10× gains in escape bandwidth Glick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Wade [2019], Bergman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2018], Maniotis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This is a necessary property to enable efficient resource disaggregation as well as handle changing bandwidth requirements of key applications such as machine learning that drastically shifts bandwidth between inter-GPUs and off-chip from inference to training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' MCMs with integrated photonics have been demonstrated in both 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='5D and 3D interposer platforms Glick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Minkenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2021], Sutono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [1998], Abrams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' They can use different die-to-die link standards such as UCIe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Active interposer platforms combine the photonic integrated circuit (PIC) and interposer into a single integrated substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The active interposer allows photonic components to be fabricated and directly integrated with through silicon vias (TSVs) and additional metal redistribution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Electronic circuits are flip-chipped on top of active interposers using copper pillars Dittrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Further work has embedded photonic switch fabrics within MCM platforms with a crosstalk suppression and extinction ratio of >50dB and on-chip loss as low <1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='8dB Glick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='5DIntegratedTransceiver ComputeNode Optical Fibers FCBumps PIC FC Bumps EIC Interposer ASIC/FPGA PCBEIO PIC 25umpitchpad array to be bumpec 25umpitchpad arraytobeb MCM C3 ElectricalloonPcBto be wirebondedto PiC EIC PICarXiv Template A PREPRINT Figure 4: Comb laser sources provide the many discrete optical frequencies for the DWDM link with up to 41% experimentally measured conversion efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This was further scaled up to support more than 100 ports with microring resonators using a scalable switch fabric that combined switching in the space domain with wavelength-selectivity to define fine-grained connectivity for node disaggregation Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Glick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1 Link Protocol We adopt CXL as our link protocol Van Doren [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' CXL is an overlay on the PCIe-Gen6 physical layer, it includes guaranteed ordering of events and is a broadly adopted industry standard with published specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' However, our study does not rely on any features of any particular protocol so alternatives such as UCIe also apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2 Link Propagation and Encoding/Decoding Latency The target reach for an intra-rack disaggregation solution is approximately 1-4 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Given the speed of light c and light propagating through optical material that has an index of refraction that is near r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='5, the effective latency of propagating through an optical fiber at nominally 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='75c is approximately 5 ns per meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Therefore, rack-scale disaggregation adds 5-15 ns to our latency budget, less than 20% of the typical DRAM latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The link latency for SERDES and photonic ring modulation is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Intra-rack fiber lengths up to 4 meters require no intervening Electrical Optical (OEO) conversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3 Bit Error Rates and FEC To achieve 10−18 BER required for memory technologies, FEC Luyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2012] will likely be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Using the lightweight FEC scheme that is proposed for CXL Van Doren [2019] and PCIe Gen6 Sharma [2020] as an example, the all-inclusive latency for FEC can be as low as 2 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Therefore, for 200 Gbps, serialization delay is 10 ns and the FEC calculations add 2-3 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' At 400 Gbps and above, the net latency for FEC would be 5 ns plus 2-3 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Of note, this approach to achieving these BER targets is achievable with less than a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1% bandwidth loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 6 100um-6 dBm 10 10 dBm S-band C-band L-band 0 Bm) 10 NO 20 30 40 1,520 1,540 1,560 1,580 1,600 Wavelength (nm)arXiv Template A PREPRINT BW (Gbps) Energy (pJ/bit) Link Gbps × Chan- nels #Links (2 TB/s es- cape) Agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Ws (2 TB/s es- cape) Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 100 30 25 × 4 160 480 Fathololoumi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2021], Agrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2016] 400 30 100 × 4 40 197 Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2015] 768 < 1 32 × 24 21 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='4 Wade [2019] 1,024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='45 16 × 64 16 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2 Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019b] 2,048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3 16 × 128 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='8 Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019b] Table 1: A range of WDM photonic link technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In terms of impact on BER, this PCIe/CXL-like correction scheme corrects all single bursts of up to 16 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Double bursts will likely be mis-corrected, but the chance of a bad flit decreases quadratically (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=', a flit BER of 10−6 becomes 10−12 as you need two error bursts per flit to fail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Each flit is protected with a strong 64-flit CRC such that the flit FIT rate (CRC escapes) is significantly less than one part per billion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Lastly, the FEC escapes become link retransmissions and the ASIC-to-ASIC connection sees close to zero errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' As a result, emerging memory fabric protocols such as CXL, which could be run over our evaluated physical links, are capable of achieving a BER rate that meets the stringent memory system requirements and minimizes performance loss due to retransmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='4 Optical Circuit Switch Technologies Motivated by minimizing latency, our vision for a disaggregated rack is to have photonically-enabled MCMs that are connected via an optical circuit switch, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Compute and memory chips would be in the center of the MCM and the edge of the MCM would contain co-packaged optical silicon in-package photonics (SiPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Switches with all-optical paths include spatial- and wave-selective approaches, shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1 Spatial Optical Switches In recent years, the primary switching cells investigated are microelectromechanical systems (MEMS) actuated couplers, Mach-Zehnder interferometers (MZIs), and microring resonators (MRRs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Taking after their free-space counterpart, photonic MEMS-actuated switches are broadband spatial switches that have demonstrated radix scaling up to 240×240 Seok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' However, MEMS switching cells generally require high driving voltages (greater than 20 V), which make them less attractive for co-integration with electronic drivers, but typically offer low inter-channel cross-talk and low optical losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Spatial switches can also use mirrors Cal, photonic integrated circuits Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2016], or tiled planar silicon photonics Seok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' MZI switches are more co-integration friendly compared to MEMS but have only been shown to scale up to 32×32 Ikeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This limit can be seen as a consequence of the higher insertion-loss scaling resulting from cascaded MZI cells, as well as the susceptibility of popular MZI topologies to first-order crosstalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The challenge for scaling-up the spatial approach is the quantization of package and MCM escape bandwidth and reduced configuration options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' For example, at 768 Gbps (the Ayar TeraPhy Wade [2019]), the number of fibers escaping the package is 21 fibers, which means the package can be connected only up to 21 different potential destinations using a spatial switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 7 arXiv Template A PREPRINT Switch Type Radix Wave- lengths per port B/W per channel (wave- length) Insertion Loss Crosstalk Mach- Zehnder based Ikeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020] 32×32 1 439 Gbps 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='8 dB 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='6 dB MEMS- actuated Seok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019b] 240×240 1 – 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='8 dB 70 dB Microring res- onator Khope et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2017], Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019b] 8×8 (128×128) 8 (128) 100 Gbps (42 Gbps) 5dB (10dB) (-35 dB) Casc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' AW- GRs Sato [2018] 370×370 370 25 Gbps 15 dB 35 dB Table 2: High-radix CMOS-compatible photonic switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2 Wavelength Selective Optical Switches and AWGRs The inherent wavelength-selectivity of MRR switching cells allows for the straightforward implementation of wavelength-selective switching (WSS) topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This enables one to establish all-to-all networks by leveraging wavelength-division multiplexing (WDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Currently, MRR-based switches with the largest radix include the 8×8 crossbar Khope et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2017] and switch-and-select Nikolova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2017], but have been experimentally emulated to include a 16×16 Clos Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The metrics in Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020] can be seen to correlate very closely with the scaling proposed in Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019b], making a practical case for the 128×128 shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' All-to-all networks via WDM signals can also be achieved by arrayed waveguide grating routers (AWGRs) Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019], Proietti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2013], Lea [2015], Terzenidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' As AWGRs are passive optical elements, no reconfiguration is possible within the routing fabric itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Instead, fast wavelength-tunable lasers must be leveraged at the transmitter of every node if it wishes to address a different destination since AWGRs shuffle the light frequencies such that one lambda goes to each endpoint from each source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' AWGRs enable us to implement an N×N all-to-all topology using just O(N) fibers (each carrying N frequencies of light) whereas an implementation using copper would require N2 wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Although the cost of fast wavelength-tunable lasers is still an ongoing research topic Dhoore, Sören and Roelkens, Günther and Morthier, Geert [2019], AWGRs are mature, commercially available, and well established in literature FSp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In AWGRs, only a limited number of ports can be practically supported due to the walk-off of passband center frequencies from the carrier wavelength grid and the worse crosstalk associated with a larger number of ports (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' A feasible implementation of AWGR-based optical switches with large N has been demonstrated utilizing cascaded small-size AWGRs Sato [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Specifically, N M × M AWGRs (front-AWGRs) are interconnected with M N × N AWGRs (rear-AWGRs) to effectively act as an MN × MN AWGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Each output port of a front-AWGR is connected to an input port of a rear-AWGR, where the interconnection pattern can be optimized with knowledge of port-specific insertion losses to minimize the worst-case insertion loss of the aggregated AWGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Further up-scaling of the switch radix can be achieved by interconnecting small K × K delivery-coupling switchs (DC-switchs) with multiple copies of the MN × MN AWGRs, yielding a KMN × KMN switching capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This architecture has been verified by hardware prototypes of 270 × 270 and 1440 × 1440 Sato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2013], Ueda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2016], showing ∼15 dB insertion loss and below −35 dB crosstalk suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In order to accommodate the 350 MCMs of our rack, a reasonable 8 arXiv Template A PREPRINT 1 5 3 2 8 4 6 7 Figure 5: With an AWGR, endpoint 1 has one wavelength directly connecting it to endpoint 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' If it desires more bandwidth, it can route through another intermediate endpoint (indirect routing) chosen in a Valiant fashion Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Teh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Here, the link from 1 to 7 is available (green) but the link from 7 to 3 is not (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The chosen path is from 1 to 6 to 3 because both links are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' configuration is KMN = 3 × 12 × 11 = 396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This results in 370 ports and 370 wavelengths per port (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Since AWGRs typically have a 25 GHz optical bandwidth if the wavelength grid is 50 GHz, with PAM4 we assume 25 Gbps per wavelength Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Bhoja [2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Wave-selective switches Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Marom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2017] can steer any subset of wavelengths to a given destination, not just all (spatial) or one (AWGR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Dynamic programming methods can avoid sending the same frequency of light from two different sources to the same destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Since this is a relatively new technology, we constructed a model shown in Table 2 that projects the performance of a larger radix switch that is comprised of smaller demonstrated building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3 Reconfiguration Time Spatial and wave-selective switches typically require centralized scheduling Teh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020] to reach a steady globally optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The reconfiguration time can range from tens of nanoseconds to tens of milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In production HPC systems, multi-node jobs start every few seconds and last from minutes to hours Michelogiannakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Also, job resource usage and communication becomes predictable early, does not change fast, and typically remains predictable throughout a job’s execution time Michelogiannakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2022, 2019], Shalf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2005], Vetter and Mueller [2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Therefore, even milliseconds of reconfiguration time is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 4 Control Logic Here we describe how we can perform indirect routing to increase point-to-point bandwidth using only per-source logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1 Indirect routing in AWGRs AWGRs dedicate exactly one wavelength between any source–destination pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' If a source–destination pair requests more bandwidth than what a single wavelength can satisfy, sources can use indirect routing an example of which is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Sources can split traffic to N intermediate destinations in parallel in order to use the bandwidth of 9 arXiv Template A PREPRINT N wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This does not consume additional power in the photonic components assuming lasers are constantly powered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Sources consider indirect paths only if the direct (single-hop) bandwidth to their desired destination does not suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' A source considers indirect destinations for which the direct bandwidth from the source is available and whose wavelengths from the intermediate hop to the desired final destination is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Among potentially multiple candidates, sources choose one in a Valiant fashion Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Teh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], Domke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This is done on a per-flow basis in order to avoid out of order packet delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This routing logic can be modelled as an allocator problem and implemented with a low latency and area penalty Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2014], Becker and Dally [2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Indirect routing relies on sources knowing which other sources attached to the same AWGR are utilizing their local wavelengths in order to identify a productive intermediate destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' For instance, in Figure 5 endpoint 1 should know whether the wavelengths from 7 to 3 and 6 to 7 are occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' For that, we rely on piggybacking where traffic between a source and a destination periodically includes the state of the sources’s wavelengths as a way to broadcast local state to the rest of the endpoints attached to the same AWGR Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In the case of a N×N AWGR, each source uses N bits to encode which of its N local wavelengths it is using with one-hot encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Even if we piggyback this information multiple times a second, the bandwidth impact is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' For instance, if we multiplex multiple flows into a wavelength and therefore denote 8 bits per wavelength, the status vector per source becomes 256 × 8 = 2048bits = 256bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' If, due to stale information, sources pick an intermediate destination whose wavelength direct to the final destination is not available, the intermediate destination performs indirect routing through a second intermediate destination, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' If no data is exchanged between a pair, thus presenting no opportunity for piggybacking, that pair can exchange a separate control message with the same information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2 Spatial and Wave-Selective Switches Spatial and wave-selective switches can use indirect routing in tandem with reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Indirect routing reduces the need for reconfiguration, but intermediate hops should be chosen among hops that already have a direct connection with the final destination;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' otherwise, the intermediate hop itself may trigger a reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The synergy between indirect routing and switch reconfiguration was explored in Teh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 5 Disaggregated Rack Design For the rest of our study, we will model an HPC rack based on a GPU-accelerated HPE/Cray EX Supercomputer Per where a rack contains 128 GPU-accelerated nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Each node of our model system contains an AMD Milan CPU that has eight memory controllers each supporting a 3200MHz DDR4 module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Therefore, each CPU has 256 GB of memory with a maximum bandwidth of 204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='8 GBps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' A compute node also has four NVIDIA Ampere A100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Each GPU supports 12 third generation NVLink links each supporting 25 GBps per direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Each GPU also has 40 GB of co-located HBM with a bandwidth of 1555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2 GBps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Each node also has four 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='5 GBps PCI Gen4 links to connect each GPU to the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The CPU also connects to four Slingshot 11 NICs with 200 Gbps per direction De Sensi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Note that our photonic disaggregation hardware is orthogonal to and thus does not impair past work related to disaggregation such as runtimes, OS support, endpoint sharing management, and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1 MCMs and Escape Bandwidth We organize chips within each rack into an MCMs package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' For simplicity, we restrict all MCMs to have the same escape bandwidth and we place chips of only the same type in MCMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' We then make conservative assumptions for next generation photonics that are entering the market today based on our analysis of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In particular, each MCM has 32 optical fibers attached to it, a conservative assumption compared to the five arrays of 24 fibers demonstrated Hosseini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Each fiber supports 64 wavelengths (channels) of 25 Gbps each for a 6400 GBps escape bandwidth per MCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' We vary the number of chips per MCM such that each chip enjoys the same escape bandwidth as in our baseline rack Per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Therefore, our photonic architecture does not restrict chip escape bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Table 3 shows the number of chips per MCM and the total number of MCMs containing chips of that type to satisfy chip escape bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Each MCM contains a controller chip that interfaces the native protocol of the disaggregated resource to the CXL protocol over the photonic links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' CXL’s overhead and its associated FEC is included in our model of the overall architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2 Optical Switches The radix and wavelengths per port of optical switches dictate number of MCMs we can fully connect optically with a single switch as well as the amount of direct (single-hop) bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' From Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='4, we pick state-of-the-art representatives of wave-selective, cascaded AWGRs, and spatial optical switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Their parameters are shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Even though spatial Seok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019b] and wave-selective switches Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020] are capable of 100 10 arXiv Template A PREPRINT Chip type Chips per MCM # MCMs per rack CPU 14 10 GPU 3 171 NIC 203 3 HBM 4 128 DDR4 27 38 Total 350 Table 3: The number of chips ((CPU, GPU, NIC, HBM, or DDR4 module) per MCM and MCMs in a rack assuming 32 fibers per MCM, 64 wavelengths of 25 Gbps per fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The target BER to and from memory is 10−18 (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Switch type State of the art Switch radix Cascaded AWGRs Sato [2018] 370 Spatial Seok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019b] 240 Wave-Selective Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020] 256 Gbps per wavelength All switches 25 Wavelengths per port Cascaded AWGRs Sato [2018] 370 Spatial Seok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019b] 240 Wave-Selective Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020] 256 Table 4: Switch configuration for our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Gbps per wavelength, most links available widely today do not support that (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In addition, we show that we can still satisfy bandwidth demands with the conservative assumption of 25 Gbps per wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' To connect our 350 MCMs using 370×370 AWGRs, we can combine MCM fibers in five groups of six and connect each group to one port of five parallel AWGRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' However, this would require each AWGR port to handle 384 wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' To respect the per port 370 wavelength limitation of our AWGR configuration but still satisfy the full escape bandwidth of MCMs, we combine the remaining 14 wavelengths along with the remaining two fibers per MCM (128 + 14 = 142 wavelengths total) that were left unconnected into an extra parallel AWGR, for a total of six parallel AWGRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' We then connect MCM fibers to AWGRs in a staggered manner such that each MCM connects to each other MCM using at least five 25 Gbps direct-path wavelengths, for a direct MCM–MCM bandwidth of 25 × 5 = 125 Gbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' For simplicity, because of their relative small difference and because wave-selective switches can also achieve configu- rations that spatial switches can, we treat both wave-selective and spatial switches as 256 ports with 256 wavelengths per port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Each MCM can connect to 2048 256 = 8 parallel switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' However, because the radix of optical switches is lower than the number of MCMs, we instantiate 11 optical switches and connect MCMs in a staggered manner such that optical switch with an index I connects to MCMs that have an index starting from (32 × I) mod 350 until (I + 255) mod 350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This way, a small number of optical switch ports are left unconnected in order to not exceed the 32 fibers per MCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Similar to AWGRs, these ports can support future larger racks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' If the switches configure appropriately, each MCM has at least three direct paths to any other MCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Each path has 256 wavelengths, thus the direct MCM bandwidth is 256 × 3 × 25 = 2304 Gbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 6 Evaluation Having previously evaluated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3 that photonic switches satisfy BER requirements, in this Section we analyze the impact of photonic-based intra-rack resource disaggregation to bandwidth, latency, and power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' We then compare against electronic switches and estimate system-wide savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1 Bandwidth Evaluation We distinguish two test cases based on Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2: (A) Six parallel AWGRs and (B) 11 parallel wave-selective switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1 Available Bandwidth Using indirect routing and switch reconfiguration, any one particular MCM can use its full escape bandwidth to reach a single destination MCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In test case (A), all wavelengths escaping an MCM can reach the same destination MCM using indirect routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In test case (B), 768 wavelengths can be configured to route directly to a destination MCM 11 arXiv Template A PREPRINT and the other 2048 − 768 = 1280 wavelengths can be configured to route indirectly through intermediate MCMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This assumes that other MCMs will not contend for bandwidth that may disrupt indirect routing or complicate switch reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' While the direct (single-hop) bandwidth between cases (A) and (B) has a large difference, case (A) always provides that direct bandwidth between MCMs whereas a spatial or wave-selective switch requires a scheduler and leaves the majority of input–output combinations unconnected at any one time, thus also has to use indirect routing to compensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Based on system profiling data of a production open-science HPC system Michelogiannakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2022], the 125 Gbps direct bandwidth between MCMs in test case (A) suffices over 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='5% of the time between CPUs and main memory (DDR4) and virtually all the time between memory and NICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In addition, the bandwidth of a single AWGR wavelength of 25 Gbps suffices 97% of the time between CPUs and memory as well as between memory and NICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This means that with a 97% probability, four of the five wavelengths between a memory and CPUs or NICs and memory pair are available to use for indirect routing in case the direct 125 Gbps bandwidth does not suffice between another memory–CPU or NIC–memory pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Therefore, the probability at any one time that the direct bandwidth does not suffice for a number of CPU–memory and NIC–memory pairs large enough such that they cannot find unused bandwidth in other pairs to use for indirect routing is multiple orders of magnitude less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1% and thus negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' To further reduce the probability, congested pairs can use direct paths from CPUs to CPUs that communicate minimally and NICs to other NICs that do not communicate at all Michelogiannakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Therefore, test case (A) satisfies bandwidth between CPUs, NICs, and main memory (DDR4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Figure 6: Average and maximum slowdown for each suite and input set size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The slowdown is for an additional 35ns of latency between the LLC and main memory from the additional photonic components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Left: in-order pipeline compute cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Right: Out of order (OOO) compute cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' For GPUs, in test case (A) with indirect routing a single GPU can use a total of 125 × 512 = 8000 GBps to access any one HBM or more in case a GPU is allocated more than one HBMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This well satisfies the 1555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2 GBps that NVIDIA Ampere A100 GPUs in our model rack Per access HBMs with today and leaves 8000 − 1555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2 = 6444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='8 GBps unused per GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In addition, in the worst case, an MCM containing three GPUs will communicate at full bandwidth (12 NVLink links of 25 GBps per each of the three GPU equals 900 GBps) to other MCMs containing GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Here, if all GPUs in the rack acts similarly, we cannot rely on indirect routing from a GPU through an intermediate GPU to reach a destination GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The direct 125 Gbps bandwidth between GPU MCMs do not suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Therefore, each GPU can use the 6444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='8 GBps of unused bandwidth to and from HBMs for indirect routing to well cover the 900 GBps bandwidth that would otherwise use NVLink GPU–GPU links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This leaves 6444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='8 − 900 = 5544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='8 GBps per GPU that can support direct HBM–HBM communication such as due to GPUDirect RDMA, indirect routing for other MCMs, or simply increase available bandwidth to memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Of note, our analysis does not use direct optical paths from GPUs to main memory (DDR4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Future protocols may use for these paths or they can be used to provide even more indirect routing bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Our analysis shows that test case (A) with AWGRs more than satisfies bandwidth demands and avoids the need for a scheduler to reconfigure spatial and wave-selective switches that would otherwise add overhead and reduce reaction time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2 Latency Evaluation Intra-rack resource disaggregation based on modern photonics increases the latency significantly less than full system disaggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' For intra-rack disaggregation we assume an additional latency between MCMs of 35 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' That additional latency covers 15 ns for electrical–optical–electrical conversion and 4 meters of photonic propagation at 5 ns per meter, which covers round-trip distance of typical two-meter tall racks (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The small impact of distance to latency 12 Average Maximum 100 1 (%) 75 umopmos 50 Percentage 25 0 Parsec Parsec Parsec NAS A NAS B NAS C Rodinia small medium largeAverage Maximum 125 100 (%) slowdown 75 Percentage s 50 25 0 Parsec Parsec Parsec NAS A NAS B NAS C Rodinia small medium largearXiv Template A PREPRINT with photonics practically makes MCMs in a rack equi-distant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' thus mitigating a traditional queuing delay versus locality tradeoff in job scheduling Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Indirect routing would increase latency by a few extra ns, but the probability of routing indirectly is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Because 35 ns is orders of magnitude lower than system-wide network latency, we do not consider the effect of the additional 35 ns to inter-rack communication through NICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1 CPU Evaluation We experimentally quantify the impact to application performance with in-order pipeline and out-of-order (OOO) compute cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In-order cores provide insight of the impact of memory latency when the compute core does not mask latency, whereas OOO cores are representative of modern cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' We use full system simulation in Gem5 Binkert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2011] of x86 compute cores running an Ubuntu 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='4 guest OS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' We configure the cache hierarchy to match the CPUs of our model HPC rack Per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' We calculate the slowdown of application execution time when we add 35 ns of latency between the LLC and main memory, compared to a baseline system with no additional latency to memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Latency is the only potential source of application slowdown since our architecture satisfies the full escape bandwidth for each chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' We evaluate the impact in three benchmark suites: PARSEC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1 Bienia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2008], NAS parallel benchmarks 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1 Bailey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [1992], and Rodinia Che et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' For PARSEC we evaluate small, medium, and large input sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' For NAS, we evaluate input sizes “A”, “B”, and “C”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' For Rodinia we use the single default input set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' These benchmark suites have been widely used and contain a large variety of computation kernels that are representative of key HPC applications such as stencils, graph processing, linear algebra, computational mathematics, grid, sorting, and many others that have been observed to be important workloads in NERSC’s systems ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='. Overall, we use 58 benchmarks to provide a wide representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' We use a single compute core to better focus on the effect of the additional latency to memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Figure 6 shows slowdown percentages for benchmarks across our three suites for an in-order core on the left and OOO core on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' As shown, NAS benchmarks are negligibly affected by the increased latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Rodinia benchmarks have an average slowdown of 15% with in-order cores and 13% for OOO cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' However, a single benchmark (NW) has a slowdown of approximately 76% for in-order cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The largest slowdown for the rest of Rodinia benchmarks across in-order and OOO cores is 12%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Finally, PARSEC benchmarks are impacted the most, but the average slowdown remains below 25% except for large inputs using OOO cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' OOO cores typically tolerate memory access latency better, but they also produce more memory accesses per unit of time compared to in-order cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Figure 7 shows slowdown for individual PARSEC benchmarks for large inputs and in-order cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' As shown, only three benchmarks exceed a 25% slowdown, while eight benchmarks have a slowdown of no more approximately 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Therefore, our experiments show that while some benchmarks (three in PARSEC and one in Rodinia) experience important slowdowns, the majority of benchmarks are impacted minimally even without mitigation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This is the case with all of the NAS benchmarks we used, eight PARSEC, and all but one Rodinia benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' For benchmarks that are more affected, there is a range of hardware and software techniques Mutlu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2006], Parcerisa and Gonzalez [2001], Mowry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [1998], Nekkalapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2008] to increase memory tolerance that we can apply to further reduce application slowdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2 Recovering Performance To gauge the effectiveness of strategies to recover application performance, we test the impact of the following remedies applied one at a time: (i) 256 miss status handling registers (MSHRs) in the LLC, (ii) doubling the LLC size with the default number of 16 MSHRs, and (iii) default LLC configuration but a strided prefetcher with a larger stride than the default four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Figure 8 shows the slowdown percentage that we were able to recover for PARSEC benchmarks through these three techniques at the best, average, and worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This figure is the only one that includes these remedies in this results of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' As shown, about a 20% performance loss for small and large inputs is recovered by average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The most effective remedy is doubling the LLC size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The reason for the smaller speedup for medium is that due the particular LLC size, memory access patterns, and input sizes in PARSEC, medium experienced a smaller benefit from a larger LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' These findings motivate future work to mitigate the latency impact of the disaggregation hardware, similar to mitigating the increased latency to access emerging memory technologies Mittal and Vetter [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3 Sensitivity to Latency To show the sensitivity of application performance to the amount of additional latency, Figure 9 shows application slowdown for 25 ns, 30 ns, and 35 ns for in-order cores (OOO cores show comparable trends).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' As shown, reducing the additional latency to 25 ns from 35 ns reduces application slowdown by as much as half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This motivates latency improvements in photonic components or shorter rack distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 13 arXiv Template A PREPRINT Figure 7: Results for individual PARSEC benchmarks with large inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Figure 8: Percentage of slowdown that we can recover with LLC modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 14 Parsec slowdown: Large inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Single in-order core 100 75 (%) easad 50 25Best case Average Worst case (%) 50 recovered 40 30 slowdown 20 Percentage 10 0 Parsec small Parsec medium Parsec largearXiv Template A PREPRINT Figure 9: Percentage slowdown for 25ns, 30ns, and 35ns of additional LLC–memory latency for in order cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The overall average slowdown across all benchmarks is approximately 13% for both in-order cores and OOO cores without architectural remedies, for large PARSEC inputs, and “B” size NAS inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This considerably less than slowdowns quoted in past work for full-system disaggregation, furthering the case for intra-rack disaggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='4 GPU Evaluation To evaluate the impact of the additional latency between GPUs and HBMs or DDR4 main memory, we extend the publicly available version of PPT-GPU Arafa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2021] toolkit to account for the additional latency between the main memory of the GPU and the LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In our evaluation, we modeled one NVIDIA A100 GPU Choquette and Gandhi [2020] running a total of 27 applications that have a total of 2133 kernels from different benchmark suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' We run 13 applications from Rodinia Che et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2009] and 10 applications from Polybench Grauer-Gray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Polybench applications are linear algebra applications that stress the GPU cache and main memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Furthermore, we run AlexNet, CifarNet, GRU, and LSTM from the Tango deep network Karki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2019] benchmark suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' We use the default input sizes and configuration that came with the benchmarks, detailed in Arafa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' We run applications using the “SASS” model, where we extract memory and instruction traces for each application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Figure 10 shows the effect of different latencies on the performance of our GPU benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' We compare performance in terms of the total predicted cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' As shown, the highest average slowdown is 24% for Polybench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The overall average slowdowns across the 27 applications is only 8%, 10%, and 12% for the 25 ns, 30 ns, and 35 ns additional latency, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' For these benchmarks, doubling the LLC size recovers an average of 8% of the performance loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='5 CPU–GPU Comparison We illustrate the difference in memory latency tolerance of in-order CPUs, OOO CPUs, and GPUs in Figure 11 for the intersection of Rodinia benchmarks that correctly ran on both CPU and GPU with their default input sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' As shown, GPUs tolerate the additional 35 ns latency significantly better with a maximum slowdown of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This is promising for resource disaggregation given the steady growth of GPU presence in HPC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3 Power Overhead We calculate the per-rack power overhead of our photonic solution for 350 MCMs with 2048 escape wavelengths from each MCM and 25 Gbps per wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' If we use a DFB laser array demonstrated in Rahimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2022] with a 11% wall plug efficiency (WPE) at 10 dDm, a total of 256 × 256 such lasers consumes 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='5 kW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' For the components and distances in our study, the required optical power per wavelength is 10 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Furthermore, 350×2048 of the modulators 15 25ns 30ns 35ns 100 75 50 25 0 Parsec par ROarXiv Template A PREPRINT Figure 10: Percentage slowdown for 25ns, 30ns, and 35ns of additional LLC–memory latency for different GPU benchmark suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Figure 11: Percentage slowdown for CPU and GPU Rodinia benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 16 25ns 30ns 35ns 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='00% % 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='00% S 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='00% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='00% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='00% Rodinia-avg Rodinia-max PBench-avg PBench-max Tango-avg Tango-max35ns in-order CPU 35nsO0OCPU 35ns GPU 80 (%) 60 slowdown ( 40 ercentage 20 ParXiv Template A PREPRINT and receivers of Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020] that consume 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='12 pJ/bit at 25 Gbps respectively result in a total additional power of 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='5 kW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Finally, the switches of Table 2 consume no more than 1 kW at the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In summary, the total power overhead taking into account parallel switches is no more than 150 kW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Our analysis assumes the components are constantly on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Considering that the maximum power consumption of a single A100 GPU is a few hundreds of Ws and our modelled rack contains 512 such GPUs, the power overhead for our photonic solution is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='4 Comparison With Electronic Switches Electronic SERDES signalling rate per wire is only 112 Gbps for a short reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Also, typical CXL or PCIe signaling rates top out at 35 GHz/wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In fact, as SERDES rates increase, the distance that those signals can reach reduces down to even a few millimeters due to the resistance and capacitance of copper wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Photonics break the reach limitations of copper and with co-packaging can achieve 4 Tbps per mm of shoreline on the chip die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Focusing on electronic switches, Rosetta De Sensi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020b] and Infiniband Katebzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020] have a measured per hop latency of no less than approximately 200 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Emerging PCIe Gen5 switches add just 10 ns per hop Vasa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2020], but only support 100 lanes per switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' To fully connect our disaggregated rack, we consider a two-level tree network with four hops (the top level is composed of an internal two-hop subnetwork).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' These four hops will be in addition to the 35 ns we previously evaluated for FEC and propagation (propagation delay is comparable between copper and photonic for rack distances), since our photonic solution uses switches with negligible traversal latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Therefore, the additional latency for disaggregation in the PCIe case becomes 85 ns compared to 35 ns for our photonic architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Finally, we also consider the latency through one hop of an Anton 3 network, which is approximately 90 ns by average Shim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2022], though scaling up to match our rack size would require multiple hops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' These latencies represent the best case for electronic packet switches because scheduler decisions or congestion can cause higher worst-case (tail) latencies that may further penalize application performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This assumes that we connect only one lane per endpoint which carries 32 Gbps for PCIe Gen5 and 29 Gbps for Anton 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This is multiple times less than the per-chip bandwidth of photonics our photonic architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Figure 12 shows the speedup of a system that implements intra-rack disaggregation with emerging photonics with an additional 35 ns latency to and from DDR4 and HBM memory compared to a similar system that uses modern electronic switches instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 85 ns is the lowest case for electronic switches and corresponds to a four-hop PCIe Gen5 network or a single-hop Anton 3 network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' As shown, for CPU benchmarks if we only take into account “medium” from PARSEC to avoid counting PARSEC benchmarks three times, the average speedup for in-order CPUs is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='7% and the maximum 76%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' For OOO compute cores, the average is 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='9% and maximum 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' For GPUs, the average and maximum are both 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' This analysis clearly shows the adverse impact of the additional latency of electronic switches and further motivates the use of photonics for intra-rack resource disaggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Furthermore, the four electronic switches of this analysis consume at least many tens of Watts of power, which is multiple orders of magnitude higher than our photonic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='5 Iso-Performance Comparison Based on our performance evaluations, in order to preserve system-wide average computational throughput as our baseline GPU-accelerated HPE/Cray EX system Per, our photonically-disaggregated system requires 13% more CPUs and 8% more GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' However, intra-rack resource disaggregation allows our rack to have an average 4× fewer memory modules and 2× fewer NICs Michelogiannakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Combining the two effects, our disaggregated rack has 1082 total modules compared to 1920 in the baseline system, a 43% reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Alternatively, we can preserve all rack resources and instead add 128 of a combination of CPUs and GPUs (with their HBMs), which is only a 7% chip increase across the rack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Doing so doubles computational throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 7 Conclusion We have designed a resource disaggregated HPC rack that uses modern photonic links and switches to meet BER and bandwidth requirements of HPC applications, has a negligible power impact, uses distributed indirect routing instead of complex switch reconfiguration, and provides a 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='9% for CPUs or 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='5% for GPUs speedup compared to a similar disaggregated rack implemented with modern electronic switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Our architecture enables a disaggregated system to preserve its performance but use 43% fewer overall chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 17 arXiv Template A PREPRINT Figure 12: Speedup of a system that uses emerging photonics to implement intra-rack resource disaggregation that adds 35 ns of additional latency to and from memory compared to a similar system that uses modern electronic switches and adds 85 ns of memory latency instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' References B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Zydek, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Selvaraj, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Gewali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Accelerating high performance computing applications: Using cpus, gpus, hybrid CPU/GPU, and fpgas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2012 13th International Conference on Parallel and Distributed Computing, Applications and Technologies, pages 337–342, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The top500 HPC list, November 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='top500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='org/green500/lists/2018/11/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Ujaldón.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Hpc accelerators with 3d memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2016 IEEE Intl Conference on Computational Science and Engineering (CSE) and IEEE Intl Conference on Embedded and Ubiquitous Computing (EUC) and 15th Intl Symposium on Distributed Computing and Applications for Business Engineering (DCABES), pages 320–328, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/CSE-EUC-DCABES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Sparsh Mittal and Jeffrey S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Vetter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' A survey of cpu-gpu heterogeneous computing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=', 47 (4), July 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISSN 0360-0300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1145/2788396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Tiwari, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Gupta, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Rogers, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Maxwell, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Rech, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Vazhkudai, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Oliveira, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Londo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' DeBardeleben, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Navaux, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Carro, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Bland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Understanding gpu errors on large-scale hpc systems and the implications for system design and operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2015 IEEE 21st International Symposium on High Performance Computer Architecture (HPCA), pages 331–342, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/HPCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='7056044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Gao and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' A survey of homogeneous and heterogeneous system architectures in high performance computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2016 IEEE International Conference on Smart Cloud (SmartCloud), pages 170–175, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/SmartCloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Milojicic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Accelerators for artificial intelligence and high-performance computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Computer, 53(2):14–22, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Asaadi and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Chapman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Comparative study of deep learning framework in hpc environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2017 New York Scientific Data Summit (NYSDS), pages 1–7, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/NYSDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='8085040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Segal, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Nasiri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Margala, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Vanderbauwhede.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' High level programming of FPGAs for HPC and data centric applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2014 IEEE High Performance Extreme Computing Conference (HPEC), pages 1–3, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/HPEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='7040979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 18 CPUtiming CPUOOO GPU 80 Photonic speedup over electronic (%) 60 40 20 xew avg avg small avg max max avg avg large Prsc Prsc Prsc PrscarXiv Template A PREPRINT Tom Hogervorst, Tong Dong Qiu, Giacomo Marchiori, Alf Birger, Markus Blatt, and Razvan Nane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Hardware acceleration of HPC computational flow dynamics using HBM-enabled FPGAs, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Lant, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Navaridas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Luján, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Goodacre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Toward fpga-based HPC: Advancing interconnect technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE Micro, 40(1):25–34, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/MM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2950655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Rob Dimond, Sebastien Racanière, and Oliver Pell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Accelerating large-scale hpc applications using fpgas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2011 IEEE 20th Symposium on Computer Arithmetic, pages 191–192, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/ARITH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Ramirez-Gargallo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Garcia-Gasulla, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Mantovani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Tensorflow on state-of-the-art HPC clusters: A machine learning use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2019 19th IEEE/ACM International Symposium on Cluster, Cloud and Grid Computing (CCGRID), pages 526–533, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/CCGRID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='00067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Venkata, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Aderholdt, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Parchman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' SharP: Towards programming extreme-scale systems with hierarchical heterogeneous memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2017 46th International Conference on Parallel Processing Workshops (ICPPW), pages 145–154, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': 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+page_content=' Just Accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Rodrigo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Östberg, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Elmroth, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Antypas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Gerber, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Ramakrishnan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Towards understanding job heterogeneity in hpc: A nersc case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2016 16th IEEE/ACM International Symposium on Cluster, Cloud and Grid Computing (CCGrid), pages 521–526, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Guleria, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Lakshmi, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Padala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Quadd: Quantifying accelerator disaggregated datacenter effi- ciency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2019 IEEE 12th International Conference on Cloud Computing (CLOUD), pages 349–357, 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/CLOUD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='00064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Peng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Pearce, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Gokhale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' On the memory underutilization: Exploring disaggregated memory on HPC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2020 IEEE 32nd International Symposium on Computer Architecture and High Performance Computing (SBAC-PAD), pages 183–190, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Taylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Facebook’s data center infrastructure: Open compute, disaggregated rack, and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2015 Optical Fiber Communications Conference and Exhibition (OFC), pages 1–1, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Huaicheng Li, Daniel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Berger, Stanko Novakovic, Lisa Hsu, Dan Ernst, Pantea Zardoshti, Monish Shah, Ishwar Agarwal, Mark D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Hill, Marcus Fontoura, and Ricardo Bianchini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' First-generation memory disaggregation for cloud platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' arXiv, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='00241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Koh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Jeon, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Huh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Disaggregated cloud memory with elastic block management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE Transactions on Computers, 68(1):39–52, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Papaioannou, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Nejabati, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Simeonidou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The benefits of a disaggregated data centre: A resource allocation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2016 IEEE Global Communications Conference (GLOBECOM), pages 1–7, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/GLOCOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='7842314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Gonzalez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Gazman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Hattink, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Palma, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Bahadori, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Rubio-Noriega, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Orosa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Glick, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Mutlu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Bergman, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Azevedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Optically connected memory for disaggregated data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2020 IEEE 32nd International Symposium on Computer Architecture and High Performance Computing (SBAC-PAD), pages 43–50, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/SBAC-PAD49847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='00017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Qixiang Cheng, Keren Bergman, Yishen Huang, Hao Yang, Meisam Bahadori, Nathan Abrams, Xiang Meng, Madeleine Glick, Yang Liu, and Michael Hochberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Silicon photonic switch topologies and routing strategies for disaggregated data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE Journal of Selected Topics in Quantum Electronics, PP:1–1, 12 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/JSTQE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2960950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Guleria, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Lakshmi, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Padala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Emf: Disaggregated gpus in datacenters for efficiency, modularity and flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2019 IEEE International Conference on Cloud Computing in Emerging Markets (CCEM), pages 1–8, 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/CCEM48484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='000-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' NERSC’s Perlmutter configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' URL https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='nersc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='gov/systems/perlmutter/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Torben Kling Petersen and John Bent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Hybrid flash arrays for hpc storage systems: An alternative to burst buffers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2017 IEEE High Performance Extreme Computing Conference (HPEC), pages 1–7, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/HPEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='8091092.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Glick, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Abrams, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Cheng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Teh, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Hung, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Jimenez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Okawachi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Meng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Johansson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Ghobadi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Dennison, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Michelogiannakis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Shalf, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Bowers, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Gaeta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Lipson, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Bergman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' PINE: Photonic integrated networked energy efficient datacenters (enlitened program) [invited].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE/OSA Journal of Optical Communications and Networking, 12(12):443–456, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1364/JOCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='402788.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 19 arXiv Template A PREPRINT Xiaotao Guo, Xuwei Xue, Bitao Pan, Fulong Yan, Georgios Exarchakos, and Nicola Calabretta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Fosda: A hybrid disaggregated hpc architecture based on distributed nanoseconds optical switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In Future computing, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISBN 978-1-61208-846-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Cheng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Andrade, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Wosinska, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Disaggregated data centers: Challenges and trade-offs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE Communications Magazine, 58(2), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Aaron Call, Jordà Polo, David Carrera, Francesc Guim, and Sujoy Sen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Disaggregating non-volatile memory for throughput-oriented genomics workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' CoRR, abs/2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='02813, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Gengchen Liu, Roberto Proietti, Marjan Fariborz, Pouya Fotouhi, Xian Xiao, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Ben Yoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Architecture and performance studies of 3d-hyper-flex-lion for reconfigurable all-to-all hpc networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE Press, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISBN 9781728199986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Xiao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Li, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Samanta, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Shang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Proietti, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Okamoto, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Yoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Foundry-enabled scalable all-to-all optical interconnects using silicon nitride arrayed waveguide router interposers and silicon photonic transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE Journal of Selected Topics in Quantum Electronics, 25(5):1–9, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/JSTQE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2910415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Tae Joon Seok, Kyungmok Kwon, Johannes Henriksson, Jianheng Luo, and Ming C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Wafer-scale silicon photonic switches beyond die size limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Optica, 6(4):490–494, Apr 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1364/OPTICA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='000490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' URL http://opg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='optica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='org/optica/abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='cfm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='URI=optica-6-4-490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Yunhong Ding, Valerija Kamchevska, Kjeld Dalgaard, Feihong Ye, Rameez Asif, Simon Gross, Michael Withford, Michael Galili, Toshio Morioka, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Oxenlowe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Reconfigurable sdm switching using novel silicon photonic integrated circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Scientific Reports, 6, 08 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1038/srep39058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Yishen Huang, Qixiang Cheng, Anthony Rizzo, and Keren Bergman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Push—pull microring-assisted space-and- wavelength selective switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=', 45(10):2696–2699, May 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1364/OL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='392482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Kevin Lim, Jichuan Chang, Trevor Mudge, Parthasarathy Ranganathan, Steven K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Reinhardt, and Thomas F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Wenisch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Disaggregated memory for expansion and sharing in blade servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' SIGARCH Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Archit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' News, 37(3):267–278, jun 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISSN 0163-5964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1145/1555815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1555789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Van Doren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Abstract - hoti 2019: Compute express link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2019 IEEE Symposium on High-Performance Interconnects (HOTI), pages 18–18, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/HOTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='00017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Zervas, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Yuan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Saljoghei, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Chen, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Mishra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Optically disaggregated data centers with minimal remote memory latency: Technologies, architectures, and resource allocation [invited].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE/OSA Journal of Optical Communications and Networking, 10(2):A270–A285, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1364/JOCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='00A270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Peter X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Gao, Akshay Narayan, Sagar Karandikar, Joao Carreira, Sangjin Han, Rachit Agarwal, Sylvia Ratnasamy, and Scott Shenker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Network requirements for resource disaggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In Proceedings of the 12th USENIX Conference on Operating Systems Design and Implementation, OSDI’16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' USENIX Association, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISBN 9781931971331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Bulent Abali, Richard J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Eickemeyer, Hubertus Franke, Chung-Sheng Li, and Marc A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Taubenblatt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Disaggregated and optically interconnected memory: when will it be cost effective?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=', 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Sangjin Han, Norbert Egi, Aurojit Panda, Sylvia Ratnasamy, Guangyu Shi, and Scott Shenker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Network support for resource disaggregation in next-generation datacenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In Proceedings of the Twelfth ACM Workshop on Hot Topics in Networks, HotNets-XII, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISBN 9781450325967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1145/2535771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2535778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Cheng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' De Andrade, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Wosinska, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Resource disaggregation versus integrated servers in data centers: Impact of internal transmission capacity limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2018 European Conference on Optical Communication (ECOC), pages 1–3, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/ECOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='8535214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Yuping Fan, Zhiling Lan, Paul Rich, William E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Allcock, Michael E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Papka, Brian Austin, and David Paul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Scheduling beyond CPUs for HPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Proceedings of the 28th International Symposium on High-Performance Parallel and Distributed Computing, Jun 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Giovanni Agosta, William Fornaciari, Giuseppe Massari, Anna Pupykina, Federico Reghenzani, and Michele Zanella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Managing heterogeneous resources in HPC systems.' 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data-centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Journal of Cloud Computing, 10, 03 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1186/s13677-021-00238-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Domeniconi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Lee, Vanamala Venkataswamy, and Swaroopa Dola.' metadata={'source': 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Inter- national Workshop on Runtime and Operating Systems for Supercomputers ROSS 2017, ROSS ’17, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISBN 9781450350860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1145/3095770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3095771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Hwu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Chang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Kim, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Dakkak, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' El Hajj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Transitioning hpc software to exascale heteroge- neous computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2015 Computational Electromagnetics International Workshop (CEM), pages 1–2, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/CEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='7237412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Yizhou Shan, Yutong Huang, Yilun Chen, and Yiying Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Legoos: A disseminated, distributed os for hardware resource disaggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In Proceedings of the 13th USENIX Conference on Operating Systems Design and Implementation, OSDI’18, page 69–87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' USENIX Association, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISBN 9781931971478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Roberto Gioiosa, Ozcelik Burcu Mutlu, Seyong Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Jeffrey Vetter, Giulio Picierro, and Marco Cesati.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The minos computing library - efficient parallel programming for extremely heterogeneous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' GPGPU@PPoPP, pages 1–10, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Khaleghzadeh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Manumachu, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Lastovetsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' A hierarchical data-partitioning algorithm for performance optimization of data-parallel applications on heterogeneous multi-accelerator numa nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE Access, 8:7861– 7876, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Zaeem Hussain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Heterogeneity aware fault tolerance for extreme scale computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' August 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' URL http: //d-scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='pitt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='edu/39456/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Jamieson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Sanaullah, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Herbordt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Benchmarking heterogeneous hpc systems including reconfigurable fabrics: Community aspirations for ideal comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2018 IEEE High Performance extreme Computing Conference (HPEC), pages 1–6, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Tang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Tiwari, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Gupta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Vazhkudai, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Effective running of end-to-end hpc workflows on emerging heterogeneous architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2017 IEEE International Conference on Cluster Computing (CLUSTER), pages 344–348, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/CLUSTER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Alexey Lastovetsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Heterogeneous parallel computing: from clusters of workstations to hierarchical hybrid platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Supercomputing Frontiers and Innovations, 1(3), 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISSN 2313-8734.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Jing Guo, Zihao Chang, Sa Wang, Haiyang Ding, Yihui Feng, Liang Mao, and Yungang Bao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Who limits the resource efficiency of my datacenter: An analysis of alibaba datacenter traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In Proceedings of the International Symposium on Quality of Service, IWQoS ’19, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISBN 9781450367783.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1145/3326285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3329074.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Patel, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Jindal, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Yen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Bastani, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Xu, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Garraghan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Workload estimation 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Vikram K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Narayana, and Tarek A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' El-Ghazawi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Efficient resource sharing through GPU virtualization on accelerated high performance computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' CoRR, abs/1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='07658, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Myeongjae Jeon, 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+page_content=' Islam, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Spreitzer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Garag, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Alduino, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Chiou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Kamyab, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Gupta, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Xie, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Appleton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Hollingsworth, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' McCargar, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Akulova, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Brown, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Jones, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Zhu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Liljeberg, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Liao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='6 tbps silicon photonics integrated circuit and 800 gbps photonic engine for switch co-packaging demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Journal of Lightwave Technology, 39(4): 1155–1161, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/JLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3039218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Erik Agrell, Magnus Karlsson, A R Chraplyvy, David J Richardson, Peter M Krummrich, Peter Winzer, Kim Roberts, Johannes Karl Fischer, Seb J Savory, Benjamin J Eggleton, Marco Secondini, Frank R Kschischang, Andrew Lord, Josep Prat, Ioannis Tomkos, John E Bowers, Sudha Srinivasan, Mae Brandt-Pearce, and Nicolas Gisin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Roadmap of optical communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Journal of Optics, 18(6):063002, may 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1088/2040-8978/18/6/063002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Wei, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Cheng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Penty, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' White, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Cunningham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 400 gigabit ethernet using advanced modulation formats: Performance, complexity, and power dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE Communications Magazine, 53(2):182–189, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/MCOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='7045407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 22 arXiv Template A PREPRINT Bok Young Kim, Yoshitomo Okawachi, Jae K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Jang, Mengjie Yu, Xingchen Ji, Yun Zhao, Chaitanya Joshi, Michal Lipson, and Alexander L.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='0 specification at 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='0 gt/s with pam-4 signaling: a low latency, high bandwidth, high reliability and cost-effective interconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2020 IEEE Symposium on High-Performance Interconnects (HOTI), pages 1–8, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/HOTI51249.' metadata={'source': 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0030-4018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='optcom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='125677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Tae Joon Seok, Kyungmok Kwon, Johannes Henriksson, Jianheng Luo, and Ming C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 240×240 wafer-scale silicon photonic switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2019 Optical Fiber Communications Conference and Exhibition (OFC), pages 1–3, 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Akhilesh S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Khope, Andrew M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Netherton, Takako Hirokawa, Nicolas Volet, Eric J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Stanton, Clint Schow, Roger Helkey, Adel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Saleh, John E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Bowers, and Rod C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Alferness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Elastic wdm optoelectronic crossbar switch with on-chip wavelength control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In Advanced Photonics 2017 (IPR, NOMA, Sensors, Networks, SPPCom, PS), page PTh1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Optica Publishing Group, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1364/PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='PTh1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Qixiang Cheng, Meisam Bahadori, Yu-Han Hung, Yishen Huang, Nathan Abrams, and Keren Bergman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Scalable microring-based silicon clos switch fabric with switch-and-select stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE Journal of Selected Topics in Quantum Electronics, 25(5):1–11, 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/JSTQE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2911421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Ken-ichi Sato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Realization and Application of Large-Scale Fast Optical Circuit Switch for Data Center Networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Journal of Lightwave Technology, 36(7):1411–1419, April 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISSN 1558-2213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/JLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2801308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' S series optical circuit switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='calient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='net/products/s-series-photonic-switch/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Dessislava Nikolova, David M Calhoun, Yang Liu, Sébastien Rumley, Ari Novack, Tom Baehr-Jones, Michael Hochberg, and Keren Bergman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Modular architecture for fully non-blocking silicon photonic switch fabric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Microsystems & nanoengineering, 3(1):1–9, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Liang Yuan Dai, Yu-Han Hung, Qixiang Cheng, and Keren Bergman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Experimental demonstration of pam-4 transmission through microring silicon photonic clos switch fabric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2020 Optical Fiber Communications Conference and Exhibition (OFC), pages 1–3, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Proietti, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Yin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Yu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Nitta, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Akella, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Mineo, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Yoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Scalable optical interconnect architecture using awgr-based tonak lion switch with limited number of wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Journal of Lightwave Technology, 31(24): 4087–4097, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/JLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2285883.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Lea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' A scalable awgr-based optical switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Journal of Lightwave Technology, 33(22):4612–4621, Nov 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISSN 0733-8724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/JLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2479296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Dhoore, Sören and Roelkens, Günther and Morthier, Geert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Fast wavelength-tunable lasers on silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, 25(6):1500908:1–1500908:8, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISSN 1077-260X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Optical circuit switch product availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='com/products/66601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Ken-ichi Sato, Hiroshi Hasegawa, Tomonobu Niwa, and Toshio Watanabe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' A large-scale wavelength routing optical switch for data center networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE Communications Magazine, 51(9):46–52, September 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISSN 1558-1896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/MCOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='6588649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Koh Ueda, Yojiro Mori, Hiroshi Hasegawa, Hiroyuki Matsuura, Kiyo Ishii, Haruhiko Kuwatsuka, Shu Namiki, Toshio Watanabe, and Ken-ichi Sato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Demonstration of 1,440×1,440 fast optical circuit switch for datacenter networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2016 21st OptoElectronics and Communications Conference (OECC) Held Jointly with 2016 International Conference on Photonics in Switching (PS), pages 1–3, July 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Sudeep Bhoja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Pam4 signaling for intra-data center and data center to data center connectivity (dci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2017 Optical Fiber Communications Conference and Exhibition (OFC), pages 1–54, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Marom, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Colbourne, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' D’errico, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Fontaine, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Ikuma, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Proietti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Zong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Rivas-Moscoso, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Tomkos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Survey of photonic switching architectures and technologies in support of spatially and spectrally flexible optical networking [invited].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE/OSA Journal of Optical Communications and Networking, 9(1):1–26, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1364/JOCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='000001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Min Yee Teh, Yu-Han Hung, George Michelogiannakis, Shijia Yan, Madeleine Glick, John Shalf, and Keren Bergman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Tago: Rethinking routing design in high performance reconfigurable networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In SC20: Inter- national Conference for High Performance Computing, Networking, Storage and Analysis, pages 1–16, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/SC41405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='00029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 23 arXiv Template A PREPRINT George 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1145/3295500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3356145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' John Shalf, Shoaib Kamil, Leonid Oliker, and David Skinner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Analyzing ultra-scale application communication requirements for a reconfigurable hybrid interconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In Proceedings of the ACM/IEEE SC2005 Conference on High Performance Networking and Computing, November 12-18, 2005, Seattle, WA, USA, CD-Rom, page 17, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Jeffrey S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Vetter and Frank Mueller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Communication characteristics of large-scale scientific applications for contempo- rary cluster architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 16th International Parallel and Distributed Processing Symposium (IPDPS 2002), 15-19 April 2002, Fort Lauderdale, FL, USA, CD-ROM/Abstracts Proceedings, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/IPDPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1015504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Jens Domke, Satoshi Matsuoka, Ivan R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Ivanov, Yuki Tsushima, Tomoya Yuki, Akihiro Nomura, Shin’ichi Miura, Nie McDonald, Dennis L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Floyd, and Nicolas Dubé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Hyperx topology: First at-scale implementation and comparison to the fat-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC ’19, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISBN 9781450362290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1145/3295500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3356140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Sheng Ma, Zhiying Wang, Natalie Enright Jerger, Li Shen, and Nong Xiao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Novel flow control for fully adaptive routing in cache-coherent nocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE Transactions on Parallel and Distributed Systems, 25(9):2397–2407, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/TPDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Daniel U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Becker and William J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Dally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Allocator implementations for network-on-chip routers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In Proceed- ings of the Conference on High Performance Computing Networking, Storage and Analysis, pages 1–12, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1145/1654059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1654112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Nan Jiang, John Kim, and William J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Dally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Indirect adaptive routing on large scale interconnection networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In Proceedings of the 36th Annual International Symposium on Computer Architecture, ISCA ’09, pages 220–231, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISBN 978-1-60558-526-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1145/1555754.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1555783.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Daniele De Sensi, Salvatore Di Girolamo, Kim H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' McMahon, Duncan Roweth, and Torsten Hoefler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' An in-depth analysis of the slingshot interconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In SC20: International Conference for High Performance Computing, Networking, Storage and Analysis, pages 1–14, 2020a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/SC41405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='00039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Kaveh Hosseini, Edwin Kok, Sergey Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Shumarayev, Chia-Pin Chiu, Arnab Sarkar, Asako Toda, Yanjing Ke, Allen Chan, Daniel Jeong, Mason Zhang, Sangeeta Raman, Thungoc Tran, Kumar Abhishek Singh, Pavan Bhargava, Chong Zhang, Haiwei Lu, Ravi Mahajan, Xiaoqian Li, Nitin Deshpande, Conor O’Keeffe, Tim Tri Hoang, Uma Krishnamoorthy, Chen Sun, Roy Meade, Vladimir Stojanovic, and Mark Wade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 8 tbps co-packaged FPGA and silicon photonics optical IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2021 Optical Fiber Communications Conference and Exhibition (OFC), pages 1–3, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Nathan Binkert, Bradford Beckmann, Gabriel Black, Steven K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Reinhardt, Ali Saidi, Arkaprava Basu, Joel Hestness, Derek R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Hower, Tushar Krishna, Somayeh Sardashti, Rathijit Sen, Korey Sewell, Muhammad Shoaib, Nilay Vaish, Mark D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Hill, and David A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Wood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The gem5 simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' SIGARCH Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Archit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' News, 39(2):1–7, aug 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISSN 0163-5964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1145/2024716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2024718.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Christian Bienia, Sanjeev Kumar, Jaswinder Pal Singh, and Kai Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The parsec benchmark suite: Characterization and architectural implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2008 International Conference on Parallel Architectures and Compilation Techniques (PACT), pages 72–81, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Bailey, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Barszcz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Dagum, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Simon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Nas parallel benchmark results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In Supercom- puting ’92:Proceedings of the 1992 ACM/IEEE Conference on Supercomputing, pages 386–393, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/SUPERC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='236665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Shuai Che, Michael Boyer, Jiayuan Meng, David Tarjan, Jeremy W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Sheaffer, Sang-Ha Lee, and Kevin Skadron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Rodinia: A benchmark suite for heterogeneous computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2009 IEEE International Symposium on Workload Characterization (IISWC), pages 44–54, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/IISWC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='5306797.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Mutlu, Hyesoon Kim, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Patt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Efficient runahead execution: Power-efficient memory latency tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE Micro, 26(1):10–20, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/MM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Parcerisa and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Gonzalez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Improving latency tolerance of multithreading through decoupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE Transactions on Computers, 50(10):1084–1094, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='956093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Mowry, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Chan, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Lo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Comparative evaluation of latency tolerance techniques for software distributed shared memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In Proceedings 1998 Fourth International Symposium on High-Performance Computer Architecture, pages 300–311, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/HPCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='650569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 24 arXiv Template A PREPRINT Satyanarayana Nekkalapu, Haitham Akkary, Komal Jothi, Renjith Retnamma, and Xiaoyu Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' A simple la- tency tolerant processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2008 IEEE International Conference on Computer Design, pages 384–389, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/ICCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='4751889.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Sparsh Mittal and Jeffrey S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Vetter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' A survey of software techniques for using non-volatile memories for storage and main memory systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Parallel Distrib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=', 27(5):1537–1550, may 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISSN 1045-9219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/TPDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2442980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Yehia Arafa, Abdel-Hameed Badawy, Ammar ElWazir, Atanu Barai, Ali Eker, Gopinath Chennupati, Nandakishore Santhi, and Stephan Eidenbenz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Hybrid, scalable, trace-driven performance modeling of GPGPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC ’21, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISBN 9781450384421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1145/3458817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3476221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Jack Choquette and Wish Gandhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' NVIDIA A100 GPU: Performance & innovation for GPU computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2020 IEEE Hot Chips 32 Symposium (HCS), pages 1–43, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/HCS49909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='9220622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Grauer-Gray, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Xu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Searles, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Ayalasomayajula, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Cavazos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Auto-tuning a high-level language targeted to gpu codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In Innovative Parallel Computing, InPar ’12, page 1–10, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/InPar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='6339595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Karki, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Keshava, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Shivakumar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Skow, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Hegde, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Jeon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Detailed characterization of deep neural networks on gpus and fpgas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In Proceedings of the 12th Workshop on General Purpose Processing Using GPUs, GPGPU ’19, page 12–21, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1145/3300053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3319418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Javad Rahimi, Joris Van Kerrebrouck, Bahawal Haq, Johan Bauwelinck, Gunther Roelkens, and Geert Morthier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Demonstration of a high-efficiency short-cavity iii-v-on-si c-band dfb laser diode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE Journal of Selected Topics in Quantum Electronics, 28(3):1–6, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/JSTQE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='3122552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Chen Sun, Daniel Jeong, Mason Zhang, Woorham Bae, Chong Zhang, Pavan Bhargava, Derek Van Orden, Shahab Ardalan, Chandarasekaran Ramamurthy, Erik Anderson, Austin Katzin, Haiwei Lu, Sidney Buchbinder, Behrooz Beheshtian, Anatoly Khilo, Michael Rust, Chen Li, Forrest Sedgwick, John Fini, Roy Meade, Vladimir Stojanovi´c, and Mark Wade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' TeraPHY: An o-band wdm electro-optic platform for low power, terabit/s optical i/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2020 IEEE Symposium on VLSI Technology, pages 1–2, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/VLSITechnology18217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='9265012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Daniele De Sensi, Salvatore Di Girolamo, Kim H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' McMahon, Duncan Roweth, and Torsten Hoefler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' An in-depth analysis of the slingshot interconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' IEEE Press, 2020b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' ISBN 9781728199986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Siavash Katebzadeh, Paolo Costa, and Boris Grot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Evaluation of an infiniband switch: Choose latency or bandwidth, but not both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2020 IEEE International Symposium on Performance Analysis of Systems and Software (ISPASS), pages 180–191, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/ISPASS48437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='00033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Mallikarjun Vasa, Chun-Lin Liao, Sanjay Kumar, Ching-Huei Chen, and Bhyrav Mutnury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Pcie gen-5 design challenges of high-speed servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' In 2020 IEEE 29th Conference on Electrical Performance of Electronic Packaging and Systems (EPEPS), pages 1–3, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='1109/EPEPS48591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content='9231458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Keun Sup Shim, Brian Greskamp, Brian Towles, Bruce Edwards, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Grossman, and David E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' Shaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' The specialized high-performance network on anton 3, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} +page_content=' 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf'} diff --git a/KtAyT4oBgHgl3EQf6Prq/content/2301.00820v1.pdf b/KtAyT4oBgHgl3EQf6Prq/content/2301.00820v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f7f26a97982fa785f5f480124fa80e4897d60d2c --- /dev/null +++ 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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf,len=539 +page_content='Leukemia Detection Based on Microscopic Blood Smear Images Using Deep Neural Networks Abdelmageed Ahmed dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Engineering Electrical and Computer Engineering University of Ottawa Cairo, Egypt ahass202@uottawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='ca Ahmed Kamal dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' biomedical engineering department Minai university Minya, Egypt ahmd654@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='com Alaa Nagy dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Engineering Electrical and Computer Engineering University of Ottawa Cairo, Egypt aelba046@uottawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='ca Daila Farghl dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' biomedical engineering department Minai university Minya, Egypt dolly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='mostafa93@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='com Abstract— In this paper we discuss a new method for detecting leukemia in microscopic blood smear images using deep neural networks to diagnose leukemia early in blood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' leukemia is considered one of the most dangerous mortality causes for a human being, the traditional process of diagnosis of leukemia in blood is complex, costly, and time- consuming, so patients could not receive medical treatment on time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Computer vision classification technique using deep learning can overcome the problems of traditional analysis of blood smears, our system for leukemia detection provides 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='3 % accuracy in classifying samples as cancerous or normal samples by taking a shot of blood smear and passing it as an input to the system that will check whether it contains cancer or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' In case of containing cancer cells, then the hematological expert passes the sample to a more complex device such as flow cytometry to generate complete information about the progress of cancer in the blood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Keywords— Leukemia cells, leukemia detection, deep neural networks, deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' INTRODUCTION Leukemia is a type of cancer affecting blood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' if it is detected late, it will result in death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Leukemia develops when the bone marrow produces an excessive number of aberrant white blood cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The normal of the blood system will be disrupted when aberrant white blood cells are in excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Hematologists can identify abnormal blood when they draw a blood sample and study it[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' However, hematologists will inspect microscopic images visually, and the process is time- consuming and tiring [1 - 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Moreover, the process requires human experts and is prone to errors due to emotional disturbance and human physical capability, which has its limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Moreover, it is not easy to get consistent results from visual inspection [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Visual inspection can only give qualitative results for further research [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Studies indicate that the majority of modern methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Use all blood-related data, such as the number of red blood cells, hemoglobin level, hematocrit level, mean corpuscular volume, and much more, as the criterion for categorizing disorders like cancer, thalassemia, Etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Expensive testing and equipment labs are required to know all information about blood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' An automatic image processing system is urgently needed and can overcome related constraints in visual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The system to be developed will be based on microscopic images to recognize leukemia cells in blood smears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The early and fast identification of the leukemia type greatly aids in providing the appropriate treatment for a particular type of leukemia [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The currently used diagnostic methods rely on analyzing immuno- phenotyping, fluorescence in situ hybridization, cytogenetic analysis, and cytochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Sophisticated and expensive laboratories are required in order to run the diagnostic methods, and it has been reported to provide a high ratio of misidentification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' with this system, more images can be processed, reduce analyzing time, exclude the influence of subjective factors, and increase the accuracy of identification process at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' In machine learning, the inspection and classification of leukemia will be based on the texture, shape, size, color, and statistical analysis of white blood cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=" In contrast, deep learning makes it much more profound and gets the whole image's exclusive features." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' This project is applied to increase efficiency globally and can simultaneously benefit and be a massive contribution to the medical and pattern recognition field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The main objective is to enhance algorithms that can extract data from human blood where human blood is the primary source to detect diseases at an earlier stage and can prevent it quickly [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' This system should be robust towards diversity among individuals, sample collection protocols, time, Etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' This automated system can produce lab results quickly, easily, and efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' DATASET Images that were used in this project were downloaded from the internet and are available in ALL IDB[6], ASH Image Bank Hematology [7], Stock photo, vectors and Royalty-free Images[8], Shutter stock[9], Atlas of Hematology [10], Atlas of blood smear analysis[11], Blue Histology and American Society of Hematology [12], This dataset is composed of 630 images, contains 480 cancer images and 150 normal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=" Data Preprocessing 1) Remove duplication As the dataset is collected from various resources, had found that there are some repetitions, some images contain a watermark, and other contains websites' logo totally about 43 images, so now the data set has become 587 images." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 2) Resizing of images As the dataset has a different distribution of size, and for training the CNN model, it was needed to make all images in the dataset has the same size, so we applied a resizing technique and make all image 256 x 256 pixels to reduce the training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' as shown in figure [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='1] 3) Filtering images Before the processing stage, we need to remove noise and enhance line structures in images [13], and this is available by applying a median filter (3 x3) and sharpening the image (3 x3) ,as shown in Fig[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=" Figure 1:(a) original image,(b) image resized by 256*256 and filtered by median and sharpen filters 4) Data augmentation Image data augmentation is a method for artificially increasing the size of a training dataset by producing altered copies of the dataset's images [14]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The capacity of fit models to generalize what they have learned to new pictures may be improved by training deep-learning neural network models on more data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Additionally, augmentation techniques can provide variants of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Through the ImageDataGenerator class, the Keras deep learning neural network framework can fit models by adding picture data [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' There are many different types of augmentation techniques, some of them as: a) Flipping An image flip means reversing the rows or columns of pixels in the case of a vertical or horizontal flip [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' b) Horizontal and Vertical Shift Augmentation A shift to an image means moving all pixels of the image in one direction, such as horizontally or vertically, while keeping the image dimensions the same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' this means that some of the pixels will be clipped off the image, and there will be a region of the image where new pixel values will have to be specified [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' c) Random Zoom Augmentation A zoom augmentation randomly enlarges the image and either interpolate or adds new pixel values around the image [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' d) Shearing Shearing will automatically crop the correct area from the sheared image so that we have an image with no black space or padding [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' e) Interpolation (Nearest) A technique for creating new data points within the range of a discrete set of existing data points is interpolation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Nearest neighbor interpolation is the most straightforward approach to interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Rather than calculate an average value by some weighting criteria or generate an intermediate value based on complicated rules, this method simply determines the "nearest" neighboring pixel and assumes its intensity value of it [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' And Fig[2] indicates a sample image with its augmented one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' (a) (b) Fig 2: (a) original image and (b) augmented image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Processing stage After augmentation processes, our data become 1550 images for cancer and 1480 for normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' To fit data to models, we divided it through coding into three data sets: training set, validation set, and test set by ratios 60%, 20%, and 20%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Then the next stage is to train the model that can be able to classify the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Our optimizing parameters are accuracy and validation accuracy: to get the best of them as possible, we trained three networks with different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' a) BasicCNN model In this model, the input images were (RGB) color images with a resolution of 128x128 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' It consists of 3 convolutional layers with max pooling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' A rectified linear unit follows each convolutional layer (relu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' We used a constant filter size (3x3), and the number of (a) (b)Filters (128), the stride of ones (equal 1), and fully connected layers trained for two categories classification using the sigmoid activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Where we classified the data set into leukemia cells or normal cells, this architect achieved 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='99% accuracy and 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='97 % validation accuracy after 17 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='3: Indicates the block diagram of the basic CNN model b) Alexnet architecture In this study, we deployed the pre-trained AlexNet to detect ALL and classify its subtypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' This architecture was proposed by Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=', nine who deployed this architecture for the ImageNet Large Scale Visual Recognition Challenge 2012,20 and won the challenge in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Input images were Red, Green, and Blue (RGB) color images with a resolution of 227 x 227 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' It consists of 5 convolutional layers with three max polling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Each convolutional layer in AlexNet architecture is followed by a rectified linear unit (ReLU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' All the parameters, including the filter size, the number of filters, and the stride for each layer, are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' we replaced the SoftMax layer with a sigmoid layer as we want to classify the input image into only two types of this architect achieved 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='35% accuracy and 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='76 % validation accuracy after 12 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Figure 4: AlexNet architecture for acute lymphoblastic leukemia subtype classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Last 2 layers are newly added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' c) Modification of model used in published paper This used a retrained model that had been used in a published paper [20], shown in figure 5, and we changed the values of the hyperparameter to become as shown in figure 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' This network contains five layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The first three layers perform feature extraction, and the other two layers (fully connected and SoftMax) classify the extracted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The input image has a size of 128x128x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' In convolution layer 1, we used a constant filter size of 5x5 and a total of 16 different filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The stride is one, and no zero-padding was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The second and third convolution layers have the same structure as the first one but a different number of filters, 32 and 64, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' We used a pooling layer with filter size two and stride 2 to decrease the volume spatially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=" During the model we learned, the mini-batch's chosen size was 128, and ReLu was used as the activation function." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' This architect gives: accuracy = 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='73 % validation accuracy = 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='64 % Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 5: The original architecture of CNN in the mentioned paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 6: Architecture of CNN after changes in hyperparameter IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' EXPREMENTL RESULT Our experiments were conducted on Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='7 with 3030 images, 60% (1818 images) of them for training, 20% (606 images) for validation, and the remaining 20% (606 images) for testing our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' In order to evaluate each model and clarify the best one, we compare them by some statistically measured parameters: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Accuracy Train accuracy For the basic CNN model, train accuracy comes to 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='99% after 17 epochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' our leukemia classifier is doing an excellent classification, as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' For AlexNet architecture, the accuracy achieved its maximum accuracy of 56% after 11 epochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' that means our model is terrible on leukemia classification as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='1 b, but the Modification of the model used in Thanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' paper [18] achieved the maximum accuracy over all models 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='73 % after ten epochs as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='3 FC Max Max Max Conv layer Conv layer Conv layer Input image pooling pooling sigmoid pooling 128*128*3 63*63*128 61*61*128 30*30*128 28*28*12814*14*128 126*126*128 No padding No paddingFullyConnected Layer Fully Connected 4096 Layer Follo wedbyRelu 1024 L1 L2 L3 256 Norn Convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='.ReLu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='.MaxPolling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='soon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='384 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='Convolution5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='Convolution2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='ImageSize=13*13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='Image Size = 13*13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='InputImageSize ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='Convolution ' 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+page_content='Stride=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='Stride=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='Stride=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='Stride=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='Stride=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='No padding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='NopaddingValidation accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='Basic CNN Model validation accuracy reaches 85% after ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='17 epochs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Therefore, we expect our model to perform with ~85% accuracy on new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' For AlexNet architecture, the accuracy achieved its maximum accuracy of 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='6% after 11 epochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' that means our model is terrible on leukemia classification, as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='1 b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' This means that we expect our model to perform with ~53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='6% accuracy on new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Nevertheless, in Modification of the model used in Thanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' paper [94] achieved the maximum validation accuracy over all models at 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='3 % after ten epochs, as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Therefore, we expect our model to perform with ~94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='3 % accuracy on new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' We notice that our train metric increases as epochs increase while the validation accuracy metric decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' That means that our model fits the training set better but slightly loses its ability to predict new data, indicating that our models are beginning to overfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='1a, curve of val acc & train acc for basic CNN model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='2b, curve of val acc & train acc for AlexNet architecture Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='2c, curve of validation accuracy & train accuracy for Modification of model used in Thanh et al paper [18] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=" Confusion Matrix A classification problem's predicted outcomes are compiled in a confusion matrix." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The count values describe the number of accurate and inaccurate predictions for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Because it is feasible to see the relationships between the classifier outputs and the real ones, this is a great alternative for reporting results in M-class classification issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' For the basic CNN model, the number of leukemia images that are predicted as leukemia is 372, the number of leukemia images that are predicted as normal is 8, the number of normal images predicted as normal is 269, and the number of normal images that are predicted as leukemia is 51, as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='3 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' These accuracies show that this model is good at predicting leukemia images but bad at predicting normal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' For AlexNet architecture, the number of leukemia images that are predicted as leukemia is 0, the number of leukemia images that are predicted as normal is 380, the number of normal images predicted as normal is 157, and the number of normal images that are predicted as leukemia is 163, as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='3 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' These accuracies show that this model is terrible at predicting normal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The number of leukemia images predicted as leukemia for the modified model used in the published paper [18] is 369, the number of leukemia images predicted as normal is 11, the number of normal images predicted as normal is 301, and the number of normal images predicted as leukemia is 19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='3 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' These accuracies show that this model has done a great job of predicting normal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='3a, Confusion matrix of basic CNN model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='3b, Confusion matrix of AlexNet Architecture trainaccvsval acc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='4 Tain 2 8 numof Epochstrain acc vs val acc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='552 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='548 accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='546 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='544 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='542 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='540 train 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='538 val 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='536 0 2 4 6 8 10 12 numofEpochs0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='98 trainaccvsval acc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='92 train 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='90 val 0 1 2 3 4 5 6 7 8 numofEpochsConfusionmatrix 320 372 280 class o(cancer) 240 True label 200 160 120 class 1(normal) 51 269 80 40 Predicted labelConfusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='matrix 320 0 380 280 class O(cancer) 240 Truelabel 200 160 120 class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='1(normal) 163 157 80 40 Predicted label Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='3c, Confusion matrix for Modification of model used in Thanh et al paper [18] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Percsision It is calculated as the proportion of accurate positive results to those that the classifier predicted to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Our CNN model has medium precision, AlexNet architecture has very low precision, and the modified version of the model used in Thanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=" 's [18] paper has good precision due to its goodness method, as shown in fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Recall It is determined by dividing the total number of pertinent samples (all samples that should have been labeled as positive) by the total number of reliable positive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' As illustrated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='4a for our CNN model, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='4b for the AlexNet architecture, and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='4c for the Thanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' paper [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The perfect model regarded recall is the third model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The first CNN model in class 1 has a high recall but low precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' This means that most of the positive examples are correctly recognized (low FN), but there are a lot of false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Nevertheless, in class 0, low recall and high precision show that we miss a lot of positive examples (high FN), but those we predict as positive are indeed positive (low FP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' F1 Score The harmonic mean of recall and accuracy is the F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The F1 score has a range of [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' It tells how accurate the classifier is (how many instances it classifies correctly) and how robust it is (it recognizes a significant number of instances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' As illustrated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='4a for our CNN model, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='4b for the AlexNet architecture, and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='4c for the Thanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' paper [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' These figures show that the modification of the model used in Thanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=" 's paper [18] is precise and robust." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Support Support is the number of samples accurately representing the response within that category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' It provides information on the precise numbers of each class in the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Figures 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='4a and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='4b for the fundamental CNN model, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='4b for the AlexNet architecture, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='8c for a modified version of the model from the Thanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' work [18] serve as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='4a, values of precision, recall, f1 score and support for our CNN model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='4b, values of precision, recall, f1 score and support for AlexNet architecture Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='4c, values of precision, recall, f1 score and support for Modification of model used in Thanh et al paper [18] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=" DISCUSSION Leukemia is a malignancy that affects the body's blood- forming tissues, including the lymphatic system and bone marrow." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' To get the most effective treatment, the patient needs early Diagnosis, so we deploy three models using the power of CNN to classify blood smears into normal and abnormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Our dataset had not been taken under the same conditions as it was collected from various resources, and it needed to be bigger to use with DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' To overcome this problem, we used the power of data augmentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' this solution was suitable for us;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' our data before augmentation was 260 images, and after augmentation became 3030 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Our optimizing parameters were accuracy and validation accuracy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' by using CNN, we trained the model: the First model consists of 3 convolutional layers with max pooling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Its accuracy was 90% and 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='97 % validation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' It was terrible with our dataset due to its few layers, so we trained another model the Second model was AlexNet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' this architecture proved its efficiency in CNN models, so we trained it with our data, input is (RGB) color images with a resolution of 227 x 227 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' It consists of 5 convolutional layers with three max polling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' These models achieved 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='35% accuracy and 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='76 % validation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' We found that it does not fit our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' So we still have the same problem of low accuracy and keep looking for another model In the last model, we used a retrained model that had been used in a published paper [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' it contain7 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The first five layers perform feature extraction, and the other two layers (fully connected and SoftMax) classify the extracted Confusionmatrix 320 369 11 280 class o(cancer) 240 True label 200 160 120 class 1(normal 19 301 80 40 cer) Predicted labelprecision recall f1-score support class 0(cancerous) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='76 78 class 1(Normal) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='91 163precision recall f1-score class 1(cancer) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='00 178 class e(normal) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='66 172precision recall f1-score support class o(cancer) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='96 373 class 1(normal) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='95 327features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The input image has a size of 128x128x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' This architect has an accuracy of 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='73 % validation accuracy is 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='64 %, finally, we found that this model fit our data VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' CONCLUSIONS In this system, we investigated the application of deep CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' We deployed a pre-trained model for detecting and classifying the blood sample into normal and abnormal samples using microscopic blood sample images and convolutional neural network classification algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The system was built by deep learning, which uses all features in microscopic images, not only examining changes of specific features as a classifier input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=" We have performed the pre- trained model in a largely augmented dataset to confirm the system's accuracy and reliability." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' By performing data augmentation, we can achieve 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='3% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The system has high accuracy, and less processing time (show results in less than 30 seconds) , minor errors, and early identification of leukemia successful in giving the patient the proper care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' And cheaper cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' The detection system was built in three parts: 1) the acquisition part, which consists of a digital camera that has been installed at the top of the eyepiece of the microscope, 2) pre-trained CNN model responsible for the classification system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 3) a graphical user interface to display the image obtained from the camera and show the classification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' FUTURE WORK Expanding the focus on classifying the subtypes of leukemia cells such as Acute Myeloid Leukemia or AML, Chronic Myeloid Leukemia or CML, Acute Lymphoid Leukemia or ALL, and Chronic Lymphoid Leukemia or CLL not only separating between cancerous and non-cancerous cells and developing a convenient environment to construct an extensive leukemia dataset as this topic of research suffer from leaks in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' REFERENCES [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=', Ritter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=', Cooper, “Segmentation and Border Identification of Cells in Images of Peripheral Blood Smear Slides”, 30th Australasian Computer Science Conference, Conference in Research and Practice in Information Technology, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 62, 2007, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 161-169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=', Sabino, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=', Costa, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=', Rizzatti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=', Zago, “A Texture Approach to Leukocyte Recognition”, Real Time Imaging, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 10, 2004, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 205-206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' [3] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=', Colunga, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=', Siordia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=', Maybank, “Leukocyte Recognition Using EMAlgorithm”, MICAI 2009, LNAI 5845, Springer Verlag Berlin Heidelberg, 2009, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 545-555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Walter,” understanding leukemia”, 2009, pp- 8-18 [5] Warren S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' McCulloch and Walter Pitts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' “Neurocomputing: Foundations of Research”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' In: edited by James A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Anderson and Edward Rosenfeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Cambridge, MA, USA: MIT Press,1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Chapter A Logical Calculus of the Ideas Immanent in Nervous Activity, page15–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='ISBN:0-262-01097-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='URL: http://dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='org/citation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content='cfm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' id=65669.' metadata={'source': 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+page_content='com/mdbloice/Augmentor [18] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' Thanh, Caleb Vununu, Sukhrob Atoev, Suk-Hwan Lee, and Ki-Ryong Kwon , “Leukemia Blood Cell Image Classification Using Convolutional Neural Network “ International Journal of Computer Theory and Engineering, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 10, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'} +page_content=' 2, April 2018' metadata={'source': 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a/LtE1T4oBgHgl3EQftAUX/content/tmp_files/2301.03371v1.pdf.txt b/LtE1T4oBgHgl3EQftAUX/content/tmp_files/2301.03371v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7bd37f11dcf36964d80bfd30420f2c5e4515530b --- /dev/null +++ b/LtE1T4oBgHgl3EQftAUX/content/tmp_files/2301.03371v1.pdf.txt @@ -0,0 +1,3133 @@ +1 +Learning Optimal Phase-Shifts of Holographic +Metasurface Transceivers +Debamita Ghosh, IITB-Monash Research Academy, IIT Bombay, India +Manjesh K. Hanawal, MLioNS Lab, IEOR, IIT Bombay, India +Nikola Zlatanov, Innopolis University, Russia +Abstract—Holographic metasurface transceivers (HMT) is +an +emerging +technology +for +enhancing +the +coverage +and +rate of wireless communication systems. However, acquiring +accurate channel state information in HMT-assisted wireless +communication systems is critical for achieving these goals. +In this paper, we propose an algorithm for learning the +optimal +phase-shifts +at +a +HMT +for +the +far-field +channel +model. +Our +proposed +algorithm +exploits +the +structure +of +the channel gains in the far-field regions and learns the +optimal +phase-shifts +in +presence +of +noise +in +the +received +signals. +We +prove +that +the +probability +that +the +optimal +phase-shifts estimated by our proposed algorithm deviate from +the true values decays exponentially in the number of pilot +signals. Extensive numerical simulations validate the theoretical +guarantees and also demonstrate significant gains as compared +to the state-of-the-art policies. +Index Terms—Holographic Metasurface Transceivers, Channel +State Information, Uniform Exploration +I. INTRODUCTION +Future wireless network technologies, namely beyond-5G +and 6G, have been focused on millimeter wave (mmWave) +and TeraHertz (THz) communications technologies as possible +solutions to the ever growing demands for higher data rates and +lower latency. However, mmWave and THz communications +have challenges that need to be addressed before this technology +is adopted [1], [2]. One such major challenge is signal +deterioration due to reflections and absorption. +A possible solution for the signal deterioration are base +stations (BSs) with massive antennas arrays that can provide +large beamforming gains and thereby compensate for the +signal deterioration [3]. However, implementing a BS with +a massive antenna array is itself challenging due to the high +hardware costs. Holographic Metasurface Transceivers (HMTs) +are introduced as a promising solution for building a massive +antenna array [4], [5]. A HMT is comprised of a large number +of metamaterial elements densely deployed into a limited +surface area in order to form a spatially continuous transceiver +aperture. These metamaterial elements at the HMT acts as +phase-shifting antennas, where each phase-shifting element +of the HMT can change the phase of transmiting/receiving +signal and thereby beamform towards desired directions where +the users are allocated [6]. Due to these continuous apertures, +HMTs can be represented as an extension of the traditional +massive antenna arrays with discrete antennas to continuous +reflecting surfaces [6]. +In this paper, we consider the HMT-assisted wireless systems +illustrated in Fig. 1, where a HMT acts as a BS that serves +multiple users. The performance of this system is dependent on +channel state information (CSI) estimates at the HMT, which +are used for accurate beamforming towards the users. The +authors in [7] and [8] have studied the effect of HMT-assisted +systems on enhancing the communication performance under +the assumption of perfect CSI. However, perfect CSI is not +available in practice. In practice, the CSI has to be estimated +via pilot signals, which results in inaccurate CSI estimates at +the HMT. +The aim of this paper is to obtain accurate CSI estimates at +the HMT, which in turn is used to set the optimal phase-shifts +at the HMT that maximize the data rate to the users when +the users are located in the far-field. To this end, we exploit +the structure of the far-field channel model between the HMT +and the users to show that the optimal phase-shifts at the +HMT can be obtained from five samples of the received pilot +signals at the HMT in a noiseless environment. We then use this +approach to develop a learning algorithm that learns the optimal +phase-shifts from the received pilot signals at the HMT in a +noisy environment. Finally, we provide theoretical guarantees +for our learning algorithm. Specifically, we prove that the +probability of the phase-shifts generated by our algorithm to +deviate by more than 𝜖 from the optimal phase-shifts is small +and decays as the number of pilot symbols increases. The +error analysis is based on tail probabilities of the non-central +Chi-squared distribution. +In summary, our main contributions are as follows: +• We propose an efficient learning algorithm for estimating +the optimal phase-shifts at an HMT in the presence of +noise for the case when the users that the HMT is serving +are located at the far-field region. +• We prove that the probability of the phase-shifts generated +by our algorithm to deviate by more than 𝜖 from the +optimal phase-shifts is small and decays exponentially as +the number of pilots used for estimation increases. +• We show numerically that the performance of the +proposed algorithm significantly outperforms existing CSI +estimation algorithms. +A. Related Works +Several channel estimation schemes, which are proposed +for the massive antenna arrays, are also applicable to the +considered HMT including exhaustive search [9], hierarchical +search [10], [11], and compressed sensing (CS) [11]. As the +exhaustive search in [9] significantly increases the training +arXiv:2301.03371v1 [eess.SP] 12 Dec 2022 + +2 +overhead, the authors in [10] and [11] proposed the hierarchical +search based on a predefined codebook as an improvement over +the exhaustive search. The hierarchical schemes, in general, +may incur high training overhead and system latency since they +require non-trivial coordination among the transmitter and the +user [11]. On the other hand, the proposed CS-based channel +estimation scheme in [11] provides trade-offs between accuracy +of estimation and training overhead at different computational +costs. +On the other hand, CSI estimation schemes developed +specifically for HMTs can be found in [12] and [13]. The +authors in [12] proposed the least-square estimation based +approach to study the channel estimation problem for the +uplink between a single user and the BS equipped with the +holographic surface with a large number of antennas. However, +the authors require an additional knowledge of antennas array +geometry to reduce the pilot overhead required by the channel +estimation, and hence the computational complexity scales +up with the number of antennas at the BS. In [13], the +authors proposed a scheme for the estimation of the far-field +channel between a HMT and a user that requires only five +pilots for perfect estimation in the noise-free environment. +In the noisy case, the authors of [13] proposed an iterative +algorithm that efficiently estimates the far-field channel. Unlike +the existing works, the training overhead and the computational +cost of the proposed scheme in [13] does not scale with the +number of phase-shifting elements at the HMT. The iterative +algorithm in [13] significantly outperforms the hierarchical +and CS based schemes. However, the authors in [13] did not +provide any theoretical guarantees on their proposed algorithm. +Motivated by [13], in this work, we propose an algorithm +which outperforms the one in [13], and, in addition, we also +provide theoretical guarantees for our proposed algorithm. +This paper is organized as follows. The system and channel +models for the HMT communication system are given in Sec. +II. The proposed algorithm for learning the optimal phase-shifts +is given in Sec. III and its theoretical guarantee is provided +in Sec. IV. Numerical evaluation of the proposed algorithm is +provided in Sec. V. Finally, Sec. VI concludes the paper. +II. SYSTEM AND CHANNEL MODELS +We consider a HMT-assisted wireless communication system, +shown in Fig. 1, where an HMT communicates with multiple +users in the mmWave band. We assume that there is a Line +of Sight (LoS) between the HMT and each user. As a result, +when modeling the far-field channel, we only take into account +the LoS path since its power is order of magnitude higher than +non-line-of-sight (NLoS) paths [14]. The NLoS components +are incorporated in the noise. We assume that the users send +orthogonal pilots to the HMT for channel estimation. Based +on the estimated CSI at the HMT to each user, the HMT sends +data to the users. Hence, the data rate from the HMT to the +users is directly dependent on the accuracy of the CSI estimates +at the HMT. Since in this paper our main goal is the accurate +CSI estimation at the HMT to each user, which in turn send +orthogonal pilots to the HMT, in the rest of the paper, we will +focus on the CSI estimation between the HMT and a typical +user. +RF Generator +Phase-shifting +Element +User 1 +User 2 +User 3 +Fig. 1: The HMT-assisted wireless communication system [13]. +A. HMT Model +The HMT has a rectangular surface of size 𝐿𝑥 × 𝐿𝑦, where +𝐿𝑥 and 𝐿𝑦 are the width and the length of the surface, +respectively. The HMT’s surface is comprised of a large +number of sub-wavelength phase-shifting elements, where +each elements is assumed to be a square of size 𝐿𝑒 × 𝐿𝑒 +and can change the phase of the transmit/receive signal +independently from rest of the elements. Let 𝑑𝑟 be the +distance between two neighboring phase-shifting elements. +The total number of phase-shifting elements of the HMT is +given by 𝑀 = 𝑀𝑥 × 𝑀𝑦, where 𝑀𝑥 = 𝐿𝑥/𝑑𝑟 and 𝑀𝑦 = 𝐿𝑦/𝑑𝑟. +Without loss of generality, we assume that the HMT lies +in the 𝑥 − 𝑦 plane of a Cartesian coordinate system, where +the center of the surface is at the origin of the coordinate +system. Assuming 𝑀𝑥 and 𝑀𝑦 are odd numbers, the position +of the (𝑚𝑥,𝑚𝑦)𝑡ℎ phase-shifting element in the Cartesian +coordinate system is given as (𝑥, 𝑦) = (𝑚𝑥𝑑𝑟,𝑚𝑦𝑑𝑟), where +𝑚𝑥 ∈ +� +− 𝑀𝑥−1 +2 +,..., 𝑀𝑥−1 +2 +� +and 𝑚𝑦 ∈ +� +− 𝑀𝑦−1 +2 +,..., 𝑀𝑦−1 +2 +� +. When +𝑀𝑥 or 𝑀𝑦 is even, the position of the (𝑚𝑥,𝑚𝑦)𝑡ℎ element can +be appropriately defined. +B. Channel Model +Consider the channel between the (𝑚𝑥,𝑚𝑦)𝑡ℎ phase-shifting +element at the HMT and the typical user. Let the beamforming +weight imposed by the (𝑚𝑥,𝑚𝑦)𝑡ℎ phase-shifting element at +the HMT be Γ𝑚𝑥𝑚𝑦 = 𝑒 𝑗𝛽𝑚𝑥 𝑚𝑦 , where 𝛽𝑚𝑥𝑚𝑦 is the phase shift +at the (𝑚𝑥,𝑚𝑦)𝑡ℎ element. Let 𝜆 denote the wavelength of +the carrier frequency, 𝑘0 = 2𝜋 +𝜆 be the wave number, 𝑑0 be the +distance between the user and the center of the HMT and +let 𝐹𝑚𝑥𝑚𝑦 denote the effect of the size and power radiation +pattern of the (𝑚𝑥,𝑚𝑦)𝑡ℎ phase-shifting element on the channel +coefficient [15]. Due to the far-field assumptions, the radiation +pattern of all the phase-shifting elements of the HMT are +identical, i.e., 𝐹𝑚𝑥𝑚𝑦 = 𝐹, ∀𝑚𝑥,𝑚𝑦 holds. Finally, let 𝜃 and +𝜙 denote the elevation and azimuth angles of the impinging +wave from the user to the center of the HMT, see Fig. 2. +Now, if the phase-shift imposed by the (𝑚𝑥,𝑚𝑦)𝑡ℎ element, +𝛽𝑚𝑥,𝑚𝑦, is set to +𝛽𝑚𝑥𝑚𝑦 = − +mod (𝑘0𝑑𝑟 (𝑚𝑥𝛽1 +𝑚𝑦𝛽2),2𝜋),∀𝑚𝑥,𝑚𝑦, + +3 +User +HMT +Fig. 2: Distance between the (𝑚𝑥,𝑚𝑦)-th phase-shifting element at +the HMT and the user [13]. +where 𝛽1 and 𝛽2 are the phase-shift parameters [13], [16], +[17], which are the only degrees of freedom within the +phase-shift 𝛽𝑚𝑥𝑚𝑦, then the HMT-user channel in the far-field +is approximated accurately by [13], [16], [17] +𝐻(𝛽1, 𝛽2) = +�√ +𝐹𝜆𝑒−𝑗𝑘0𝑑0 +4𝜋𝑑0 +� +𝐿𝑥𝐿𝑦 ×sinc +� +𝐾𝑥𝜋(𝛼1 − 𝛽1) +� +×sinc +� +𝐾𝑦𝜋(𝛼2 − 𝛽2) +� +, +(1) +where 𝐾𝑥 = 𝐿𝑥 +𝜆 ,𝐾𝑦 = 𝐿𝑦 +𝜆 ,𝛼1 = sin(𝜃) cos(𝜙),𝛼2 = sin(𝜃) sin(𝜙), +and sinc(𝑥) = sin(𝑥) +𝑥 +. Please note that 𝛼1 ∈ [−1,1] and 𝛼2 ∈ +[−1,1], and their values depend on the location of the user, +i.e., on 𝜃 and 𝜙. +From (1), it is clear that the absolute value of the HMT-user +channel is maximized when the two sinc functions attain +their maximum values, which occurs when the phase-shifting +parameters, 𝛽1 and 𝛽2, are set to 𝛽1 = 𝛼1 and 𝛽2 = 𝛼2, where +(𝛼1,𝛼2) are unknown to the HMT since they depend on the +location of the user. Therefore, in the far-field case, the problem +of finding the optimal phase-shifts of the elements at the HMT +reduces to estimating the two parameters, 𝛼1 and 𝛼2 at the +HMT. +Remark 1. Fig. 3 shows an example of |𝐻(𝛽1, 𝛽2)| as a +function of (𝛽1, 𝛽2). As can be seen from Fig. 3, the graph +of |𝐻(𝛽1, 𝛽2)| hits zero periodically and has several lobes. +The optimal value (𝛼1,𝛼2) = (0.68,−0.45) is attained at the +central lobe which has the highest peak and is attained for +(𝛽∗ +1, 𝛽∗ +2) = (𝛼1,𝛼2) = (0.68,−0.45). +III. PROPOSED CHANNEL ESTIMATION STRATEGY +In this section, we propose an algorithm that estimates the +optimal phase-shifting parameters 𝛽1 and 𝛽2 that maximize +|𝐻(𝛽1, 𝛽2)| in (1) in the presence of noise. +A. Problem Formulation +In the channel estimation procedure, the user sends a +pilot symbol 𝑥𝑝 = +√ +𝑃 to the HMT, where 𝑃 is the pilot +transmit power. Then, the received signal at the HMT for +fixed phase-shifting parameters (𝛽1, 𝛽2), denoted by 𝑦(𝛽1, 𝛽2), +is given by +𝑦(𝛽1, 𝛽2) = +√ +𝑃 × 𝐻(𝛽1, 𝛽2) + 𝜁, +(2) +Fig. 3: |𝐻(𝛽1, 𝛽2)| v/s (𝛽1, 𝛽2) for values of (𝛼1,𝛼2) = (0.68,−0.45). +where 𝜁 is the complex-valued additive white Gaussian noise +(AWGN) with zero mean and variance 𝜎2 at the HMT. The +received signal in (2) is then squared in order to obtain the +received signal squared, denoted by 𝑟(𝛽1, 𝛽2), and given by +𝑟(𝛽1, 𝛽2) = |𝑦(𝛽1, 𝛽2)|2 = +��� +√ +𝑃 × 𝐻(𝛽1, 𝛽2) + 𝜁 +��� +2 +. +(3) +Objective: Our goal is to identify the optimal phase-shifting +parameters, denoted by (𝛽∗ +1, 𝛽∗ +2), at the HMT that maximizes +𝑟(𝛽1, 𝛽2) given by (3). Specifically, we aim to solve the +following optimisation problem +(𝛽∗ +1, 𝛽∗ +2) = argmax +𝛽1∈[−1,1] +𝛽2∈[−1,1] +𝑟(𝛽1, 𝛽2). +(4) +The expected value of 𝑟(𝛽1, 𝛽2), denoted by 𝜇(𝛽1, 𝛽2), is given +by +𝜇(𝛽1, 𝛽2) = E [𝑟(𝛽1, 𝛽2)] += +��� +√ +𝑃 × 𝐻(𝛽1, 𝛽2) +��� +2 ++ 𝜎2. +(5) +Using (5), the optimization problem in (4) can be written +equivalently as +(𝛽∗ +1, 𝛽∗ +2) = argmax +𝛽1∈[−1,1] +𝛽2∈[−1,1] +𝜇(𝛽1, 𝛽2). +(6) +In order to obtain an intuition on how to solve (6), we first +assume that 𝜇(𝛽1, 𝛽2) in (5) is known perfectly at the HMT for +five specific values of the pair (𝛽1, 𝛽2). Later, we use the same +intuition to solve (6) when 𝜇(𝛽1, 𝛽2) are not known perfectly +but can be estimated. +B. The Optimal Phase-Shifting Parameters When 𝜇(𝛽1, 𝛽2) +Are Known In Advance +For notational convenience, let us define the set B as +B = +� +(𝛽0 +1, 𝛽0 +2), (𝛽0 +1 +𝑣, 𝛽0 +2), (𝛽0 +1 −𝑣, 𝛽0 +2), +(𝛽0 +1, 𝛽0 +2 + 𝑤), (𝛽0 +1, 𝛽0 +2 − 𝑤) +� +. +(7) +The set B is comprised of five pairs of the phase-shifting +parameters (𝛽1, 𝛽2), where 𝛽0 +1 and 𝛽0 +2 are some initial arbitrarily +selected phase-shifting parameters, 𝑣 and 𝑤 are numbers chosen +such that 𝐾𝑥𝑣 ∈ N and 𝐾𝑦𝑤 ∈ N hold, where N is the set of +natural numbers. Please note that for a selected (𝛽0 +1, 𝛽0 +2) and a + +1 +(α_1,α2)=(0.68,-0.45) +0.8 +[H(β1, β2) /2 +0.5 +0.6 +0.4 +0.4 +β1 +01 +0.8 +0.6 +-0.6 +0.2 +0.4 +0.8 +0.2 +2 +β2 +0.4 +14 +chosen 𝑣 and 𝑤, if +��𝛽0 +1 ±𝑣 +�� ≥ 1 then we set +��𝛽0 +1 ±𝑣 +�� = 1. In the +same way, if +��𝛽0 +2 ± 𝑤 +�� ≥ 1 then we set +��𝛽0 +2 ± 𝑤 +�� = 1. +Theorem 1. If the HMT can obtain 𝜇(𝛽0 +1, 𝛽0 +2), 𝜇(𝛽0 +1 + 𝑣, 𝛽0 +2), +𝜇(𝛽0 +1 − 𝑣, 𝛽0 +2), 𝜇(𝛽0 +1, 𝛽0 +2 + 𝑤) and 𝜇(𝛽0 +1, 𝛽0 +2 − 𝑤), i.e., obtain +𝜇(𝛽1, 𝛽2) for the five phase-shifting parameters in (𝛽1, 𝛽2) ∈ B +given in (7), then the optimal phase-shifting parameters 𝛽∗ +1 +and 𝛽∗ +2, which are the solutions of (6), are given by +𝛽∗ +1 = +� +𝛼(𝑖) +1 +𝛼( 𝑗) +1 +2 +: +min +𝑖∈{1,2}, 𝑗 ∈{3,4} +���𝛼(𝑖) +1 −𝛼( 𝑗) +1 +��� +� +, +(8) +where +𝛼(1)/(2) +1 += 𝛽0 +1 + +𝑣 +1± +√︄���� +𝜇(𝛽0 +1,𝛽0 +2)−𝜎2 +𝜇(𝛽0 +1+𝑣,𝛽0 +2)−𝜎2 +���� +𝛼(3)/(4) +1 += 𝛽0 +1 − +𝑣 +1± +√︄���� +𝜇(𝛽0 +1,𝛽0 +2)−𝜎2 +𝜇(𝛽0 +1−𝑣,𝛽0 +2)−𝜎2 +���� +and +𝛽∗ +2 = +� +𝛼(𝑖) +2 +𝛼( 𝑗) +2 +2 +: +min +𝑖∈{1,2}, 𝑗 ∈{3,4} +���𝛼(𝑖) +2 −𝛼( 𝑗) +2 +��� +� +, +(9) +where +𝛼(1)/(2) +2 += 𝛽0 +2 + +𝑣 +1± +√︄���� +𝜇(𝛽0 +1,𝛽0 +2)−𝜎2 +𝜇(𝛽0 +1,𝛽0 +2+𝑤)−𝜎2 +���� +𝛼(3)/(4) +2 += 𝛽0 +2 − +𝑣 +1± +√︄���� +𝜇(𝛽0 +1,𝛽0 +2)−𝜎2 +𝜇(𝛽0 +1,𝛽0 +2−𝑤)−𝜎2 +���� +Proof. By using (5) and (1) for any (𝛽1, 𝛽2) = (𝛽0 +1, 𝛽0 +2), we +have the following +𝜇(𝛽0 +1, 𝛽0 +2) − 𝜎2 = +����� +√ +𝑃 +�√ +𝐹𝜆𝑒− 𝑗𝑘0𝑑0 +4𝜋𝑑0 +� +𝐿𝑥𝐿𝑦sinc +� +𝐾𝑥(𝛼1 − 𝛽0 +1) +� +×sinc +� +𝐾𝑦(𝛼2 − 𝛽0 +2) +������ +2 +. +(10) +For (𝛽1, 𝛽2) = (𝛽0 +1 +𝑣, 𝛽0 +2), where 𝑣 is any arbitrary parameter +such that 𝐾𝑥𝑣 ∈ N and +��𝛽0 +1 ±𝑣 +�� ≤ 1 holds, we have +𝜇(𝛽0 +1 +𝑣, 𝛽0 +2) − 𝜎2 = +����� +√ +𝑃 +�√ +𝐹𝜆𝑒−𝑗𝑘0𝑑0 +4𝜋𝑑0 +� +𝐿𝑥𝐿𝑦 +×sinc +� +𝐾𝑦(𝛼2 − 𝛽0 +2) +������ +2 +. +(11) +Dividing (10) by (11), we obtain +𝜇(𝛽0 +1, 𝛽0 +2) − 𝜎2 +𝜇(𝛽0 +1 +𝑣, 𝛽0 +2) − 𝜎2 = +����sinc +� +𝐾𝑥(𝛼1 − 𝛽0 +1) +����� +2 +����sinc +� +𝐾𝑥(𝛼1 − 𝛽0 +1 −𝑣) +����� +2 +𝜇(𝛽0 +1, 𝛽0 +2) − 𝜎2 +𝜇(𝛽0 +1 +𝑣, 𝛽0 +2) − 𝜎2 = +���� +sin(𝐾𝑥 𝜋(𝛼1−𝛽0 +1)) +𝐾𝑥 𝜋(𝛼1−𝛽0 +1) +���� +2 +���� +sin(𝐾𝑥 𝜋(𝛼1−𝛽0 +1−𝑣)) +𝐾𝑥 𝜋(𝛼1−𝛽0 +1−𝑣) +���� +2 . +(12) +If +𝑣 +is +selected +such +that +𝐾𝑥𝑣 ∈ N, +then +we +have +����sin +� +𝐾𝑥𝜋(𝛼1 − 𝛽0 +1 ±𝑣) +����� = +����sin +� +𝐾𝑥𝜋(𝛼1 − 𝛽0 +1) +�����. As a result, +(12) is simplified to +𝜇(𝛽0 +1, 𝛽0 +2) − 𝜎2 +𝜇(𝛽0 +1 +𝑣, 𝛽0 +2) − 𝜎2 = +����� +𝛼1 − 𝛽0 +1 −𝑣 +𝛼1 − 𝛽0 +1 +����� +2 +. +(13) +Since +𝜇(𝛽1, 𝛽2) ≥ 𝜎2, it follows that +𝜇(𝛽1, 𝛽2) − 𝜎2 = +��𝜇(𝛽1, 𝛽2) − 𝜎2�� always holds, for all (𝛽1, 𝛽2) ∈ B. Using this +fact, (13) can be written equivalently as +� +� +������ +𝜇(𝛽0 +1, 𝛽0 +2) − 𝜎2 +𝜇(𝛽0 +1 +𝑣, 𝛽0 +2) − 𝜎2 +����� = +����� +𝛼1 − 𝛽0 +1 −𝑣 +𝛼1 − 𝛽0 +1 +�����. +(14) +By solving the nonlinear equation in (14) w.r.t. the unknown +𝛼1, we obtain two solutions for 𝛼1, denoted by 𝛼(1) +1 +and 𝛼(2) +1 , +given by +𝛼(1)/(2) +1 += 𝛽0 +1 + +𝑣 +1± +√︂��� +𝜇(𝛽0 +1,𝛽0 +2)−𝜎2 +𝜇(𝛽0 +1+𝑣,𝛽0 +2)−𝜎2 +��� +. +(15) +It is not known which of the two values 𝛼(1) +1 +and 𝛼(2) +1 +is +equal to 𝛼1. To identify the correct solution for 𝛼1 of the +two solutions given by (15), we need the value of 𝜇(𝛽1, 𝛽2) +for (𝛽1, 𝛽2) = (𝛽0 +1 − 𝑣, 𝛽0 +2). Following the same procedure as +for (10)-(15), but now by using the values of 𝜇(𝛽1, 𝛽2) for +(𝛽1, 𝛽2) = (𝛽0 +1, 𝛽0 +2) and (𝛽1, 𝛽2) = (𝛽0 +1 −𝑣, 𝛽0 +2), we obtain +� +� +������ +𝜇(𝛽0 +1, 𝛽0 +2) − 𝜎2 +𝜇(𝛽0 +1 −𝑣, 𝛽0 +2) − 𝜎2 +����� = +����� +𝛼1 − 𝛽0 +1 +𝑣 +𝛼1 − 𝛽0 +1 +�����. +(16) +By solving (16), we obtain +𝛼(3)/(4) +1 += 𝛽0 +1 − +𝑣 +1± +√︂��� +𝜇(𝛽0 +1,𝛽0 +2)−𝜎2 +𝜇(𝛽0 +1−𝑣,𝛽0 +2)−𝜎2 +��� +. +(17) +One of the solutions in (15) is identical to one of the solutions +in (17). Therefore, using (15) and (17), the correct solution of +𝛼1 can be obtained as1 +𝛼1 = +� +𝛼(𝑖) +1 +𝛼( 𝑗) +1 +2 +: +min +𝑖∈{1,2}, 𝑗 ∈{3,4} +���𝛼(𝑖) +1 −𝛼( 𝑗) +1 +��� +� +. +(18) +In order to obtain 𝛼2, we need the value of 𝜇(𝛽1, 𝛽2) for +(𝛽1, 𝛽2) = (𝛽0 +1, 𝛽0 +2), which we already have, and for (𝛽1, 𝛽2) = +(𝛽0 +1, 𝛽0 +2 +𝑤), where 𝑤 is selected such that 𝐾𝑦𝑤 ∈ N, +��𝛽0 +2 ± 𝑤 +�� ≤ +1 and +��sin(𝐾𝑦𝜋(𝛼2 − 𝛽0 +2 ± 𝑤)) +�� = +��sin(𝐾𝑦𝜋(𝛼2 − 𝛽0 +2)) +��. Then, +similar to (10)-(14), we use the values of 𝜇(𝛽1, 𝛽2) for +(𝛽1, 𝛽2) = (𝛽0 +1, 𝛽0 +2) and (𝛽1, 𝛽2) = (𝛽0 +1, 𝛽0 +2 + 𝑤) to obtain +1Note +that +𝛼1 +can +also +be +written +equivalently +as +𝛼1 = +� +𝛼(1) +1 +, 𝛼(2) +1 +� � � +𝛼(3) +1 +, 𝛼(4) +1 +� +. However, the expression in (18) is more +convenient for the case when the values of 𝜇(𝛽1, 𝛽2) need to be estimated. + +5 +� +� +������ +𝜇(𝛽0 +1, 𝛽0 +2) − 𝜎2 +𝜇(𝛽0 +1, 𝛽0 +2 + 𝑤) − 𝜎2 +����� = +����� +𝛼2 − 𝛽0 +2 − 𝑤 +𝛼2 − 𝛽0 +2 +�����. +(19) +By solving the nonlinear equation (19), we obtain two solutions +for 𝛼2, denoted by 𝛼(1) +2 +and 𝛼(2) +2 , given by +𝛼(1)/(2) +2 += 𝛽0 +2 + +𝑤 +1± +√︂��� +𝜇(𝛽0 +1,𝛽0 +2)−𝜎2 +𝜇(𝛽0 +1,𝛽0 +2+𝑤)−𝜎2 +��� +. +(20) +To identify the correct solution for 𝛼2 of the two given in +(20), we need the value of 𝜇(𝛽1, 𝛽2) for (𝛽1, 𝛽2) = (𝛽0 +1, 𝛽0 +2 −𝑤). +Again, following the procedure from (10)-(15), by using the +values of 𝜇(𝛽1, 𝛽2) for (𝛽0 +1, 𝛽0 +2) and (𝛽0 +1, 𝛽0 +2 − 𝑤), we obtain +𝛼(3)/(4) +2 += 𝛽0 +2 − +𝑤 +1± +√︂��� +𝜇(𝛽0 +1,𝛽0 +2)−𝜎2 +𝜇(𝛽0 +1,𝛽0 +2−𝑤)−𝜎2 +��� +. +(21) +One of the solutions in (20) is exactly same as the solutions +of (21). Therefore, using (20) and (21), the correct solution of +𝛼2 can be obtained as +𝛼2 = +� +𝛼(𝑖) +2 +𝛼( 𝑗) +2 +2 +: +min +𝑖∈{1,2}, 𝑗 ∈{3,4} +���𝛼(𝑖) +2 −𝛼( 𝑗) +2 +��� +� +. +(22) +Finally, by setting 𝛽∗ +1 = 𝛼1 and 𝛽∗ +2 = 𝛼2, where 𝛼1 and 𝛼2 are +given by (18) and (22), respectively, we obtain (8) and (9). +■ +Remark 2. In [13, Sec. IV.A], the authors proposed the channel +estimation strategy under the assumption that there is no noise +in the system. However, in the noisy case, we proposed an +estimation scheme based on the assumption that 𝜇(𝛽1, 𝛽2) for +any of the phase-shifting parameters (𝛽1, 𝛽2) ∈ B are perfectly +known at the HMT. +However, in practice the exact values of 𝜇(𝛽1, 𝛽2) for any of +the phase-shifting parameters (𝛽1, 𝛽2) ∈ B cannot be known in +advance at the HMT, and therefore they need to be estimated +using pilot symbols. In the following, we propose an algorithm +that estimates 𝜇(𝛽1, 𝛽2) for the phase-shifting parameters in +B and then uses the estimated values of 𝜇(𝛽1, 𝛽2) to find the +optimal phase-shifting parameters (𝛽∗ +1, 𝛽∗ +2) in the presence of +noise. +C. Estimation Of The Optimal Phase-Shifting Parameters In +The Noisy Case +The user sends in total 𝑁 number of pilot signals to the +HMT for the estimation of the five values of 𝜇(𝛽1, 𝛽2) for the +five pairs of (𝛽1, 𝛽2) ∈ B. As a result, the proposed algorithm +works in five epochs. In the 𝑘𝑡ℎ epoch, for 𝑘 = 1,2,..., the user +transmits +� 𝑁 +5 +� number of pilots to the HMT. The HMT sets +(𝛽1, 𝛽2) to the 𝑘𝑡ℎ element in B, and collects +� 𝑁 +5 +� samples +of the received signal squared, given by (3). Then 𝜇(𝛽1, 𝛽2), +for (𝛽1, 𝛽2) being the 𝑘𝑡ℎ elements in B, is estimated as +ˆ𝜇(𝛽1, 𝛽2) = +1 +⌊𝑁/5⌋ +⌊𝑁 /5⌋ +∑︁ +𝑖=1 +𝑟𝑖(𝛽1, 𝛽2), +(23) +where 𝑟𝑖(𝛽1, 𝛽2) is the 𝑖𝑡ℎ sample of 𝑟(𝛽1, 𝛽2) in (3). +Next, we replace 𝜇(𝛽1, 𝛽2) in (15), (17), (20), and (21) +by ˆ𝜇(𝛽1, 𝛽2), ∀(𝛽1, 𝛽2) ∈ B, and thereby obtain our estimates +for 𝛽∗ +1 and 𝛽∗ +2, denoted by ˆ𝛽∗ +1 and ˆ𝛽∗ +2. The pseudo-code of +the proposed algorithm is given in Two-Stage Phase-Shifts +Estimation Algorithm below. We note that the choice of the +Two-Stage Phase-Shifts Estimation Algorithm +1: Input: 𝑁,B,𝜎2. +2: ***Stage 1: Uniform Exploration *** +3: for 𝑘 = 1 to 5 do +4: +HMT sets (𝛽1, 𝛽2) to the 𝑘𝑡ℎ pair in B. +5: +User sends ⌊𝑁/5⌋ number of pilots to the HMT. +6: +For the 𝑖𝑡ℎ pilot, the HMT receives 𝑟𝑖(𝛽1, 𝛽2), given by +(3), for 𝑖 = 1,2,..., ⌊𝑁/5⌋ . +7: +The HMT computes ˆ𝜇𝑘 (𝛽1, 𝛽2) using (23). +8: end for +9: ***Stage +2: +Estimate +Optimal +Phase-Shifting +Parameters*** +10: Obtain ˆ𝛽∗ +1 as +ˆ𝛽∗ +1 = +� +ˆ𝛼(𝑖) +1 + ˆ𝛼( 𝑗) +1 +2 +: +min +𝑖∈{1,2}, 𝑗 ∈{3,4} +��� ˆ𝛼(𝑖) +1 − ˆ𝛼( 𝑗) +1 +��� +� +, +(24) +where ˆ𝛼(1)/(2) +1 +is obtained by replacing the value of +𝜇(𝛽1, 𝛽2) by ˆ𝜇(𝛽1, 𝛽2) in (15), and ˆ𝛼(3)/(4) +1 +is obtained +by replacing the value of 𝜇(𝛽1, 𝛽2) by ˆ𝜇(𝛽1, 𝛽2) in (17). +11: Obtain ˆ𝛽∗ +2 as +ˆ𝛽∗ +2 = +� +ˆ𝛼(𝑖) +2 + ˆ𝛼( 𝑗) +2 +2 +: +min +𝑖∈{1,2}, 𝑗 ∈{3,4} +��� ˆ𝛼(𝑖) +2 − ˆ𝛼( 𝑗) +2 +��� +� +, +(25) +where ˆ𝛼(1)/(2) +2 +is obtained by replacing the value of +𝜇(𝛽1, 𝛽2) by ˆ𝜇(𝛽1, 𝛽2) in (20), and ˆ𝛼(3)/(4) +2 +is obtained +by replacing the value of 𝜇(𝛽1, 𝛽2) by ˆ𝜇(𝛽1, 𝛽2) in (21). +12: Output: ˆ𝛽∗ +1 and ˆ𝛽∗ +2. +13: Phase-shifts at HMT Set the phase-shift of the (𝑚𝑥,𝑚𝑦)𝑡ℎ +element at the HMT to +𝛽𝑚𝑥𝑚𝑦 = − +mod (𝑘0𝑑𝑟 (𝑚𝑥 ˆ𝛽∗ +1 +𝑚𝑦 ˆ𝛽∗ +2),2𝜋). +initial (𝛽0 +1, 𝛽0 +2) in the set B was arbitrary. The values of (𝛽0 +1, 𝛽0 +2) +can effect the estimation error. In general, if the values (𝛽0 +1, 𝛽0 +2) +are closer to the (𝛼1,𝛼2), the better the estimation will be. A +good choice for (𝛽0 +1, 𝛽0 +2) is given in [13, Sec. V.C], which leads +to faster learning of (𝛼1,𝛼2). +IV. THEORETICAL GUARANTEES FOR THE PROPOSED +ALGORITHM +In the section, we bound the probability that the estimates, +obtained from the proposed Two-Stage Phase-Shifts Estimation +Algorithm Algorithm, deviate from the true values of (𝛼1,𝛼2) +by an amount 0 ≤ 𝜖 ≤ 1. In particular, we upper bound the +following error probability +P +�� +ˆ𝛽∗ +1 −𝛼1 +�2 ++ +� +ˆ𝛽∗ +2 −𝛼2 +�2 +≥ 𝜖 +� +. +(26) + +6 +We use the following results to upper bound the error probability +in (26). +Lemma 1. Let {𝑋𝑛} be a sequence of random variables (RVs) +on a probability space. Let 𝑋 be a RV defined on the same +probability space. Then, the following holds +P{|𝑋𝑛 − 𝑋𝑚| ≥ 𝜖} ≤ P +� +|𝑋𝑛 − 𝑋| ≥ 𝜖 +2 +� ++P +� +|𝑋𝑚 − 𝑋| ≥ 𝜖 +2 +� +. +Proof. The proof is given in the Appendix A. +■ +Let 𝜒2 +𝑝(𝜆) denote a non-central Chi-squared distribution with +𝑝 degrees of freedom and non-centrality parameter 𝜆. +Lemma 2. Let 𝑋 = +2 +𝜎2 𝑟(𝛽1, 𝛽2), where 𝑟(𝛽1, 𝛽2) is given by +(3), and let 𝜆1 = +2 +𝜎2 +��� +√ +𝑃𝐻(𝛽1, 𝛽2) +��� +2 +. Then, 𝑋 is distributed as +𝜒2 +2(𝜆1), i.e., 𝑋 ∼ 𝜒2 +2(𝜆1). Furthermore, if 𝑋𝑖 for 𝑖 = 1,2,...,𝑛 +are 𝑛 independently and identically distributed (i.i.d.) RVs of +𝜒2 +2(𝜆1), then +𝑛 +∑︁ +𝑖=1 +𝑋𝑖 ∼ 𝜒2 +2𝑛(𝑛𝜆1). +Proof. The proof is given in the Appendix B. +■ +The following theorem provides an upper bound on the error +probability in (26). +Theorem 2. Let us perform uniform exploration on the set B +given in (7). For any 0 ≤ 𝜖 ≤ 1, the error probability in (26) +is upper bounded as +P +�� +ˆ𝛽∗ +1 −𝛼1 +�2 ++ +� +ˆ𝛽∗ +2 −𝛼2 +�2 +≥ 𝜖 +� +≤ 4 +� +𝑒− 𝑛 +32 +� 𝜖 𝜆2 +1+𝜆2 +�2 ++ 𝑒− 𝑛 +32 +� 𝜖 𝜆3 +1+𝜆3 +�2 ++ 𝑒− 𝑛 +32 +� 𝜖 𝜆4 +1+𝜆4 +�2 ++ 𝑒− 𝑛 +32 +� 𝜖 𝜆5 +1+𝜆5 +�2� +, +(27) +where +𝜆1 = +2 +��� +√ +𝑃𝐻(𝛽0 +1, 𝛽0 +2) +��� +2 +𝜎2 +, +𝜆2 = +2 +��� +√ +𝑃𝐻(𝛽0 +1 +𝑣, 𝛽0 +2) +��� +2 +𝜎2 +, +𝜆3 = +2 +��� +√ +𝑃𝐻(𝛽0 +1 −𝑣, 𝛽0 +2) +��� +2 +𝜎2 +, +𝜆4 = +2 +��� +√ +𝑃𝐻(𝛽0 +1, 𝛽0 +2 + 𝑤) +��� +2 +𝜎2 +, +𝜆5 = +2 +��� +√ +𝑃𝐻(𝛽0 +1, 𝛽0 +2 − 𝑤) +��� +2 +𝜎2 +. +Proof. Let us denote the estimate of 𝜇(𝛽0 +1, 𝛽0 +2) by ˆ𝜇(𝛽0 +1, 𝛽0 +2) +which is given by +ˆ𝜇(𝛽0 +1, 𝛽0 +2) = 1 +𝑛 +𝑛 +∑︁ +𝑖=1 +𝑟𝑖(𝛽0 +1, 𝛽0 +2) = 𝜎2 +2𝑛 +𝑛 +∑︁ +𝑖=1 +𝑋𝑖. +Using Lemma 2, we have +ˆ𝜇1 := 2𝑛 +𝜎2 ˆ𝜇(𝛽0 +1, 𝛽0 +2) ∼ 𝜒2 +2𝑛(𝑛𝜆1) +(28) +ˆ𝜇2 := 2𝑛 +𝜎2 ˆ𝜇(𝛽0 +1 +𝑣, 𝛽0 +2) ∼ 𝜒2 +2𝑛(𝑛𝜆2) +(29) +ˆ𝜇3 := 2𝑛 +𝜎2 ˆ𝜇(𝛽0 +1 −𝑣, 𝛽0 +2) ∼ 𝜒2 +2𝑛(𝑛𝜆3) +(30) +ˆ𝜇4 := 2𝑛 +𝜎2 ˆ𝜇(𝛽0 +1, 𝛽0 +2 + 𝑤) ∼ 𝜒2 +2𝑛(𝑛𝜆4) +(31) +ˆ𝜇5 := 2𝑛 +𝜎2 ˆ𝜇(𝛽0 +1, 𝛽0 +2 − 𝑤) ∼ 𝜒2 +2𝑛(𝑛𝜆5) +(32) +where 𝜆1,𝜆2,𝜆3,𝜆4 and 𝜆5 is given in Theorem 2. +The random variables ˆ𝜇1, ˆ𝜇2, ˆ𝜇3, ˆ𝜇4, and ˆ𝜇5 are mutually +independent, since they are sampled at different epochs. The +estimated optimal phase-shifting parameters ( ˆ𝛽∗ +1, ˆ𝛽∗ +2), are given +by (24) and (25), where the values of ˆ𝛼(1) +1 , ˆ𝛼(2) +1 , ˆ𝛼(3) +1 , ˆ𝛼(4) +1 , +and, ˆ𝛼(1) +2 , ˆ𝛼(2) +2 , ˆ𝛼(3) +2 , and ˆ𝛼(4) +2 +are given by +ˆ𝛼(1)/(2) +1 += 𝛽0 +1 + +𝑣 +1± +√︄���� +ˆ𝜇(𝛽0 +1,𝛽0 +2)−𝜎2 +ˆ𝜇(𝛽0 +1+𝑣,𝛽0 +2)−𝜎2 +���� +(33) +ˆ𝛼(3)/(4) +1 += 𝛽0 +1 − +𝑣 +1± +√︄���� +ˆ𝜇(𝛽0 +1,𝛽0 +2)−𝜎2 +ˆ𝜇(𝛽0 +1−𝑣,𝛽0 +2)−𝜎2 +���� +(34) +ˆ𝛼(1)/(2) +2 += 𝛽0 +2 + +𝑤 +1± +√︄���� +ˆ𝜇(𝛽0 +1,𝛽0 +2)−𝜎2 +ˆ𝜇(𝛽0 +1,𝛽0 +2+𝑤)−𝜎2 +���� +(35) +ˆ𝛼(3)/(4) +2 += 𝛽0 +2 − +𝑤 +1± +√︄���� +ˆ𝜇(𝛽0 +1,𝛽0 +2)−𝜎2 +ˆ𝜇(𝛽0 +1,𝛽0 +2−𝑤)−𝜎2 +���� +. +(36) +By inserting (28), (29), (30), (31), and (32) into (33), (34), +(35) and (36), we obtain +ˆ𝛼(1)/(2) +1 += 𝛽0 +1 + +𝑣 +1± +√︂��� ˆ𝜇1−2𝑛 +ˆ𝜇2−2𝑛 +��� +(37) +ˆ𝛼(3)/(4) +1 += 𝛽0 +1 − +𝑣 +1± +√︂��� ˆ𝜇1−2𝑛 +ˆ𝜇3−2𝑛 +��� +(38) +ˆ𝛼(1)/(2) +2 += 𝛽0 +2 + +𝑤 +1± +√︂��� ˆ𝜇1−2𝑛 +ˆ𝜇4−2𝑛 +��� +(39) +ˆ𝛼(3)/(4) +2 += 𝛽0 +2 − +𝑤 +1± +√︂��� ˆ𝜇1−2𝑛 +ˆ𝜇5−2𝑛 +��� +. +(40) +Let us denote +𝐼 := P +��� ˆ𝛽∗ +1 −𝛼1 +�� ≥ +√︂ +𝜖 +2 +� +𝐼𝐼 := P +��� ˆ𝛽∗ +2 −𝛼2 +�� ≥ +√︂ +𝜖 +2 +� +. +Now, applying Lemma 1 in (26), we obtain +P +�� +ˆ𝛽∗ +1 −𝛼1 +�2 ++ +� +ˆ𝛽∗ +2 −𝛼2 +�2 +≥ 𝜖 +� +≤ P +�� +ˆ𝛽∗ +1 −𝛼1 +�2 +≥ 𝜖 +2 +� ++P +�� +ˆ𝛽∗ +2 −𝛼2 +�2 +≥ 𝜖 +2 +� +≤ 𝐼 + 𝐼𝐼. +(41) +We upper bound each of the term in right-hand side of (41). +We begin with the first term P +��� ˆ𝛽∗ +1 −𝛼1 +�� ≥ √︁ 𝜖 +2 +� +, denoted as I. +G Step 1: Upper bound on I +From (24), we have + +7 +P +��� ˆ𝛽∗ +1 −𝛼1 +�� ≥ +√︂ +𝜖 +2 +� += P +� ������ +ˆ𝛼(1) +1 ++ ˆ𝛼(3) +1 +2 +−𝛼1 +����� ≥ +√︂ +𝜖 +2 +� +� ������ +ˆ𝛼(1) +1 ++ ˆ𝛼(4) +1 +2 +−𝛼1 +����� ≥ +√︂ +𝜖 +2 +� +� ������ +ˆ𝛼(2) +1 ++ ˆ𝛼(3) +1 +2 +−𝛼1 +����� ≥ +√︂ +𝜖 +2 +� +� ������ +ˆ𝛼(2) +1 ++ ˆ𝛼(4) +1 +2 +−𝛼1 +����� ≥ +√︂ +𝜖 +2 +� � +≤ P +������ +ˆ𝛼(1) +1 ++ ˆ𝛼(3) +1 +2 +−𝛼1 +����� ≥ +√︂ +𝜖 +2 +� ++P +������ +ˆ𝛼(1) +1 ++ ˆ𝛼(4) +1 +2 +−𝛼1 +����� ≥ +√︂ +𝜖 +2 +� ++P +������ +ˆ𝛼(2) +1 ++ ˆ𝛼(3) +1 +2 +−𝛼1 +����� ≥ +√︂ +𝜖 +2 +� ++P +������ +ˆ𝛼(2) +1 ++ ˆ𝛼(4) +1 +2 +−𝛼1 +����� ≥ +√︂ +𝜖 +2 +� += +∑︁ +𝑖=1,2 +𝑗=3,4 +P +����� +� +ˆ𝛼(𝑖) +1 −𝛼1 +� ++ +� +ˆ𝛼( 𝑗) +1 +−𝛼1 +����� ≥ 2 +√︂ +𝜖 +2 +� +≤ +∑︁ +𝑖=1,2 +𝑗=3,4 +� +P +���� ˆ𝛼(𝑖) +1 −𝛼1 +��� ≥ +√︂ +𝜖 +2 +� ++P +���� ˆ𝛼( 𝑗) +1 +−𝛼1 +��� ≥ +√︂ +𝜖 +2 +� � += 2 +� +P +���� ˆ𝛼(1) +1 +−𝛼1 +��� ≥ +√︂ +𝜖 +2 +� ++P +���� ˆ𝛼(2) +1 +−𝛼1 +��� ≥ +√︂ +𝜖 +2 +� ++P +���� ˆ𝛼(3) +1 +−𝛼1 +��� ≥ +√︂ +𝜖 +2 +� ++P +���� ˆ𝛼(4) +1 +−𝛼1 +��� ≥ +√︂ +𝜖 +2 +� � +, +(42) +where we applied the union bound to get the first inequality +and applied Lemma 1 for the second inequality. We now +bound each term in (42) separately. +® Upper bound of P +���� ˆ𝜶(1) +1 +−𝜶1 +��� ≥ +√︁ 𝝐 +2 +� +: Substituting +the +values +of +ˆ𝛼(1) +1 , +as +given +by +(37), +in +P +���� ˆ𝜶(1) +1 +−𝜶1 +��� ≥ +√︁ 𝝐 +2 +� +, we obtain +P +���� ˆ𝛼(1) +1 +−𝛼1) +��� ≥ +√︂ +𝜖 +2 +� += P +������ +������ +��������� +𝑣 +1+ +√︂��� ˆ𝜇1−2𝑛 +ˆ𝜇2−2𝑛 +��� +− (𝛼1 − 𝛽0 +1) +��������� +≥ +√︂ +𝜖 +2 +������ +������ +. +(43) +Note that the following holds. +��������� +𝑣 +1+ +√︂��� ˆ𝜇1−2𝑛 +ˆ𝜇2−2𝑛 +��� +− (𝛼1 − 𝛽0 +1) +��������� +≤ +��������� +𝑣 +1+ +√︂��� ˆ𝜇1−2𝑛 +ˆ𝜇2−2𝑛 +��� +��������� ++ +�����𝛼1 − 𝛽0 +1 +�����. +(44) +By applying (44) in (43), we obtain +P +���� ˆ𝛼(1) +1 +−𝛼1) +��� ≥ +√︂ +𝜖 +2 +� +≤ P +������ +������ +��������� +1 +1+ +√︂��� ˆ𝜇1−2𝑛 +ˆ𝜇2−2𝑛 +��� +��������� +≥ 1 +𝑣 +�√︂ +𝜖 +2 − +�����𝛼1 − 𝛽0 +1 +����� +������� +������ +. +(45) +For the RVs ˆ𝜇1 and ˆ𝜇2, +1 +1+ +√︂��� +ˆ𝜇1−2𝑛 +ˆ𝜇2−2𝑛 +��� +is always positive. +Using this fact in (45), we obtain +P +���� ˆ𝛼(1) +1 +−𝛼1) +��� ≥ +√︂ +𝜖 +2 +� +≤ P +������ +������ +1 +1+ +√︂��� ˆ𝜇1−2𝑛 +ˆ𝜇2−2𝑛 +��� +≥ 1 +𝑣 +�√︂ +𝜖 +2 − +�����𝛼1 − 𝛽0 +1 +����� +������� +������ +. +Let 𝑎 = 1 +𝑣 +�√︁ 𝜖 +2 − +��𝛼1 − 𝛽0 +1 +�� +� +. We have +P +���� ˆ𝛼(1) +1 +−𝛼1) +��� ≥ +√︂ +𝜖 +2 +� +≤ P +�� +�� +1+ +√︄���� +ˆ𝜇1 −2𝑛 +ˆ𝜇2 −2𝑛 +���� ≤ 1 +𝑎 +�� +�� += P +����� +ˆ𝜇1 −2𝑛 +ˆ𝜇2 −2𝑛 +���� ≤ +� +1− 1 +𝑎 +�2� +. +(46) +® Upper bound of P +���� ˆ𝜶(2) +1 +−𝜶1 +��� ≥ +√︁ 𝝐 +2 +� +: Substituting +the +values +of +ˆ𝛼(2) +1 , +as +given +by +(37), +in +P +���� ˆ𝜶(2) +1 +−𝜶1 +��� ≥ +√︁ 𝝐 +2 +� +, we obtain +P +���� ˆ𝛼(2) +1 +−𝛼1) +��� ≥ +√︂ +𝜖 +2 +� += P +������ +������ +��������� +𝑣 +1− +√︂��� ˆ𝜇1−2𝑛 +ˆ𝜇2−2𝑛 +��� +− (𝛼1 − 𝛽0 +1) +��������� +≥ +√︂ +𝜖 +2 +������ +������ +. +(47) +Note that the following holds. +��������� +𝑣 +1− +√︂��� ˆ𝜇1−2𝑛 +ˆ𝜇2−2𝑛 +��� +− (𝛼1 − 𝛽0 +1) +��������� +≤ +��������� +𝑣 +1− +√︂��� ˆ𝜇1−2𝑛 +ˆ𝜇2−2𝑛 +��� +��������� ++ +�����𝛼1 − 𝛽0 +1 +�����. +(48) +By applying (48) in the right-hand side of (47), we + +8 +obtain +P +���� ˆ𝛼(2) +1 +−𝛼1) +��� ≥ +√︂ +𝜖 +2 +� +≤ P +��� +��� +������ +1− +√︄���� +ˆ𝜇1 −2𝑛 +ˆ𝜇2 −2𝑛 +���� +������ +≤ 1 +𝑎 +��� +��� += P +�� +1− 1 +𝑎 +�2 +≤ +���� +ˆ𝜇1 −2𝑛 +ˆ𝜇2 −2𝑛 +���� ≤ +� +1+ 1 +𝑎 +�2� +≤ P +����� +ˆ𝜇1 −2𝑛 +ˆ𝜇2 −2𝑛 +���� ≤ +� +1+ 1 +𝑎 +�2� +−P +����� +ˆ𝜇1 −2𝑛 +ˆ𝜇2 −2𝑛 +���� ≤ +� +1− 1 +𝑎 +�2� +. +(49) +® Upper bound of P +���� ˆ𝜶(3) +1 +−𝜶1 +��� ≥ +√︁ 𝝐 +2 +� +: Substituting +the +values +of +ˆ𝛼(3) +1 , +as +given +by +(38), +in +P +���� ˆ𝜶(3) +1 +−𝜶1 +��� ≥ +√︁ 𝝐 +2 +� +and +following +similar +steps +to bound P +���� ˆ𝜶(1) +1 +−𝜶1 +��� ≥ +√︁ 𝝐 +2 +� +, we obtain +P +���� ˆ𝛼(3) +1 +−𝛼1) +��� ≥ +√︂ +𝜖 +2 +� +≤ P +����� +ˆ𝜇1 −2𝑛 +ˆ𝜇3 −2𝑛 +���� ≤ +� +1− 1 +𝑎 +�2� +. +(50) +® Upper bound of P +���� ˆ𝜶(4) +1 +−𝜶1 +��� ≥ +√︁ 𝝐 +2 +� +: Substituting +the +values +of +ˆ𝛼(4) +1 , +as +given +by +(38), +in +P +���� ˆ𝜶(4) +1 +−𝜶1 +��� ≥ +√︁ 𝝐 +2 +� +and +following +similar +steps +to bound P +���� ˆ𝜶(2) +2 +−𝜶1 +��� ≥ +√︁ 𝝐 +2 +� +, we obtain +P +���� ˆ𝛼(4) +1 +−𝛼1) +��� ≥ +√︂ +𝜖 +2 +� +≤ P +����� +ˆ𝜇1 −2𝑛 +ˆ𝜇3 −2𝑛 +���� ≤ +� +1+ 1 +𝑎 +�2� +−P +����� +ˆ𝜇1 −2𝑛 +ˆ𝜇3 −2𝑛 +���� ≤ +� +1− 1 +𝑎 +�2� +. +(51) +By inserting the bounds (46), (49), (50) and (51) to (42) +we obtain +P +��� ˆ𝛽∗ +1 −𝛼1 +�� ≥ +√︂ +𝜖 +2 +� +≤ 2 +� +P +����� +ˆ𝜇2 −2𝑛 +ˆ𝜇1 −2𝑛 +���� ≥ 𝛾1 +� ++P +����� +ˆ𝜇3 −2𝑛 +ˆ𝜇1 −2𝑛 +���� ≥ 𝛾1 +�� +, +(52) +where we set 𝛾1 = +� +1 +1+(1/𝑎) +�2 +. +We next upper bound each term on the right-hand side of +(52). The bounds are derived using the properties of the +sub-exponential distributions which we introduce below. +G Step 2: Sub-exponential Distributions and its Tail +Bound +Definition IV.1 (sub-exponential distribution). A RV 𝑋 +with mean 𝜇 is said to be sub-exponential with parameters +(𝜈,𝛼), for 𝛼 > 0, if +E +� +exp +� +𝑡(𝑋 − 𝜇) +�� +≤ exp +�𝑡2𝜈2 +2 +� +, for |𝑡| < 1 +𝛼 . +Theorem +3 +([18]). Let +𝑋𝑘 +for +𝑘 = 1,2,...,𝑛 be +independent RVs where 𝑋𝑘 is sub-exponential with +parameters +(𝜈𝑘,𝑏𝑘), and mean +𝜇𝑘 = E [𝑋𝑘]. Then +𝑛� +𝑘=1 +(𝑋𝑘 − 𝜇𝑘) is a sub-exponential RV with parameters +(𝜈∗,𝑏∗) where +𝑏∗ = +max +𝑘=1,2...,𝑛𝑏𝑘, +and +𝜈∗ = +� +� 𝑛 +∑︁ +𝑘=1 +𝜈2 +𝑘. +Furthermore, its tail probability can be bounded as +P +������ +1 +𝑛 +𝑛 +∑︁ +𝑘=1 +(𝑋𝑘 − 𝜇𝑘) +����� ≥ 𝑡 +� +≤ +��� +��� +2𝑒 +− +𝑛𝑡2 +2(𝜈2∗ /𝑛) , +for 0 ≤ 𝑡 ≤ +𝜈2 +∗ +𝑛𝑏∗ +2𝑒− 𝑛𝑡 +2𝑏∗ , +for 𝑡 ≥ +𝜈2 +∗ +𝑛𝑏∗ . +Proof. The proof is given in Appendix C. +■ +Corollary +1. +Let +𝑋𝑘 +for +𝑘 = 1,2...,𝑛 +be +i.i.d. +sub-exponential RVs with parameters (2(2+2𝑎),4) each +with mean 2+ 𝑎. Then, +P +������ +1 +𝑛 +𝑛 +∑︁ +𝑘=1 +(𝑋𝑘 − 𝜇𝑘) +����� ≥ 𝑡 +� +≤ 2𝑒 +− +𝑛𝑡2 +8(2+2𝑎)2 , +for 𝑡 > 0. +Proof. The proof is given in then Appendix D. +■ +We use Corollary 1 to upper bound of the right-hand +side terms in (52). The following lemma establishes +the connection between the non-central chi-squared +distribution and the sub-exponential distributions. +Lemma 3. Let 𝑋 ∼ 𝜒2 +𝑝(𝑎). Then, 𝑋 is sub-exponential +with parameters �2(𝑝 +2𝑎),4�. +Proof. The proof is given in then Appendix E. +■ +G Step 3: Upper Bounding Eq. (52) +– Recall that ˆ𝜇1 ∼ 𝜒2 +2𝑛(𝑛𝜆1) and ˆ𝜇2 ∼ 𝜒2 +2𝑛(𝑛𝜆2). Let +𝑓 ˆ𝜇1 denote the pdf of ˆ𝜇1. We upper bound the term +P +���� ˆ𝜇2−2𝑛 +ˆ𝜇1−2𝑛 +��� ≥ 𝛾1 +� +as follows +P +����� +ˆ𝜇2 −2𝑛 +ˆ𝜇1 −2𝑛 +���� ≥ 𝛾1 +� += +∞ +∫ +0 +P +����� ˆ𝜇2 −2𝑛 +���� ≥ 𝛾1 +����𝑢 −2𝑛 +���� +� +𝑓 ˆ𝜇1(𝑢)𝑑𝑢 += +∞ +∫ +0 +P +����� ˆ𝜇2 −2𝑛 −𝑛𝜆2 +𝑛𝜆2 +���� ≥ 𝛾1 +����𝑢 −2𝑛 +���� +� +𝑓 ˆ𝜇1(𝑢)𝑑𝑢 +≤ +∞ +∫ +0 +P +�1 +𝑛 +���� ˆ𝜇2 −𝑛(2+𝜆2) +���� ≥ 𝛾1|𝑢 −2𝑛| −𝑛𝜆2 +𝑛 +� +𝑓 ˆ𝜇1(𝑢)𝑑𝑢 +(53) +Note that, if 𝛾1 |𝑢−2𝑛|−𝑛𝜆2 +𝑛 +< 0, then P +���� ˆ𝜇2−2𝑛 +ˆ𝜇1−2𝑛 +��� ≥ 𝛾1 +� +≤ +1 as P +� +1 +𝑛 +���� ˆ𝜇2 −𝑛(2+𝜆2) +���� ≥ 𝛾1 |𝑢−2𝑛|−𝑛𝜆2 +𝑛 +� += 1, which is +trivial. + +9 +For 𝛾1 |𝑢−2𝑛|−𝑛𝜆2 +𝑛 +≥ 0, using the assumption 0 ≤ 𝜖 ≤ 1 in +(53), we have +P +����� +ˆ𝜇2 −2𝑛 +ˆ𝜇1 −2𝑛 +���� ≥ 𝛾1 +� +≤ +∞ +∫ +0 +P +�1 +𝑛 +���� ˆ𝜇2 −𝑛(2+𝜆2) +���� ≥ 𝜖 +� 𝛾1|𝑢 −2𝑛| −𝑛𝜆2 +𝑛 +� � +× 𝑓 ˆ𝜇1(𝑢)𝑑𝑢. +(54) +The last inequality follows from Lemma 1. Let +𝑡1 := 𝑡1(𝑢) = 𝜖 +� +𝛾1 |𝑢−2𝑛|−𝑛𝜆2 +𝑛 +� +. As E [ ˆ𝜇2] = 2𝑛 +𝑛𝜆2, by +applying Corollary 1, we obtain +P +�1 +𝑛 +���� ˆ𝜇2 −𝑛(2+𝜆2) +���� ≥ 𝑡1 +� +≤ 2𝑒 +− +𝑛𝑡2 +1 +8(2+2𝜆2)2 , +𝑡1 ≥ 0. +(55) +By applying (55) to (54), we obtain +P +����� ˆ𝜇2 −2𝑛 +���� ≥ 𝛾1 +���� ˆ𝜇1 −2𝑛 +���� +� +≤ +∞ +∫ +0 +2𝑒 +− +𝑛𝑡2 +1 +8(2+2𝜆2)2 𝑓 ˆ𝜇1(𝑢)𝑑𝑢, += +2𝑛 +∫ +0 +2𝑒 +− +𝑛 +� +𝜖𝑛 +� +𝛾1 (2𝑛−𝑢)−𝑛𝜆2 +��2 +8(2+2𝜆2)2 +𝑓 ˆ𝜇1(𝑢)𝑑𝑢 ++ +∞ +∫ +2𝑛 +2𝑒 +− +𝑛 +� +𝜖𝑛 +� +𝛾1 (𝑢−2𝑛)−𝑛𝜆2 +��2 +8(2+2𝜆2)2 +𝑓 ˆ𝜇1 (𝑢)𝑑𝑢. +(56) +For 0 ≤ 𝑢 ≤ 2𝑛, we have +2𝑒 +− +𝑛 +� +𝜖𝑛 +� +𝛾1 (2𝑛−𝑢)−𝑛𝜆2 +��2 +8(2+2𝜆2)2 +≤ 2𝑒 +− +𝑛� +𝜖 𝜆2 +�2 +8(2+2𝜆2)2 . +(57) +For 2𝑛 ≤ 𝑢 ≤ ∞, we have +2𝑒 +− +𝑛 +� +𝜖𝑛 +� +𝛾1 (𝑢−2𝑛)−𝑛𝜆2 +��2 +8(2+2𝜆2)2 +≤ 2𝑒 +− +𝑛� +𝜖 𝜆2 +�2 +8(2+2𝜆2)2 . +(58) +Using (57) and (58) in (56), we obtain +P +����� ˆ𝜇2 −2𝑛 +���� ≥ 𝛾1 +���� ˆ𝜇1 −2𝑛 +���� +� +≤ 2𝑒 +− +𝑛� +𝜖 𝜆2 +�2 +8(2+2𝜆2)2 P{0 < ˆ𝜇1 < 2𝑛} ++2𝑒 +− +𝑛� +𝜖 𝜆2 +�2 +8(2+2𝜆2)2 P{ ˆ𝜇1 > 2𝑛} +P +����� ˆ𝜇2 −2𝑛 +���� ≥ 𝛾1 +���� ˆ𝜇1 −2𝑛 +���� +� +≤ 2𝑒 +− +𝑛� +𝜖 𝜆2 +�2 +8(2+2𝜆2)2 . +(59) +– We next upper bound P +���� ˆ𝜇3−2𝑛 +ˆ𝜇1−2𝑛 +��� ≥ 𝛾1 +� +. Set 𝑡2 = +𝜖 +� +𝛾1 |𝑢−2𝑛|−𝑛𝜆3 +𝑛 +� +. Recall that ˆ𝜇3 ∼ 𝜒2 +2𝑛(𝑛𝜆3). Following +the steps similar to the derivation of the bound in (59), +we obtain +P +����� ˆ𝜇3 −2𝑛 +���� ≥ 𝛾1 +���� ˆ𝜇1 −2𝑛 +���� +� +≤ 2𝑒 +− +𝑛� +𝜖 𝜆3 +�2 +8(2+2𝜆3)2 . +(60) +Combining (59) and (60) we obtain the following upper +bound on (52) +P +��� ˆ𝛽∗ +1 −𝛼1 +�� ≥ +√︂ +𝜖 +2 +� +≤ 4 +� +𝑒− 𝑛 +32 +� 𝜖 𝜆2 +1+𝜆2 +�2 ++ 𝑒− 𝑛 +32 +� 𝜖 𝜆3 +1+𝜆3 +�2� +. (61) +G Step 4: Upper bound on II +By following the same steps for deriving the upper bound +of P +��� ˆ𝛽∗ +1 −𝛼1 +�� ≥ √︁ 𝜖 +2 +� +, we can obtain the following bound +P +��� ˆ𝛽∗ +2 −𝛼2 +�� ≥ +√︂ +𝜖 +2 +� +≤ 4 +� +𝑒− 𝑛 +32 +� 𝜖 𝜆4 +1+𝜆4 +�2 ++ 𝑒− 𝑛 +32 +� 𝜖 𝜆5 +1+𝜆5 +�2� +. (62) +Combining (61) and (62), we obtain the required upper +bound in (27). +■ +V. NUMERICAL SIMULATIONS +We estimate the initial value of (𝛽0 +1, 𝛽0 +2) as given in [13, Sec. +V.C]. Based on the the initial value of (𝛽0 +1, 𝛽0 +2) we set B as +given in (7), where 𝑣 and 𝑤 are selected such that 𝐾𝑥𝑣 ∈ N and +𝐾𝑦𝑤 ∈ N respectively. In addition to the LoS path, we assume +that there are 4 NLoS path components due to scatters between +the user and the HMT. The elevation and azimuth angles of +each NLoS path from these scatters to the center of HMT follow +the uniform distribution, i.e., 𝑈(0,2𝜋). Moreover, we consider +the path coefficient of each NLoS path as a complex Gaussian +distribution, i.e., 𝐶𝑁(0,𝜎2 +𝑠 ), where 𝜎2 +𝑠 is 20 dB weaker than +the power of the LoS component [19]. The system parameters +for numerical simulations are listed in Table I. +TABLE I: A list of system parameters for numerical simulations +Parameters +Values +Description +𝑓𝑐 +30 GHz +Carrier frequency +𝜆 +1 cm +Wavelength +𝐿𝑥 +1 m +Width of the HMT +𝐿𝑦 +1 m +Length of the HMT +𝑑𝑟 +𝜆/4 +Unit element spacing +𝐿𝑒 +𝑑𝑟 +Width +and +length +of +each +phase-shifting element +𝑃 +20 dBm +Transmission power of the HMT +during data transmission +𝜎2 +-115 dBm +Noise power for 200 KHz 2 +A. Comparison +Between +the +Proposed +Algorithm +and +Benchmark Scheme +According to the approximated channel model, where the +phase-shift parameters at the HMT are given by 𝛽1 and 𝛽2, +the achieved data rate at the user of the HMT-assisted wireless +communication system is given by +𝑅(𝛽1, 𝛽2) = 𝑙𝑜𝑔2 +� +1+ 𝑃|𝐻(𝛽1, 𝛽2)|2 +𝜎2 +� +, +(63) + +10 +where 𝑃 is the transmission power at the HMT. The HMT +uses the acquired CSI during the channel estimation period to +maximize the received data rate by the user. Hence, we consider +the achieved data rate by the user, using the acquired CSI as +a performance metric. We applied the proposed algorithm in +two different cases when the distance between the user and +the center of the HMT (𝑑0 = 200 m and when 𝑑0 = 10 m. We +compared our proposed algorithm with two benchmarks, the +proposed algorithm in [13] and the oracle scheme where 𝛼1 +and 𝛼2 are estimated perfectly and thereby the maximum rate +is achieved. +In Fig. 4, we compared the achievable rates, given by (63), +of the proposed scheme and the benchmark schemes. We +considered both 𝑑0 = 200 m and 𝑑0 = 10 m regions of the +HMT with respect to the transmit power of the pilot signals +when the number of the pilot signals is fixed to 23. For all +the algorithms we use the same number of pilots, i.e. 23, +for both the cases. Our proposed algorithm uses four pilots +in each epoch and there are five epochs, which makes the +total number of pilots equals to 20. We require additional three +number of pilots to estimate (𝛽0 +1, 𝛽0 +2). We run the simulation for +1000 times. We see that in both cases, the proposed Two-Stage +Phase-Shifts Estimation Algorithm gives higher rates than other +two benchmark schemes. +20 +10 +0 +10 +20 +30 +40 +Power of pilot signals (in dBm) +15 +20 +25 +30 +35 +40 +45 +50 +Achievable Rate (in bits/symbol) +Maximum Rate d0=200 +Proposed Algorithm d0=200 +Algorithm from [13] d0=200 +Maximum Rate d0=10 +Proposed Algorithm d0=10 +Algorithm from [13] d0=10 +Fig. 4: Achievable rate vs. the transmit power of the pilot signals +(in dBm). +B. Convergence of The Proposed Algorithm +We now numerically evaluate the convergence of the upper +bound of the proposed algorithm, given by (27). We also +compare the actual probability, given by (26), that we obtain +by simulations. +In Fig. 5, we show the convergence property of the error +probability and its upper bound of the proposed algorithm +for increasing values of 𝜖 = {0.01,0.05,0.1} when the power +of the pilot signal is 𝑃 = 10 dBm and 𝑑0 = 200 m. We run +the simulation for 1000 times. We see that for each value +of 𝜖, the proposed algorithm converges towards zero as we +increase the number of pilots. Moreover, the upper bound of +the error probability also converges to the error probability as +the number of pilot signals increases. +2This setting corresponds to the noise power spectrum density at the HMT +is −174 dBm/Hz and signal bandwidth is 200 KHz, assuming the noise figure +of each user to be 6 dB [8]. +0 +500 +1000 +1500 +2000 +2500 +3000 +Number of Pilots +10 +1 +100 +Error Probability +Proposed Scheme, = 0.01 +Upper Bound, = 0.01 +Proposed Scheme, = 0.05 +Upper Bound, = 0.05 +Proposed Scheme, = 0.1 +Upper Bound, = 0.1 +Fig. 5: Error Probability Bound v/s Number of Pilots for 𝜖 = +{0.01,0.05,0.1} for 𝑃 = 10 dBm. +In Fig. 6, we compare the convergence property of the error +probability of the proposed algorithm with respect to 𝜖 = 0.05 +and 𝑑0 = 200 m for different levels of power of the pilot signals, +𝑃 = {5,10,20} dBm. As we increase the power of the pilot +signals, the estimation accuracy of 𝛼1 and 𝛼2 increases and +hence the error probability decreases. This is so because, as +we increase the power of pilot signals the received signals will +be less noisy which increases the chances of estimating the 𝛼1 +and 𝛼2 more accurately. +0 +500 +1000 +1500 +2000 +2500 +3000 +Number of Pilots +10 +2 +10 +1 +100 +Error Probability +Proposed Scheme, Pilot Power = 5 dBm +Upper Bound, Pilot Power = 5 dBm +Proposed Scheme, Pilot Power = 10 dBm +Upper Bound, Pilot Power = 10 dBm +Proposed Scheme, Pilot Power = 20 dBm +Upper Bound, Pilot Power = 20 dBm +Fig. 6: Error Probability Bound v/s Number of Pilots for 𝜖 = 0.05 +for 𝑃 = {5,10,20} dBm. +VI. CONCLUSION +We investigated the problem of estimation of the optimal +phase-shift at the HMT-assisted wireless communication system +in a noisy environment. We proposed a learning algorithm to +estimate the optimal phase-shifting parameters and showed that +the probability that the phase-shifting parameters generated by +the proposed algorithm to deviate by more than 𝜖 from the +optimal values decay exponentially fast as the number of pilots +grows. Our proposed algorithm exploited structural properties +of the channel gains in the far-field regions. + +11 +APPENDIX +A. Proof of Proposition 1 +Proof. Let us define the following events. +𝐴𝑛,𝑚 = |𝑋𝑛 − 𝑋𝑚| > 𝜖, +𝐴𝑛 = |𝑋𝑛 − 𝑋| > 𝜖 +2, +and +𝐴𝑚 = |𝑋𝑚 − 𝑋| > 𝜖 +2 +By the triangle inequality, we have +|𝑋𝑛 − 𝑋𝑚| ≤ |𝑋𝑛 − 𝑋| + |𝑋𝑚 − 𝑋|. +(64) +Using (64), the event 𝐴𝑛,𝑚 can be written as +|𝑋𝑛 − 𝑋𝑚| ≥ 𝜖 =⇒ |𝑋𝑛 − 𝑋| + |𝑋𝑚 − 𝑋| ≥ 𝜖 +Therefore, we have +𝐴𝑛,𝑚 ⊂ {|𝑋𝑛 − 𝑋| + |𝑋 − 𝑋𝑚| > 𝜖} +⊂ +� +|𝑋𝑛 − 𝑋| > 𝜖 +2 +� +|𝑋 − 𝑋𝑚| > 𝜖 +2 +� +(65) +Note that for any two events 𝐴 and 𝐵 where 𝐴 ⊂ 𝐵, then +P{𝐴} ≤ P{𝐵}. We use this fact in (65), and we get +P{|𝑋𝑛 − 𝑋𝑚| > 𝜖} ≤ P +� +|𝑋𝑛 − 𝑋| > 𝜖 +2 +� ++P +� +|𝑋𝑚 − 𝑋| > 𝜖 +2 +� +■ +B. Proof of Lemma 2 +Proof. We consider 𝑟(𝛽1, 𝛽2) as given in (3) which comprises +of two complex-valued factors +√ +𝑃 × 𝐻(𝛽1, 𝛽2) (see (1)) and 𝜁. +Write 𝜁 = 𝑛1 + 𝑗𝑛2, where 𝑛1 and 𝑛2 follows 𝑁 +� +0, 𝜎2 +2 +� +and +are independent, and write +√ +𝑃 × 𝐻(𝛽1, 𝛽2) = 𝑎 + 𝑗𝑏, where 𝑎 +and 𝑏 are real values. Therefore, +𝑟(𝛽1, 𝛽2) = |𝑦(𝛽1, 𝛽2)|2 = (𝑎 +𝑛1)2 + (𝑏 +𝑛2)2. +(66) +Note that 𝑎+𝑛1 +𝜎/ +√ +2 ∼ 𝑁 +� +𝑎 +𝜎/ +√ +2,1 +� +and 𝑏+𝑛2 +𝜎/ +√ +2 ∼ 𝑁 +� +𝑏 +𝜎/ +√ +2,1 +� +and they +are independent. Therefore, +2 +𝜎2 +� +(𝑎 +𝑛1)2 + (𝑏 +𝑛2)2 +� +∼ 𝜒2 +2 +� 2 +𝜎2 +� +𝑎2 + 𝑏2�� +. +(67) +Applying (67) in (66), we get 𝑋 = +2 +𝜎2 𝑟(𝛽1, 𝛽2) ∼ 𝜒2 +2 (𝜆1) , +where 𝜆1 = +2 +𝜎2 +��� +√ +𝑃 × 𝐻(𝛽1, 𝛽2) +��� +2 +. The second part of the lemma +follows from the additive property of non-central Chi-squared +distribution of the sum of 𝑛 i.i.d. RVs of 𝜒2 +2 (𝜆1) . +■ +C. Proof of Theorem 3 +As 𝑋𝑘,∀𝑘 are independent, applying the definition IV.1, the +moment generating function of +𝑛� +𝑘=1 +(𝑋𝑘 − 𝜇𝑘) is given by +E +� +𝑒 +𝑡 +𝑛� +𝑘=1 +(𝑋𝑘−𝜇𝑘) +� +≤ 𝑒 +𝜆2 +2 +𝑛� +𝑘=1 +𝜈2 +𝑘, +∀|𝑡| < �� +� +1 +max +𝑘=1,2,...,𝑛𝑏𝑘 +�� +� +. +Since the moment generating functions uniquely determines +the distribution, comparing with the definition IV.1, it follows +that +𝑛� +𝑘=1 +(𝑋𝑘 − 𝜇𝑘) is a sub-exponential (𝜈∗,𝑏∗) random variable, +where +𝑏∗ = +max +𝑘=1,2...,𝑛𝑏𝑘 +and +𝜈∗ = +� +� 𝑛 +∑︁ +𝑘=1 +𝜈2 +𝑘. +To prove the second part of the Theorem we use the following +tail bound on a sub-exponential distribution proved in [18]. +Proposition 1 ([18] Proposition 2.9). Let 𝑋 is sub-exponential +random variable with parameters (𝜈,𝑏) and E [𝑋] = 𝜇. Then +P{|𝑋 − 𝜇| ≥ 𝑡} ≤ +� +2𝑒− 𝑡2 +2𝜈2 , +if 0 ≤ 𝑡 ≤ 𝜈2 +𝑏 +2𝑒− 𝑡 +2𝑏 , +if 𝑡 ≥ 𝜈2 +𝑏 . +The claim immediately follows by applying the above result on +𝑍𝑛 := +𝑛� +𝑘=1 +(𝑋𝑘 − 𝜇𝑘), which is sub-exponential (𝜈∗,𝑏∗), where +𝑏∗ = +max +𝑘=1,2...,𝑛𝑏𝑘 and 𝜈∗ = +√︂ 𝑛� +𝑘=1 +𝜈2 +𝑘. +D. Proof of Corollary 1 +From Theorem 3, �𝑛 +𝑘=1(𝑋𝑘 −𝜇𝑘) is sub-exponential (𝜈∗,𝑏∗), +where 𝑏∗ = 4 and 𝜈∗ = 2√𝑛(2 + 2𝑎). Using the parameters +(𝜈∗,𝑏∗) = (2√𝑛(2+2𝑎),4) in Proposition 1, we get the required +upper bound as +P +������ +1 +𝑛 +𝑛 +∑︁ +𝑘=1 +(𝑋𝑘 − 𝜇𝑘) +����� ≥ 𝑡 +� +≤ +�� +�� +2𝑒 +− +𝑛𝑡2 +8(2+2𝑎)2 , +0 ≤ 𝑡 ≤ (2+2𝑎)2 +2𝑒− 𝑛𝑡 +8 , +𝑡 ≥ (2+2𝑎)2 +P +������ +1 +𝑛 +𝑛 +∑︁ +𝑘=1 +(𝑋𝑘 − 𝜇𝑘) +����� ≥ 𝑡 +� +≤ 2𝑒 +− +𝑛𝑡2 +8(2+2𝑎)2 , +𝑡 > 0. +E. Proof of Lemma 3 +If 𝑋 ∼ 𝜒2 +𝑝(𝑎), then according to [20], the moment-generating +function (MGF) of 𝑋 is given by +E [exp{𝑡(𝑋 − (𝑝 + 𝑎)}] = 𝑒−𝑡 ( 𝑝+𝑎)E +� +𝑒𝑡𝑋� += 𝑒−𝑡 ( 𝑝+𝑎)𝑒 +𝑎𝑡 +1−2𝑡 +(1−2𝑡) 𝑝/2 += 𝑒 +2𝑎𝑡2 +1−2𝑡 +𝑒−𝑝𝑡 +(1−2𝑡) 𝑝/2 , +for 𝑡 < 1 +2. +(68) +By following some calculus, refer [21], [18, Example 2.8], we +obtain +𝑒−𝑝𝑡 +(1−2𝑡) 𝑝/2 ≤ 𝑒2𝑝𝑡2, +for |𝑡| ≤ 1 +4. +(69) +For |𝑡| ≤ 1 +4, we have +𝑒 +2𝑎𝑡2 +1−2𝑡 ≤ 𝑒4𝑎𝑡2. +(70) +Applying (69) and (70) to (68), we obtain +E [exp{𝑡(𝑋 − (𝑝 + 𝑎)}] ≤ 𝑒2( 𝑝+2𝑎)𝑡2, +∀|𝑡| ≤ 1 +4. +(71) +Therefore, by (71), 𝑋 is Sub-exponential distribution with +parameters �2(𝑝 +2𝑎),4�. +REFERENCES +[1] Z. Wan, Z. Gao, F. Gao, M. Di Renzo, and M.-S. Alouini, “Terahertz +massive mimo with holographic reconfigurable intelligent surfaces,” IEEE +Transactions on Communications, 2021. + +12 +[2] V. Jamali, A. M. Tulino, G. Fischer, R. R. Müller, and R. Schober, +“Intelligent surface-aided transmitter architectures for millimeter-wave +ultra massive mimo systems,” IEEE Open Journal of the Communications +Society, vol. 2, pp. 144–167, 2020. +[3] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive +mimo for next generation wireless systems,” IEEE communications +magazine, vol. 52, no. 2, pp. 186–195, 2014. +[4] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless +network via joint active and passive beamforming,” IEEE Transactions +on Wireless Communications, vol. 18, no. 11, pp. 5394–5409, 2019. +[5] C. Huang, S. Hu, G. C. Alexandropoulos, A. Zappone, C. Yuen, R. Zhang, +M. Di Renzo, and M. Debbah, “Holographic mimo surfaces for 6g +wireless networks: Opportunities, challenges, and trends,” IEEE Wireless +Communications, vol. 27, no. 5, pp. 118–125, 2020. +[6] S. Hu, F. Rusek, and O. Edfors, “Beyond massive mimo: The potential +of data transmission with large intelligent surfaces,” IEEE Transactions +on Signal Processing, vol. 66, no. 10, pp. 2746–2758, 2018. +[7] I. Yoo and D. R. Smith, “Holographic metasurface antennas for uplink +massive mimo systems,” arXiv preprint arXiv:2108.12513, 2021. +[8] H. Zhang, N. Shlezinger, F. Guidi, D. Dardari, M. F. Imani, and Y. C. +Eldar, “Beam focusing for near-field multi-user mimo communications,” +IEEE Transactions on Wireless Communications, 2022. +[9] F. Dai and J. Wu, “Efficient broadcasting in ad hoc wireless networks +using directional antennas,” IEEE Transactions on Parallel and +Distributed Systems, vol. 17, no. 4, pp. 335–347, 2006. +[10] Z. Xiao, T. He, P. Xia, and X.-G. Xia, “Hierarchical codebook design +for beamforming training in millimeter-wave communication,” IEEE +Transactions on Wireless Communications, vol. 15, no. 5, pp. 3380–3392, +2016. +[11] K. Chen and C. Qi, “Beam training based on dynamic hierarchical +codebook for millimeter wave massive mimo,” IEEE Communications +Letters, vol. 23, no. 1, pp. 132–135, 2018. +[12] Ö. T. Demir, E. Björnson, and L. Sanguinetti, “Channel modeling and +channel estimation for holographic massive mimo with planar arrays,” +IEEE Wireless Communications Letters, vol. 11, no. 5, pp. 997–1001, +2022. +[13] M. Ghermezcheshmeh, V. Jamali, H. Gacanin, and N. Zlatanov, +“Channel estimation for large intelligent surface-based transceiver using +a parametric channel model,” arXiv preprint arXiv:2112.02874, 2021. +[14] M. R. Akdeniz, Y. Liu, M. K. Samimi, S. Sun, S. Rangan, T. S. Rappaport, +and E. Erkip, “Millimeter wave channel modeling and cellular capacity +evaluation,” IEEE journal on selected areas in communications, vol. 32, +no. 6, pp. 1164–1179, 2014. +[15] S. W. Ellingson, “Path loss in reconfigurable intelligent surface-enabled +channels,” in 2021 IEEE 32nd Annual International Symposium on +Personal, Indoor and Mobile Radio Communications (PIMRC). +IEEE, +2021, pp. 829–835. +[16] K. T. Selvan and R. Janaswamy, “Fraunhofer and fresnel distances: +Unified +derivation +for +aperture +antennas.” +IEEE +Antennas +and +Propagation Magazine, vol. 59, no. 4, pp. 12–15, 2017. +[17] M. Najafi, V. Jamali, R. Schober, and H. V. Poor, “Physics-based modeling +and scalable optimization of large intelligent reflecting surfaces,” IEEE +Transactions on Communications, vol. 69, no. 4, pp. 2673–2691, 2020. +[18] M. J. Wainwright, High-dimensional statistics: A non-asymptotic +viewpoint. +Cambridge University Press, 2019, vol. 48. +[19] W. Wang and W. Zhang, “Joint beam training and positioning for +intelligent reflecting surfaces assisted millimeter wave communications,” +IEEE Transactions on Wireless Communications, vol. 20, no. 10, pp. +6282–6297, 2021. +[20] A. +E. +El-Sayed, +A. +I. +Sahar, +and +Y. +A. +Yassmen, +“Moment +generating function of the unbalanced non-central chi-square distribution,” +International Journal of Engineering, vol. 4, no. 3, p. 8269, 2013. +[21] M. Ghosh, “Exponential tail bounds for chisquared random variables,” +Journal of Statistical Theory and Practice, vol. 15, no. 2, pp. 1–6, 2021. + diff --git a/LtE1T4oBgHgl3EQftAUX/content/tmp_files/load_file.txt b/LtE1T4oBgHgl3EQftAUX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c71f669a3a18203d4d9e415e8a25b3f671ea0a0a --- /dev/null +++ b/LtE1T4oBgHgl3EQftAUX/content/tmp_files/load_file.txt @@ -0,0 +1,765 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf,len=764 +page_content='1 Learning Optimal Phase-Shifts of Holographic Metasurface Transceivers Debamita Ghosh, IITB-Monash Research Academy, IIT Bombay, India Manjesh K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Hanawal, MLioNS Lab, IEOR, IIT Bombay, India Nikola Zlatanov, Innopolis University, Russia Abstract—Holographic metasurface transceivers (HMT) is an emerging technology for enhancing the coverage and rate of wireless communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' However, acquiring accurate channel state information in HMT-assisted wireless communication systems is critical for achieving these goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' In this paper, we propose an algorithm for learning the optimal phase-shifts at a HMT for the far-field channel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Our proposed algorithm exploits the structure of the channel gains in the far-field regions and learns the optimal phase-shifts in presence of noise in the received signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We prove that the probability that the optimal phase-shifts estimated by our proposed algorithm deviate from the true values decays exponentially in the number of pilot signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Extensive numerical simulations validate the theoretical guarantees and also demonstrate significant gains as compared to the state-of-the-art policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Index Terms—Holographic Metasurface Transceivers, Channel State Information, Uniform Exploration I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' INTRODUCTION Future wireless network technologies, namely beyond-5G and 6G, have been focused on millimeter wave (mmWave) and TeraHertz (THz) communications technologies as possible solutions to the ever growing demands for higher data rates and lower latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' However, mmWave and THz communications have challenges that need to be addressed before this technology is adopted [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' One such major challenge is signal deterioration due to reflections and absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' A possible solution for the signal deterioration are base stations (BSs) with massive antennas arrays that can provide large beamforming gains and thereby compensate for the signal deterioration [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' However, implementing a BS with a massive antenna array is itself challenging due to the high hardware costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Holographic Metasurface Transceivers (HMTs) are introduced as a promising solution for building a massive antenna array [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' A HMT is comprised of a large number of metamaterial elements densely deployed into a limited surface area in order to form a spatially continuous transceiver aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' These metamaterial elements at the HMT acts as phase-shifting antennas, where each phase-shifting element of the HMT can change the phase of transmiting/receiving signal and thereby beamform towards desired directions where the users are allocated [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Due to these continuous apertures, HMTs can be represented as an extension of the traditional massive antenna arrays with discrete antennas to continuous reflecting surfaces [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' In this paper, we consider the HMT-assisted wireless systems illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 1, where a HMT acts as a BS that serves multiple users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The performance of this system is dependent on channel state information (CSI) estimates at the HMT, which are used for accurate beamforming towards the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The authors in [7] and [8] have studied the effect of HMT-assisted systems on enhancing the communication performance under the assumption of perfect CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' However, perfect CSI is not available in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' In practice, the CSI has to be estimated via pilot signals, which results in inaccurate CSI estimates at the HMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The aim of this paper is to obtain accurate CSI estimates at the HMT, which in turn is used to set the optimal phase-shifts at the HMT that maximize the data rate to the users when the users are located in the far-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' To this end, we exploit the structure of the far-field channel model between the HMT and the users to show that the optimal phase-shifts at the HMT can be obtained from five samples of the received pilot signals at the HMT in a noiseless environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We then use this approach to develop a learning algorithm that learns the optimal phase-shifts from the received pilot signals at the HMT in a noisy environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Finally, we provide theoretical guarantees for our learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Specifically, we prove that the probability of the phase-shifts generated by our algorithm to deviate by more than 𝜖 from the optimal phase-shifts is small and decays as the number of pilot symbols increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The error analysis is based on tail probabilities of the non-central Chi-squared distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' In summary, our main contributions are as follows: We propose an efficient learning algorithm for estimating the optimal phase-shifts at an HMT in the presence of noise for the case when the users that the HMT is serving are located at the far-field region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We prove that the probability of the phase-shifts generated by our algorithm to deviate by more than 𝜖 from the optimal phase-shifts is small and decays exponentially as the number of pilots used for estimation increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We show numerically that the performance of the proposed algorithm significantly outperforms existing CSI estimation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Related Works Several channel estimation schemes, which are proposed for the massive antenna arrays, are also applicable to the considered HMT including exhaustive search [9], hierarchical search [10], [11], and compressed sensing (CS) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' As the exhaustive search in [9] significantly increases the training arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='03371v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='SP] 12 Dec 2022 2 overhead, the authors in [10] and [11] proposed the hierarchical search based on a predefined codebook as an improvement over the exhaustive search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The hierarchical schemes, in general, may incur high training overhead and system latency since they require non-trivial coordination among the transmitter and the user [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' On the other hand, the proposed CS-based channel estimation scheme in [11] provides trade-offs between accuracy of estimation and training overhead at different computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' On the other hand, CSI estimation schemes developed specifically for HMTs can be found in [12] and [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The authors in [12] proposed the least-square estimation based approach to study the channel estimation problem for the uplink between a single user and the BS equipped with the holographic surface with a large number of antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' However, the authors require an additional knowledge of antennas array geometry to reduce the pilot overhead required by the channel estimation, and hence the computational complexity scales up with the number of antennas at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' In [13], the authors proposed a scheme for the estimation of the far-field channel between a HMT and a user that requires only five pilots for perfect estimation in the noise-free environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' In the noisy case, the authors of [13] proposed an iterative algorithm that efficiently estimates the far-field channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Unlike the existing works, the training overhead and the computational cost of the proposed scheme in [13] does not scale with the number of phase-shifting elements at the HMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The iterative algorithm in [13] significantly outperforms the hierarchical and CS based schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' However, the authors in [13] did not provide any theoretical guarantees on their proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Motivated by [13], in this work, we propose an algorithm which outperforms the one in [13], and, in addition, we also provide theoretical guarantees for our proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The system and channel models for the HMT communication system are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The proposed algorithm for learning the optimal phase-shifts is given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' III and its theoretical guarantee is provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Numerical evaluation of the proposed algorithm is provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Finally, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' VI concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' SYSTEM AND CHANNEL MODELS We consider a HMT-assisted wireless communication system, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 1, where an HMT communicates with multiple users in the mmWave band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We assume that there is a Line of Sight (LoS) between the HMT and each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' As a result, when modeling the far-field channel, we only take into account the LoS path since its power is order of magnitude higher than non-line-of-sight (NLoS) paths [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The NLoS components are incorporated in the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We assume that the users send orthogonal pilots to the HMT for channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Based on the estimated CSI at the HMT to each user, the HMT sends data to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Hence, the data rate from the HMT to the users is directly dependent on the accuracy of the CSI estimates at the HMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Since in this paper our main goal is the accurate CSI estimation at the HMT to each user, which in turn send orthogonal pilots to the HMT, in the rest of the paper, we will focus on the CSI estimation between the HMT and a typical user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' RF Generator Phase-shifting Element User 1 User 2 User 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 1: The HMT-assisted wireless communication system [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' HMT Model The HMT has a rectangular surface of size 𝐿𝑥 × 𝐿𝑦, where 𝐿𝑥 and 𝐿𝑦 are the width and the length of the surface, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The HMT’s surface is comprised of a large number of sub-wavelength phase-shifting elements, where each elements is assumed to be a square of size 𝐿𝑒 × 𝐿𝑒 and can change the phase of the transmit/receive signal independently from rest of the elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Let 𝑑𝑟 be the distance between two neighboring phase-shifting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The total number of phase-shifting elements of the HMT is given by 𝑀 = 𝑀𝑥 × 𝑀𝑦, where 𝑀𝑥 = 𝐿𝑥/𝑑𝑟 and 𝑀𝑦 = 𝐿𝑦/𝑑𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Without loss of generality, we assume that the HMT lies in the 𝑥 − 𝑦 plane of a Cartesian coordinate system, where the center of the surface is at the origin of the coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Assuming 𝑀𝑥 and 𝑀𝑦 are odd numbers, the position of the (𝑚𝑥,𝑚𝑦)𝑡ℎ phase-shifting element in the Cartesian coordinate system is given as (𝑥, 𝑦) = (𝑚𝑥𝑑𝑟,𝑚𝑦𝑑𝑟), where 𝑚𝑥 ∈ � − 𝑀𝑥−1 2 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=', 𝑀𝑥−1 2 � and 𝑚𝑦 ∈ � − 𝑀𝑦−1 2 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=', 𝑀𝑦−1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' When 𝑀𝑥 or 𝑀𝑦 is even, the position of the (𝑚𝑥,𝑚𝑦)𝑡ℎ element can be appropriately defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Channel Model Consider the channel between the (𝑚𝑥,𝑚𝑦)𝑡ℎ phase-shifting element at the HMT and the typical user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Let the beamforming weight imposed by the (𝑚𝑥,𝑚𝑦)𝑡ℎ phase-shifting element at the HMT be Γ𝑚𝑥𝑚𝑦 = 𝑒 𝑗𝛽𝑚𝑥 𝑚𝑦 , where 𝛽𝑚𝑥𝑚𝑦 is the phase shift at the (𝑚𝑥,𝑚𝑦)𝑡ℎ element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Let 𝜆 denote the wavelength of the carrier frequency, 𝑘0 = 2𝜋 𝜆 be the wave number, 𝑑0 be the distance between the user and the center of the HMT and let 𝐹𝑚𝑥𝑚𝑦 denote the effect of the size and power radiation pattern of the (𝑚𝑥,𝑚𝑦)𝑡ℎ phase-shifting element on the channel coefficient [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Due to the far-field assumptions, the radiation pattern of all the phase-shifting elements of the HMT are identical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=', 𝐹𝑚𝑥𝑚𝑦 = 𝐹, ∀𝑚𝑥,𝑚𝑦 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Finally, let 𝜃 and 𝜙 denote the elevation and azimuth angles of the impinging wave from the user to the center of the HMT, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Now, if the phase-shift imposed by the (𝑚𝑥,𝑚𝑦)𝑡ℎ element, 𝛽𝑚𝑥,𝑚𝑦, is set to 𝛽𝑚𝑥𝑚𝑦 = − mod (𝑘0𝑑𝑟 (𝑚𝑥𝛽1 +𝑚𝑦𝛽2),2𝜋),∀𝑚𝑥,𝑚𝑦, 3 User HMT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 2: Distance between the (𝑚𝑥,𝑚𝑦)-th phase-shifting element at the HMT and the user [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' where 𝛽1 and 𝛽2 are the phase-shift parameters [13], [16], [17], which are the only degrees of freedom within the phase-shift 𝛽𝑚𝑥𝑚𝑦, then the HMT-user channel in the far-field is approximated accurately by [13], [16], [17] 𝐻(𝛽1, 𝛽2) = �√ 𝐹𝜆𝑒−𝑗𝑘0𝑑0 4𝜋𝑑0 � 𝐿𝑥𝐿𝑦 ×sinc � 𝐾𝑥𝜋(𝛼1 − 𝛽1) � ×sinc � 𝐾𝑦𝜋(𝛼2 − 𝛽2) � , (1) where 𝐾𝑥 = 𝐿𝑥 𝜆 ,𝐾𝑦 = 𝐿𝑦 𝜆 ,𝛼1 = sin(𝜃) cos(𝜙),𝛼2 = sin(𝜃) sin(𝜙), and sinc(𝑥) = sin(𝑥) 𝑥 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Please note that 𝛼1 ∈ [−1,1] and 𝛼2 ∈ [−1,1], and their values depend on the location of the user, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=', on 𝜃 and 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' From (1), it is clear that the absolute value of the HMT-user channel is maximized when the two sinc functions attain their maximum values, which occurs when the phase-shifting parameters, 𝛽1 and 𝛽2, are set to 𝛽1 = 𝛼1 and 𝛽2 = 𝛼2, where (𝛼1,𝛼2) are unknown to the HMT since they depend on the location of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Therefore, in the far-field case, the problem of finding the optimal phase-shifts of the elements at the HMT reduces to estimating the two parameters, 𝛼1 and 𝛼2 at the HMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 3 shows an example of |𝐻(𝛽1, 𝛽2)| as a function of (𝛽1, 𝛽2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' As can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 3, the graph of |𝐻(𝛽1, 𝛽2)| hits zero periodically and has several lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The optimal value (𝛼1,𝛼2) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='68,−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='45) is attained at the central lobe which has the highest peak and is attained for (𝛽∗ 1, 𝛽∗ 2) = (𝛼1,𝛼2) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='68,−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' PROPOSED CHANNEL ESTIMATION STRATEGY In this section, we propose an algorithm that estimates the optimal phase-shifting parameters 𝛽1 and 𝛽2 that maximize |𝐻(𝛽1, 𝛽2)| in (1) in the presence of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Problem Formulation In the channel estimation procedure, the user sends a pilot symbol 𝑥𝑝 = √ 𝑃 to the HMT, where 𝑃 is the pilot transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Then, the received signal at the HMT for fixed phase-shifting parameters (𝛽1, 𝛽2), denoted by 𝑦(𝛽1, 𝛽2), is given by 𝑦(𝛽1, 𝛽2) = √ 𝑃 × 𝐻(𝛽1, 𝛽2) + 𝜁, (2) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 3: |𝐻(𝛽1, 𝛽2)| v/s (𝛽1, 𝛽2) for values of (𝛼1,𝛼2) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='68,−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' where 𝜁 is the complex-valued additive white Gaussian noise (AWGN) with zero mean and variance 𝜎2 at the HMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The received signal in (2) is then squared in order to obtain the received signal squared, denoted by 𝑟(𝛽1, 𝛽2), and given by 𝑟(𝛽1, 𝛽2) = |𝑦(𝛽1, 𝛽2)|2 = ��� √ 𝑃 × 𝐻(𝛽1, 𝛽2) + 𝜁 ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (3) Objective: Our goal is to identify the optimal phase-shifting parameters, denoted by (𝛽∗ 1, 𝛽∗ 2), at the HMT that maximizes 𝑟(𝛽1, 𝛽2) given by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Specifically, we aim to solve the following optimisation problem (𝛽∗ 1, 𝛽∗ 2) = argmax 𝛽1∈[−1,1] 𝛽2∈[−1,1] 𝑟(𝛽1, 𝛽2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (4) The expected value of 𝑟(𝛽1, 𝛽2), denoted by 𝜇(𝛽1, 𝛽2), is given by 𝜇(𝛽1, 𝛽2) = E [𝑟(𝛽1, 𝛽2)] = ��� √ 𝑃 × 𝐻(𝛽1, 𝛽2) ��� 2 + 𝜎2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (5) Using (5), the optimization problem in (4) can be written equivalently as (𝛽∗ 1, 𝛽∗ 2) = argmax 𝛽1∈[−1,1] 𝛽2∈[−1,1] 𝜇(𝛽1, 𝛽2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (6) In order to obtain an intuition on how to solve (6), we first assume that 𝜇(𝛽1, 𝛽2) in (5) is known perfectly at the HMT for five specific values of the pair (𝛽1, 𝛽2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Later, we use the same intuition to solve (6) when 𝜇(𝛽1, 𝛽2) are not known perfectly but can be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The Optimal Phase-Shifting Parameters When 𝜇(𝛽1, 𝛽2) Are Known In Advance For notational convenience, let us define the set B as B = � (𝛽0 1, 𝛽0 2), (𝛽0 1 +𝑣, 𝛽0 2), (𝛽0 1 −𝑣, 𝛽0 2), (𝛽0 1, 𝛽0 2 + 𝑤), (𝛽0 1, 𝛽0 2 − 𝑤) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (7) The set B is comprised of five pairs of the phase-shifting parameters (𝛽1, 𝛽2), where 𝛽0 1 and 𝛽0 2 are some initial arbitrarily selected phase-shifting parameters, 𝑣 and 𝑤 are numbers chosen such that 𝐾𝑥𝑣 ∈ N and 𝐾𝑦𝑤 ∈ N hold, where N is the set of natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Please note that for a selected (𝛽0 1, 𝛽0 2) and a 1 (α_1,α2)=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='68,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='45) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='8 [H(β1, β2) /2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='4 β1 01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 2 β2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='4 14 chosen 𝑣 and 𝑤, if ��𝛽0 1 ±𝑣 �� ≥ 1 then we set ��𝛽0 1 ±𝑣 �� = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' In the same way, if ��𝛽0 2 ± 𝑤 �� ≥ 1 then we set ��𝛽0 2 ± 𝑤 �� = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' If the HMT can obtain 𝜇(𝛽0 1, 𝛽0 2), 𝜇(𝛽0 1 + 𝑣, 𝛽0 2), 𝜇(𝛽0 1 − 𝑣, 𝛽0 2), 𝜇(𝛽0 1, 𝛽0 2 + 𝑤) and 𝜇(𝛽0 1, 𝛽0 2 − 𝑤), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' obtain 𝜇(𝛽1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 𝛽2) for the five phase-shifting parameters in (𝛽1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 𝛽2) ∈ B given in (7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' then the optimal phase-shifting parameters 𝛽∗ 1 and 𝛽∗ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' which are the solutions of (6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' are given by 𝛽∗ 1 = � 𝛼(𝑖) 1 +𝛼( 𝑗) 1 2 : min 𝑖∈{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 𝑗 ∈{3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='4} ���𝛼(𝑖) 1 −𝛼( 𝑗) 1 ��� � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (8) where 𝛼(1)/(2) 1 = 𝛽0 1 + 𝑣 1± √︄���� 𝜇(𝛽0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝛽0 2)−𝜎2 𝜇(𝛽0 1+𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝛽0 2)−𝜎2 ���� 𝛼(3)/(4) 1 = 𝛽0 1 − 𝑣 1± √︄���� 𝜇(𝛽0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝛽0 2)−𝜎2 𝜇(𝛽0 1−𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝛽0 2)−𝜎2 ���� and 𝛽∗ 2 = � 𝛼(𝑖) 2 +𝛼( 𝑗) 2 2 : min 𝑖∈{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 𝑗 ∈{3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='4} ���𝛼(𝑖) 2 −𝛼( 𝑗) 2 ��� � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (9) where 𝛼(1)/(2) 2 = 𝛽0 2 + 𝑣 1± √︄���� 𝜇(𝛽0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝛽0 2)−𝜎2 𝜇(𝛽0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝛽0 2+𝑤)−𝜎2 ���� 𝛼(3)/(4) 2 = 𝛽0 2 − 𝑣 1± √︄���� 𝜇(𝛽0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝛽0 2)−𝜎2 𝜇(𝛽0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝛽0 2−𝑤)−𝜎2 ���� Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' By using (5) and (1) for any (𝛽1, 𝛽2) = (𝛽0 1, 𝛽0 2), we have the following 𝜇(𝛽0 1, 𝛽0 2) − 𝜎2 = ����� √ 𝑃 �√ 𝐹𝜆𝑒− 𝑗𝑘0𝑑0 4𝜋𝑑0 � 𝐿𝑥𝐿𝑦sinc � 𝐾𝑥(𝛼1 − 𝛽0 1) � ×sinc � 𝐾𝑦(𝛼2 − 𝛽0 2) ������ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (10) For (𝛽1, 𝛽2) = (𝛽0 1 +𝑣, 𝛽0 2), where 𝑣 is any arbitrary parameter such that 𝐾𝑥𝑣 ∈ N and ��𝛽0 1 ±𝑣 �� ≤ 1 holds, we have 𝜇(𝛽0 1 +𝑣, 𝛽0 2) − 𝜎2 = ����� √ 𝑃 �√ 𝐹𝜆𝑒−𝑗𝑘0𝑑0 4𝜋𝑑0 � 𝐿𝑥𝐿𝑦 ×sinc � 𝐾𝑦(𝛼2 − 𝛽0 2) ������ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (11) Dividing (10) by (11), we obtain 𝜇(𝛽0 1, 𝛽0 2) − 𝜎2 𝜇(𝛽0 1 +𝑣, 𝛽0 2) − 𝜎2 = ����sinc � 𝐾𝑥(𝛼1 − 𝛽0 1) ����� 2 ����sinc � 𝐾𝑥(𝛼1 − 𝛽0 1 −𝑣) ����� 2 𝜇(𝛽0 1, 𝛽0 2) − 𝜎2 𝜇(𝛽0 1 +𝑣, 𝛽0 2) − 𝜎2 = ���� sin(𝐾𝑥 𝜋(𝛼1−𝛽0 1)) 𝐾𝑥 𝜋(𝛼1−𝛽0 1) ���� 2 ���� sin(𝐾𝑥 𝜋(𝛼1−𝛽0 1−𝑣)) 𝐾𝑥 𝜋(𝛼1−𝛽0 1−𝑣) ���� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (12) If 𝑣 is selected such that 𝐾𝑥𝑣 ∈ N, then we have ����sin � 𝐾𝑥𝜋(𝛼1 − 𝛽0 1 ±𝑣) ����� = ����sin � 𝐾𝑥𝜋(𝛼1 − 𝛽0 1) �����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' As a result, (12) is simplified to 𝜇(𝛽0 1, 𝛽0 2) − 𝜎2 𝜇(𝛽0 1 +𝑣, 𝛽0 2) − 𝜎2 = ����� 𝛼1 − 𝛽0 1 −𝑣 𝛼1 − 𝛽0 1 ����� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (13) Since 𝜇(𝛽1, 𝛽2) ≥ 𝜎2, it follows that 𝜇(𝛽1, 𝛽2) − 𝜎2 = ��𝜇(𝛽1, 𝛽2) − 𝜎2�� always holds, for all (𝛽1, 𝛽2) ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Using this fact, (13) can be written equivalently as � � ������ 𝜇(𝛽0 1, 𝛽0 2) − 𝜎2 𝜇(𝛽0 1 +𝑣, 𝛽0 2) − 𝜎2 ����� = ����� 𝛼1 − 𝛽0 1 −𝑣 𝛼1 − 𝛽0 1 �����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (14) By solving the nonlinear equation in (14) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' the unknown 𝛼1, we obtain two solutions for 𝛼1, denoted by 𝛼(1) 1 and 𝛼(2) 1 , given by 𝛼(1)/(2) 1 = 𝛽0 1 + 𝑣 1± √︂��� 𝜇(𝛽0 1,𝛽0 2)−𝜎2 𝜇(𝛽0 1+𝑣,𝛽0 2)−𝜎2 ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (15) It is not known which of the two values 𝛼(1) 1 and 𝛼(2) 1 is equal to 𝛼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' To identify the correct solution for 𝛼1 of the two solutions given by (15), we need the value of 𝜇(𝛽1, 𝛽2) for (𝛽1, 𝛽2) = (𝛽0 1 − 𝑣, 𝛽0 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Following the same procedure as for (10)-(15), but now by using the values of 𝜇(𝛽1, 𝛽2) for (𝛽1, 𝛽2) = (𝛽0 1, 𝛽0 2) and (𝛽1, 𝛽2) = (𝛽0 1 −𝑣, 𝛽0 2), we obtain � � ������ 𝜇(𝛽0 1, 𝛽0 2) − 𝜎2 𝜇(𝛽0 1 −𝑣, 𝛽0 2) − 𝜎2 ����� = ����� 𝛼1 − 𝛽0 1 +𝑣 𝛼1 − 𝛽0 1 �����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (16) By solving (16), we obtain 𝛼(3)/(4) 1 = 𝛽0 1 − 𝑣 1± √︂��� 𝜇(𝛽0 1,𝛽0 2)−𝜎2 𝜇(𝛽0 1−𝑣,𝛽0 2)−𝜎2 ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (17) One of the solutions in (15) is identical to one of the solutions in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Therefore, using (15) and (17), the correct solution of 𝛼1 can be obtained as1 𝛼1 = � 𝛼(𝑖) 1 +𝛼( 𝑗) 1 2 : min 𝑖∈{1,2}, 𝑗 ∈{3,4} ���𝛼(𝑖) 1 −𝛼( 𝑗) 1 ��� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (18) In order to obtain 𝛼2, we need the value of 𝜇(𝛽1, 𝛽2) for (𝛽1, 𝛽2) = (𝛽0 1, 𝛽0 2), which we already have, and for (𝛽1, 𝛽2) = (𝛽0 1, 𝛽0 2 +𝑤), where 𝑤 is selected such that 𝐾𝑦𝑤 ∈ N, ��𝛽0 2 ± 𝑤 �� ≤ 1 and ��sin(𝐾𝑦𝜋(𝛼2 − 𝛽0 2 ± 𝑤)) �� = ��sin(𝐾𝑦𝜋(𝛼2 − 𝛽0 2)) ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Then, similar to (10)-(14), we use the values of 𝜇(𝛽1, 𝛽2) for (𝛽1, 𝛽2) = (𝛽0 1, 𝛽0 2) and (𝛽1, 𝛽2) = (𝛽0 1, 𝛽0 2 + 𝑤) to obtain 1Note that 𝛼1 can also be written equivalently as 𝛼1 = � 𝛼(1) 1 , 𝛼(2) 1 � � � 𝛼(3) 1 , 𝛼(4) 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' However, the expression in (18) is more convenient for the case when the values of 𝜇(𝛽1, 𝛽2) need to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 5 � � ������ 𝜇(𝛽0 1, 𝛽0 2) − 𝜎2 𝜇(𝛽0 1, 𝛽0 2 + 𝑤) − 𝜎2 ����� = ����� 𝛼2 − 𝛽0 2 − 𝑤 𝛼2 − 𝛽0 2 �����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (19) By solving the nonlinear equation (19), we obtain two solutions for 𝛼2, denoted by 𝛼(1) 2 and 𝛼(2) 2 , given by 𝛼(1)/(2) 2 = 𝛽0 2 + 𝑤 1± √︂��� 𝜇(𝛽0 1,𝛽0 2)−𝜎2 𝜇(𝛽0 1,𝛽0 2+𝑤)−𝜎2 ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (20) To identify the correct solution for 𝛼2 of the two given in (20), we need the value of 𝜇(𝛽1, 𝛽2) for (𝛽1, 𝛽2) = (𝛽0 1, 𝛽0 2 −𝑤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Again, following the procedure from (10)-(15), by using the values of 𝜇(𝛽1, 𝛽2) for (𝛽0 1, 𝛽0 2) and (𝛽0 1, 𝛽0 2 − 𝑤), we obtain 𝛼(3)/(4) 2 = 𝛽0 2 − 𝑤 1± √︂��� 𝜇(𝛽0 1,𝛽0 2)−𝜎2 𝜇(𝛽0 1,𝛽0 2−𝑤)−𝜎2 ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (21) One of the solutions in (20) is exactly same as the solutions of (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Therefore, using (20) and (21), the correct solution of 𝛼2 can be obtained as 𝛼2 = � 𝛼(𝑖) 2 +𝛼( 𝑗) 2 2 : min 𝑖∈{1,2}, 𝑗 ∈{3,4} ���𝛼(𝑖) 2 −𝛼( 𝑗) 2 ��� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (22) Finally, by setting 𝛽∗ 1 = 𝛼1 and 𝛽∗ 2 = 𝛼2, where 𝛼1 and 𝛼2 are given by (18) and (22), respectively, we obtain (8) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ■ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' In [13, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='A], the authors proposed the channel estimation strategy under the assumption that there is no noise in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' However, in the noisy case, we proposed an estimation scheme based on the assumption that 𝜇(𝛽1, 𝛽2) for any of the phase-shifting parameters (𝛽1, 𝛽2) ∈ B are perfectly known at the HMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' However, in practice the exact values of 𝜇(𝛽1, 𝛽2) for any of the phase-shifting parameters (𝛽1, 𝛽2) ∈ B cannot be known in advance at the HMT, and therefore they need to be estimated using pilot symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' In the following, we propose an algorithm that estimates 𝜇(𝛽1, 𝛽2) for the phase-shifting parameters in B and then uses the estimated values of 𝜇(𝛽1, 𝛽2) to find the optimal phase-shifting parameters (𝛽∗ 1, 𝛽∗ 2) in the presence of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Estimation Of The Optimal Phase-Shifting Parameters In The Noisy Case The user sends in total 𝑁 number of pilot signals to the HMT for the estimation of the five values of 𝜇(𝛽1, 𝛽2) for the five pairs of (𝛽1, 𝛽2) ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' As a result, the proposed algorithm works in five epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' In the 𝑘𝑡ℎ epoch, for 𝑘 = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=', the user transmits � 𝑁 5 � number of pilots to the HMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The HMT sets (𝛽1, 𝛽2) to the 𝑘𝑡ℎ element in B, and collects � 𝑁 5 � samples of the received signal squared, given by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Then 𝜇(𝛽1, 𝛽2), for (𝛽1, 𝛽2) being the 𝑘𝑡ℎ elements in B, is estimated as ˆ𝜇(𝛽1, 𝛽2) = 1 ⌊𝑁/5⌋ ⌊𝑁 /5⌋ ∑︁ 𝑖=1 𝑟𝑖(𝛽1, 𝛽2), (23) where 𝑟𝑖(𝛽1, 𝛽2) is the 𝑖𝑡ℎ sample of 𝑟(𝛽1, 𝛽2) in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Next, we replace 𝜇(𝛽1, 𝛽2) in (15), (17), (20), and (21) by ˆ𝜇(𝛽1, 𝛽2), ∀(𝛽1, 𝛽2) ∈ B, and thereby obtain our estimates for 𝛽∗ 1 and 𝛽∗ 2, denoted by ˆ𝛽∗ 1 and ˆ𝛽∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The pseudo-code of the proposed algorithm is given in Two-Stage Phase-Shifts Estimation Algorithm below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We note that the choice of the Two-Stage Phase-Shifts Estimation Algorithm 1: Input: 𝑁,B,𝜎2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 2: ***Stage 1: Uniform Exploration *** 3: for 𝑘 = 1 to 5 do 4: HMT sets (𝛽1, 𝛽2) to the 𝑘𝑡ℎ pair in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 5: User sends ⌊𝑁/5⌋ number of pilots to the HMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 6: For the 𝑖𝑡ℎ pilot, the HMT receives 𝑟𝑖(𝛽1, 𝛽2), given by (3), for 𝑖 = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=', ⌊𝑁/5⌋ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 7: The HMT computes ˆ𝜇𝑘 (𝛽1, 𝛽2) using (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 8: end for 9: ***Stage 2: Estimate Optimal Phase-Shifting Parameters*** 10: Obtain ˆ𝛽∗ 1 as ˆ𝛽∗ 1 = � ˆ𝛼(𝑖) 1 + ˆ𝛼( 𝑗) 1 2 : min 𝑖∈{1,2}, 𝑗 ∈{3,4} ��� ˆ𝛼(𝑖) 1 − ˆ𝛼( 𝑗) 1 ��� � , (24) where ˆ𝛼(1)/(2) 1 is obtained by replacing the value of 𝜇(𝛽1, 𝛽2) by ˆ𝜇(𝛽1, 𝛽2) in (15), and ˆ𝛼(3)/(4) 1 is obtained by replacing the value of 𝜇(𝛽1, 𝛽2) by ˆ𝜇(𝛽1, 𝛽2) in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 11: Obtain ˆ𝛽∗ 2 as ˆ𝛽∗ 2 = � ˆ𝛼(𝑖) 2 + ˆ𝛼( 𝑗) 2 2 : min 𝑖∈{1,2}, 𝑗 ∈{3,4} ��� ˆ𝛼(𝑖) 2 − ˆ𝛼( 𝑗) 2 ��� � , (25) where ˆ𝛼(1)/(2) 2 is obtained by replacing the value of 𝜇(𝛽1, 𝛽2) by ˆ𝜇(𝛽1, 𝛽2) in (20), and ˆ𝛼(3)/(4) 2 is obtained by replacing the value of 𝜇(𝛽1, 𝛽2) by ˆ𝜇(𝛽1, 𝛽2) in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 12: Output: ˆ𝛽∗ 1 and ˆ𝛽∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 13: Phase-shifts at HMT Set the phase-shift of the (𝑚𝑥,𝑚𝑦)𝑡ℎ element at the HMT to 𝛽𝑚𝑥𝑚𝑦 = − mod (𝑘0𝑑𝑟 (𝑚𝑥 ˆ𝛽∗ 1 +𝑚𝑦 ˆ𝛽∗ 2),2𝜋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' initial (𝛽0 1, 𝛽0 2) in the set B was arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The values of (𝛽0 1, 𝛽0 2) can effect the estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' In general, if the values (𝛽0 1, 𝛽0 2) are closer to the (𝛼1,𝛼2), the better the estimation will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' A good choice for (𝛽0 1, 𝛽0 2) is given in [13, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='C], which leads to faster learning of (𝛼1,𝛼2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' THEORETICAL GUARANTEES FOR THE PROPOSED ALGORITHM In the section, we bound the probability that the estimates, obtained from the proposed Two-Stage Phase-Shifts Estimation Algorithm Algorithm, deviate from the true values of (𝛼1,𝛼2) by an amount 0 ≤ 𝜖 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' In particular, we upper bound the following error probability P �� ˆ𝛽∗ 1 −𝛼1 �2 + � ˆ𝛽∗ 2 −𝛼2 �2 ≥ 𝜖 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (26) 6 We use the following results to upper bound the error probability in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Let {𝑋𝑛} be a sequence of random variables (RVs) on a probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Let 𝑋 be a RV defined on the same probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Then, the following holds P{|𝑋𝑛 − 𝑋𝑚| ≥ 𝜖} ≤ P � |𝑋𝑛 − 𝑋| ≥ 𝜖 2 � +P � |𝑋𝑚 − 𝑋| ≥ 𝜖 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The proof is given in the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ■ Let 𝜒2 𝑝(𝜆) denote a non-central Chi-squared distribution with 𝑝 degrees of freedom and non-centrality parameter 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Let 𝑋 = 2 𝜎2 𝑟(𝛽1, 𝛽2), where 𝑟(𝛽1, 𝛽2) is given by (3), and let 𝜆1 = 2 𝜎2 ��� √ 𝑃𝐻(𝛽1, 𝛽2) ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Then, 𝑋 is distributed as 𝜒2 2(𝜆1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=', 𝑋 ∼ 𝜒2 2(𝜆1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Furthermore, if 𝑋𝑖 for 𝑖 = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=',𝑛 are 𝑛 independently and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=') RVs of 𝜒2 2(𝜆1), then 𝑛 ∑︁ 𝑖=1 𝑋𝑖 ∼ 𝜒2 2𝑛(𝑛𝜆1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The proof is given in the Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ■ The following theorem provides an upper bound on the error probability in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Let us perform uniform exploration on the set B given in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' For any 0 ≤ 𝜖 ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' the error probability in (26) is upper bounded as P �� ˆ𝛽∗ 1 −𝛼1 �2 + � ˆ𝛽∗ 2 −𝛼2 �2 ≥ 𝜖 � ≤ 4 � 𝑒− 𝑛 32 � 𝜖 𝜆2 1+𝜆2 �2 + 𝑒− 𝑛 32 � 𝜖 𝜆3 1+𝜆3 �2 + 𝑒− 𝑛 32 � 𝜖 𝜆4 1+𝜆4 �2 + 𝑒− 𝑛 32 � 𝜖 𝜆5 1+𝜆5 �2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (27) where 𝜆1 = 2 ��� √ 𝑃𝐻(𝛽0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 𝛽0 2) ��� 2 𝜎2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 𝜆2 = 2 ��� √ 𝑃𝐻(𝛽0 1 +𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 𝛽0 2) ��� 2 𝜎2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 𝜆3 = 2 ��� √ 𝑃𝐻(𝛽0 1 −𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 𝛽0 2) ��� 2 𝜎2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 𝜆4 = 2 ��� √ 𝑃𝐻(𝛽0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 𝛽0 2 + 𝑤) ��� 2 𝜎2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 𝜆5 = 2 ��� √ 𝑃𝐻(𝛽0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 𝛽0 2 − 𝑤) ��� 2 𝜎2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Let us denote the estimate of 𝜇(𝛽0 1, 𝛽0 2) by ˆ𝜇(𝛽0 1, 𝛽0 2) which is given by ˆ𝜇(𝛽0 1, 𝛽0 2) = 1 𝑛 𝑛 ∑︁ 𝑖=1 𝑟𝑖(𝛽0 1, 𝛽0 2) = 𝜎2 2𝑛 𝑛 ∑︁ 𝑖=1 𝑋𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Using Lemma 2, we have ˆ𝜇1 := 2𝑛 𝜎2 ˆ𝜇(𝛽0 1, 𝛽0 2) ∼ 𝜒2 2𝑛(𝑛𝜆1) (28) ˆ𝜇2 := 2𝑛 𝜎2 ˆ𝜇(𝛽0 1 +𝑣, 𝛽0 2) ∼ 𝜒2 2𝑛(𝑛𝜆2) (29) ˆ𝜇3 := 2𝑛 𝜎2 ˆ𝜇(𝛽0 1 −𝑣, 𝛽0 2) ∼ 𝜒2 2𝑛(𝑛𝜆3) (30) ˆ𝜇4 := 2𝑛 𝜎2 ˆ𝜇(𝛽0 1, 𝛽0 2 + 𝑤) ∼ 𝜒2 2𝑛(𝑛𝜆4) (31) ˆ𝜇5 := 2𝑛 𝜎2 ˆ𝜇(𝛽0 1, 𝛽0 2 − 𝑤) ∼ 𝜒2 2𝑛(𝑛𝜆5) (32) where 𝜆1,𝜆2,𝜆3,𝜆4 and 𝜆5 is given in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The random variables ˆ𝜇1, ˆ𝜇2, ˆ𝜇3, ˆ𝜇4, and ˆ𝜇5 are mutually independent, since they are sampled at different epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The estimated optimal phase-shifting parameters ( ˆ𝛽∗ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ˆ𝛽∗ 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' are given by (24) and (25),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' where the values of ˆ𝛼(1) 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ˆ𝛼(2) 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ˆ𝛼(3) 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ˆ𝛼(4) 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ˆ𝛼(1) 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ˆ𝛼(2) 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ˆ𝛼(3) 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' and ˆ𝛼(4) 2 are given by ˆ𝛼(1)/(2) 1 = 𝛽0 1 + 𝑣 1± √︄���� ˆ𝜇(𝛽0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝛽0 2)−𝜎2 ˆ𝜇(𝛽0 1+𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝛽0 2)−𝜎2 ���� (33) ˆ𝛼(3)/(4) 1 = 𝛽0 1 − 𝑣 1± √︄���� ˆ𝜇(𝛽0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝛽0 2)−𝜎2 ˆ𝜇(𝛽0 1−𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝛽0 2)−𝜎2 ���� (34) ˆ𝛼(1)/(2) 2 = 𝛽0 2 + 𝑤 1± √︄���� ˆ𝜇(𝛽0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝛽0 2)−𝜎2 ˆ𝜇(𝛽0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝛽0 2+𝑤)−𝜎2 ���� (35) ˆ𝛼(3)/(4) 2 = 𝛽0 2 − 𝑤 1± √︄���� ˆ𝜇(𝛽0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝛽0 2)−𝜎2 ˆ𝜇(𝛽0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝛽0 2−𝑤)−𝜎2 ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (36) By inserting (28), (29), (30), (31), and (32) into (33), (34), (35) and (36), we obtain ˆ𝛼(1)/(2) 1 = 𝛽0 1 + 𝑣 1± √︂��� ˆ𝜇1−2𝑛 ˆ𝜇2−2𝑛 ��� (37) ˆ𝛼(3)/(4) 1 = 𝛽0 1 − 𝑣 1± √︂��� ˆ𝜇1−2𝑛 ˆ𝜇3−2𝑛 ��� (38) ˆ𝛼(1)/(2) 2 = 𝛽0 2 + 𝑤 1± √︂��� ˆ𝜇1−2𝑛 ˆ𝜇4−2𝑛 ��� (39) ˆ𝛼(3)/(4) 2 = 𝛽0 2 − 𝑤 1± √︂��� ˆ𝜇1−2𝑛 ˆ𝜇5−2𝑛 ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (40) Let us denote 𝐼 := P ��� ˆ𝛽∗ 1 −𝛼1 �� ≥ √︂ 𝜖 2 � 𝐼𝐼 := P ��� ˆ𝛽∗ 2 −𝛼2 �� ≥ √︂ 𝜖 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Now, applying Lemma 1 in (26), we obtain P �� ˆ𝛽∗ 1 −𝛼1 �2 + � ˆ𝛽∗ 2 −𝛼2 �2 ≥ 𝜖 � ≤ P �� ˆ𝛽∗ 1 −𝛼1 �2 ≥ 𝜖 2 � +P �� ˆ𝛽∗ 2 −𝛼2 �2 ≥ 𝜖 2 � ≤ 𝐼 + 𝐼𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (41) We upper bound each of the term in right-hand side of (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We begin with the first term P ��� ˆ𝛽∗ 1 −𝛼1 �� ≥ √︁ 𝜖 2 � , denoted as I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' G Step 1: Upper bound on I From (24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='��� ˆ𝛽∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 −𝛼1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='�� ≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='√︂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝜖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='= P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='� ������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='ˆ𝛼(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='+ ˆ𝛼(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='−𝛼1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='����� ≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='√︂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝜖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='� ������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='ˆ𝛼(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='+ ˆ𝛼(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='−𝛼1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='����� ≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='√︂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝜖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='� ������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='ˆ𝛼(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='+ ˆ𝛼(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='−𝛼1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='����� ≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='√︂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝜖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='� ������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='ˆ𝛼(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='+ ˆ𝛼(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='−𝛼1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='����� ≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='√︂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝜖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='≤ P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='ˆ𝛼(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='+ ˆ𝛼(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='−𝛼1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='����� ≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='√︂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝜖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='+P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='ˆ𝛼(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='+ ˆ𝛼(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='−𝛼1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='����� ≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='√︂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝜖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='+P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='ˆ𝛼(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='+ ˆ𝛼(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='−𝛼1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='����� ≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='√︂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝜖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='+P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='ˆ𝛼(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='+ ˆ𝛼(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='−𝛼1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='����� ≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='√︂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝜖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='∑︁ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='𝑖=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 𝑗=3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='4 P ����� � ˆ𝛼(𝑖) 1 −𝛼1 � + � ˆ𝛼( 𝑗) 1 −𝛼1 ����� ≥ 2 √︂ 𝜖 2 � ≤ ∑︁ 𝑖=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='2 𝑗=3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='4 � P ���� ˆ𝛼(𝑖) 1 −𝛼1 ��� ≥ √︂ 𝜖 2 � +P ���� ˆ𝛼( 𝑗) 1 −𝛼1 ��� ≥ √︂ 𝜖 2 � � = 2 � P ���� ˆ𝛼(1) 1 −𝛼1 ��� ≥ √︂ 𝜖 2 � +P ���� ˆ𝛼(2) 1 −𝛼1 ��� ≥ √︂ 𝜖 2 � +P ���� ˆ𝛼(3) 1 −𝛼1 ��� ≥ √︂ 𝜖 2 � +P ���� ˆ𝛼(4) 1 −𝛼1 ��� ≥ √︂ 𝜖 2 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (42) where we applied the union bound to get the first inequality and applied Lemma 1 for the second inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We now bound each term in (42) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ® Upper bound of P ���� ˆ𝜶(1) 1 −𝜶1 ��� ≥ √︁ 𝝐 2 � : Substituting the values of ˆ𝛼(1) 1 , as given by (37), in P ���� ˆ𝜶(1) 1 −𝜶1 ��� ≥ √︁ 𝝐 2 � , we obtain P ���� ˆ𝛼(1) 1 −𝛼1) ��� ≥ √︂ 𝜖 2 � = P ������ ������ ��������� 𝑣 1+ √︂��� ˆ𝜇1−2𝑛 ˆ𝜇2−2𝑛 ��� − (𝛼1 − 𝛽0 1) ��������� ≥ √︂ 𝜖 2 ������ ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (43) Note that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ��������� 𝑣 1+ √︂��� ˆ𝜇1−2𝑛 ˆ𝜇2−2𝑛 ��� − (𝛼1 − 𝛽0 1) ��������� ≤ ��������� 𝑣 1+ √︂��� ˆ𝜇1−2𝑛 ˆ𝜇2−2𝑛 ��� ��������� + �����𝛼1 − 𝛽0 1 �����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (44) By applying (44) in (43), we obtain P ���� ˆ𝛼(1) 1 −𝛼1) ��� ≥ √︂ 𝜖 2 � ≤ P ������ ������ ��������� 1 1+ √︂��� ˆ𝜇1−2𝑛 ˆ𝜇2−2𝑛 ��� ��������� ≥ 1 𝑣 �√︂ 𝜖 2 − �����𝛼1 − 𝛽0 1 ����� ������� ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (45) For the RVs ˆ𝜇1 and ˆ𝜇2, 1 1+ √︂��� ˆ𝜇1−2𝑛 ˆ𝜇2−2𝑛 ��� is always positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Using this fact in (45), we obtain P ���� ˆ𝛼(1) 1 −𝛼1) ��� ≥ √︂ 𝜖 2 � ≤ P ������ ������ 1 1+ √︂��� ˆ𝜇1−2𝑛 ˆ𝜇2−2𝑛 ��� ≥ 1 𝑣 �√︂ 𝜖 2 − �����𝛼1 − 𝛽0 1 ����� ������� ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Let 𝑎 = 1 𝑣 �√︁ 𝜖 2 − ��𝛼1 − 𝛽0 1 �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We have P ���� ˆ𝛼(1) 1 −𝛼1) ��� ≥ √︂ 𝜖 2 � ≤ P �� �� 1+ √︄���� ˆ𝜇1 −2𝑛 ˆ𝜇2 −2𝑛 ���� ≤ 1 𝑎 �� �� = P ����� ˆ𝜇1 −2𝑛 ˆ𝜇2 −2𝑛 ���� ≤ � 1− 1 𝑎 �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (46) ® Upper bound of P ���� ˆ𝜶(2) 1 −𝜶1 ��� ≥ √︁ 𝝐 2 � : Substituting the values of ˆ𝛼(2) 1 , as given by (37), in P ���� ˆ𝜶(2) 1 −𝜶1 ��� ≥ √︁ 𝝐 2 � , we obtain P ���� ˆ𝛼(2) 1 −𝛼1) ��� ≥ √︂ 𝜖 2 � = P ������ ������ ��������� 𝑣 1− √︂��� ˆ𝜇1−2𝑛 ˆ𝜇2−2𝑛 ��� − (𝛼1 − 𝛽0 1) ��������� ≥ √︂ 𝜖 2 ������ ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (47) Note that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ��������� 𝑣 1− √︂��� ˆ𝜇1−2𝑛 ˆ𝜇2−2𝑛 ��� − (𝛼1 − 𝛽0 1) ��������� ≤ ��������� 𝑣 1− √︂��� ˆ𝜇1−2𝑛 ˆ𝜇2−2𝑛 ��� ��������� + �����𝛼1 − 𝛽0 1 �����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (48) By applying (48) in the right-hand side of (47), we 8 obtain P ���� ˆ𝛼(2) 1 −𝛼1) ��� ≥ √︂ 𝜖 2 � ≤ P ��� ��� ������ 1− √︄���� ˆ𝜇1 −2𝑛 ˆ𝜇2 −2𝑛 ���� ������ ≤ 1 𝑎 ��� ��� = P �� 1− 1 𝑎 �2 ≤ ���� ˆ𝜇1 −2𝑛 ˆ𝜇2 −2𝑛 ���� ≤ � 1+ 1 𝑎 �2� ≤ P ����� ˆ𝜇1 −2𝑛 ˆ𝜇2 −2𝑛 ���� ≤ � 1+ 1 𝑎 �2� −P ����� ˆ𝜇1 −2𝑛 ˆ𝜇2 −2𝑛 ���� ≤ � 1− 1 𝑎 �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (49) ® Upper bound of P ���� ˆ𝜶(3) 1 −𝜶1 ��� ≥ √︁ 𝝐 2 � : Substituting the values of ˆ𝛼(3) 1 , as given by (38), in P ���� ˆ𝜶(3) 1 −𝜶1 ��� ≥ √︁ 𝝐 2 � and following similar steps to bound P ���� ˆ𝜶(1) 1 −𝜶1 ��� ≥ √︁ 𝝐 2 � , we obtain P ���� ˆ𝛼(3) 1 −𝛼1) ��� ≥ √︂ 𝜖 2 � ≤ P ����� ˆ𝜇1 −2𝑛 ˆ𝜇3 −2𝑛 ���� ≤ � 1− 1 𝑎 �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (50) ® Upper bound of P ���� ˆ𝜶(4) 1 −𝜶1 ��� ≥ √︁ 𝝐 2 � : Substituting the values of ˆ𝛼(4) 1 , as given by (38), in P ���� ˆ𝜶(4) 1 −𝜶1 ��� ≥ √︁ 𝝐 2 � and following similar steps to bound P ���� ˆ𝜶(2) 2 −𝜶1 ��� ≥ √︁ 𝝐 2 � , we obtain P ���� ˆ𝛼(4) 1 −𝛼1) ��� ≥ √︂ 𝜖 2 � ≤ P ����� ˆ𝜇1 −2𝑛 ˆ𝜇3 −2𝑛 ���� ≤ � 1+ 1 𝑎 �2� −P ����� ˆ𝜇1 −2𝑛 ˆ𝜇3 −2𝑛 ���� ≤ � 1− 1 𝑎 �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (51) By inserting the bounds (46), (49), (50) and (51) to (42) we obtain P ��� ˆ𝛽∗ 1 −𝛼1 �� ≥ √︂ 𝜖 2 � ≤ 2 � P ����� ˆ𝜇2 −2𝑛 ˆ𝜇1 −2𝑛 ���� ≥ 𝛾1 � +P ����� ˆ𝜇3 −2𝑛 ˆ𝜇1 −2𝑛 ���� ≥ 𝛾1 �� , (52) where we set 𝛾1 = � 1 1+(1/𝑎) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We next upper bound each term on the right-hand side of (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The bounds are derived using the properties of the sub-exponential distributions which we introduce below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' G Step 2: Sub-exponential Distributions and its Tail Bound Definition IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 (sub-exponential distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' A RV 𝑋 with mean 𝜇 is said to be sub-exponential with parameters (𝜈,𝛼), for 𝛼 > 0, if E � exp � 𝑡(𝑋 − 𝜇) �� ≤ exp �𝑡2𝜈2 2 � , for |𝑡| < 1 𝛼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Theorem 3 ([18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Let 𝑋𝑘 for 𝑘 = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=',𝑛 be independent RVs where 𝑋𝑘 is sub-exponential with parameters (𝜈𝑘,𝑏𝑘), and mean 𝜇𝑘 = E [𝑋𝑘].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Then 𝑛� 𝑘=1 (𝑋𝑘 − 𝜇𝑘) is a sub-exponential RV with parameters (𝜈∗,𝑏∗) where 𝑏∗ = max 𝑘=1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=',𝑛𝑏𝑘, and 𝜈∗ = � � 𝑛 ∑︁ 𝑘=1 𝜈2 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Furthermore, its tail probability can be bounded as P ������ 1 𝑛 𝑛 ∑︁ 𝑘=1 (𝑋𝑘 − 𝜇𝑘) ����� ≥ 𝑡 � ≤ ��� ��� 2𝑒 − 𝑛𝑡2 2(𝜈2∗ /𝑛) , for 0 ≤ 𝑡 ≤ 𝜈2 ∗ 𝑛𝑏∗ 2𝑒− 𝑛𝑡 2𝑏∗ , for 𝑡 ≥ 𝜈2 ∗ 𝑛𝑏∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The proof is given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ■ Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Let 𝑋𝑘 for 𝑘 = 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=',𝑛 be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' sub-exponential RVs with parameters (2(2+2𝑎),4) each with mean 2+ 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Then, P ������ 1 𝑛 𝑛 ∑︁ 𝑘=1 (𝑋𝑘 − 𝜇𝑘) ����� ≥ 𝑡 � ≤ 2𝑒 − 𝑛𝑡2 8(2+2𝑎)2 , for 𝑡 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The proof is given in then Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ■ We use Corollary 1 to upper bound of the right-hand side terms in (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The following lemma establishes the connection between the non-central chi-squared distribution and the sub-exponential distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Let 𝑋 ∼ 𝜒2 𝑝(𝑎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Then, 𝑋 is sub-exponential with parameters �2(𝑝 +2𝑎),4�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The proof is given in then Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ■ G Step 3: Upper Bounding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (52) – Recall that ˆ𝜇1 ∼ 𝜒2 2𝑛(𝑛𝜆1) and ˆ𝜇2 ∼ 𝜒2 2𝑛(𝑛𝜆2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Let 𝑓 ˆ𝜇1 denote the pdf of ˆ𝜇1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We upper bound the term P ���� ˆ𝜇2−2𝑛 ˆ𝜇1−2𝑛 ��� ≥ 𝛾1 � as follows P ����� ˆ𝜇2 −2𝑛 ˆ𝜇1 −2𝑛 ���� ≥ 𝛾1 � = ∞ ∫ 0 P ����� ˆ𝜇2 −2𝑛 ���� ≥ 𝛾1 ����𝑢 −2𝑛 ���� � 𝑓 ˆ𝜇1(𝑢)𝑑𝑢 = ∞ ∫ 0 P ����� ˆ𝜇2 −2𝑛 −𝑛𝜆2 +𝑛𝜆2 ���� ≥ 𝛾1 ����𝑢 −2𝑛 ���� � 𝑓 ˆ𝜇1(𝑢)𝑑𝑢 ≤ ∞ ∫ 0 P �1 𝑛 ���� ˆ𝜇2 −𝑛(2+𝜆2) ���� ≥ 𝛾1|𝑢 −2𝑛| −𝑛𝜆2 𝑛 � 𝑓 ˆ𝜇1(𝑢)𝑑𝑢 (53) Note that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' if 𝛾1 |𝑢−2𝑛|−𝑛𝜆2 𝑛 < 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' then P ���� ˆ𝜇2−2𝑛 ˆ𝜇1−2𝑛 ��� ≥ 𝛾1 � ≤ 1 as P � 1 𝑛 ���� ˆ𝜇2 −𝑛(2+𝜆2) ���� ≥ 𝛾1 |𝑢−2𝑛|−𝑛𝜆2 𝑛 � = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' which is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 9 For 𝛾1 |𝑢−2𝑛|−𝑛𝜆2 𝑛 ≥ 0, using the assumption 0 ≤ 𝜖 ≤ 1 in (53), we have P ����� ˆ𝜇2 −2𝑛 ˆ𝜇1 −2𝑛 ���� ≥ 𝛾1 � ≤ ∞ ∫ 0 P �1 𝑛 ���� ˆ𝜇2 −𝑛(2+𝜆2) ���� ≥ 𝜖 � 𝛾1|𝑢 −2𝑛| −𝑛𝜆2 𝑛 � � × 𝑓 ˆ𝜇1(𝑢)𝑑𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (54) The last inequality follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Let 𝑡1 := 𝑡1(𝑢) = 𝜖 � 𝛾1 |𝑢−2𝑛|−𝑛𝜆2 𝑛 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' As E [ ˆ𝜇2] = 2𝑛 +𝑛𝜆2, by applying Corollary 1, we obtain P �1 𝑛 ���� ˆ𝜇2 −𝑛(2+𝜆2) ���� ≥ 𝑡1 � ≤ 2𝑒 − 𝑛𝑡2 1 8(2+2𝜆2)2 , 𝑡1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (55) By applying (55) to (54), we obtain P ����� ˆ𝜇2 −2𝑛 ���� ≥ 𝛾1 ���� ˆ𝜇1 −2𝑛 ���� � ≤ ∞ ∫ 0 2𝑒 − 𝑛𝑡2 1 8(2+2𝜆2)2 𝑓 ˆ𝜇1(𝑢)𝑑𝑢, = 2𝑛 ∫ 0 2𝑒 − 𝑛 � 𝜖𝑛 � 𝛾1 (2𝑛−𝑢)−𝑛𝜆2 ��2 8(2+2𝜆2)2 𝑓 ˆ𝜇1(𝑢)𝑑𝑢 + ∞ ∫ 2𝑛 2𝑒 − 𝑛 � 𝜖𝑛 � 𝛾1 (𝑢−2𝑛)−𝑛𝜆2 ��2 8(2+2𝜆2)2 𝑓 ˆ𝜇1 (𝑢)𝑑𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (56) For 0 ≤ 𝑢 ≤ 2𝑛, we have 2𝑒 − 𝑛 � 𝜖𝑛 � 𝛾1 (2𝑛−𝑢)−𝑛𝜆2 ��2 8(2+2𝜆2)2 ≤ 2𝑒 − 𝑛� 𝜖 𝜆2 �2 8(2+2𝜆2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (57) For 2𝑛 ≤ 𝑢 ≤ ∞, we have 2𝑒 − 𝑛 � 𝜖𝑛 � 𝛾1 (𝑢−2𝑛)−𝑛𝜆2 ��2 8(2+2𝜆2)2 ≤ 2𝑒 − 𝑛� 𝜖 𝜆2 �2 8(2+2𝜆2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (58) Using (57) and (58) in (56), we obtain P ����� ˆ𝜇2 −2𝑛 ���� ≥ 𝛾1 ���� ˆ𝜇1 −2𝑛 ���� � ≤ 2𝑒 − 𝑛� 𝜖 𝜆2 �2 8(2+2𝜆2)2 P{0 < ˆ𝜇1 < 2𝑛} +2𝑒 − 𝑛� 𝜖 𝜆2 �2 8(2+2𝜆2)2 P{ ˆ𝜇1 > 2𝑛} P ����� ˆ𝜇2 −2𝑛 ���� ≥ 𝛾1 ���� ˆ𝜇1 −2𝑛 ���� � ≤ 2𝑒 − 𝑛� 𝜖 𝜆2 �2 8(2+2𝜆2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (59) – We next upper bound P ���� ˆ𝜇3−2𝑛 ˆ𝜇1−2𝑛 ��� ≥ 𝛾1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Set 𝑡2 = 𝜖 � 𝛾1 |𝑢−2𝑛|−𝑛𝜆3 𝑛 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Recall that ˆ𝜇3 ∼ 𝜒2 2𝑛(𝑛𝜆3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Following the steps similar to the derivation of the bound in (59), we obtain P ����� ˆ𝜇3 −2𝑛 ���� ≥ 𝛾1 ���� ˆ𝜇1 −2𝑛 ���� � ≤ 2𝑒 − 𝑛� 𝜖 𝜆3 �2 8(2+2𝜆3)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (60) Combining (59) and (60) we obtain the following upper bound on (52) P ��� ˆ𝛽∗ 1 −𝛼1 �� ≥ √︂ 𝜖 2 � ≤ 4 � 𝑒− 𝑛 32 � 𝜖 𝜆2 1+𝜆2 �2 + 𝑒− 𝑛 32 � 𝜖 𝜆3 1+𝜆3 �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (61) G Step 4: Upper bound on II By following the same steps for deriving the upper bound of P ��� ˆ𝛽∗ 1 −𝛼1 �� ≥ √︁ 𝜖 2 � , we can obtain the following bound P ��� ˆ𝛽∗ 2 −𝛼2 �� ≥ √︂ 𝜖 2 � ≤ 4 � 𝑒− 𝑛 32 � 𝜖 𝜆4 1+𝜆4 �2 + 𝑒− 𝑛 32 � 𝜖 𝜆5 1+𝜆5 �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (62) Combining (61) and (62), we obtain the required upper bound in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ■ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' NUMERICAL SIMULATIONS We estimate the initial value of (𝛽0 1, 𝛽0 2) as given in [13, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Based on the the initial value of (𝛽0 1, 𝛽0 2) we set B as given in (7), where 𝑣 and 𝑤 are selected such that 𝐾𝑥𝑣 ∈ N and 𝐾𝑦𝑤 ∈ N respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' In addition to the LoS path, we assume that there are 4 NLoS path components due to scatters between the user and the HMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The elevation and azimuth angles of each NLoS path from these scatters to the center of HMT follow the uniform distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=', 𝑈(0,2𝜋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Moreover, we consider the path coefficient of each NLoS path as a complex Gaussian distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=', 𝐶𝑁(0,𝜎2 𝑠 ), where 𝜎2 𝑠 is 20 dB weaker than the power of the LoS component [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The system parameters for numerical simulations are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' TABLE I: A list of system parameters for numerical simulations Parameters Values Description 𝑓𝑐 30 GHz Carrier frequency 𝜆 1 cm Wavelength 𝐿𝑥 1 m Width of the HMT 𝐿𝑦 1 m Length of the HMT 𝑑𝑟 𝜆/4 Unit element spacing 𝐿𝑒 𝑑𝑟 Width and length of each phase-shifting element 𝑃 20 dBm Transmission power of the HMT during data transmission 𝜎2 115 dBm Noise power for 200 KHz 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Comparison Between the Proposed Algorithm and Benchmark Scheme According to the approximated channel model, where the phase-shift parameters at the HMT are given by 𝛽1 and 𝛽2, the achieved data rate at the user of the HMT-assisted wireless communication system is given by 𝑅(𝛽1, 𝛽2) = 𝑙𝑜𝑔2 � 1+ 𝑃|𝐻(𝛽1, 𝛽2)|2 𝜎2 � , (63) 10 where 𝑃 is the transmission power at the HMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The HMT uses the acquired CSI during the channel estimation period to maximize the received data rate by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Hence, we consider the achieved data rate by the user, using the acquired CSI as a performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We applied the proposed algorithm in two different cases when the distance between the user and the center of the HMT (𝑑0 = 200 m and when 𝑑0 = 10 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We compared our proposed algorithm with two benchmarks, the proposed algorithm in [13] and the oracle scheme where 𝛼1 and 𝛼2 are estimated perfectly and thereby the maximum rate is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 4, we compared the achievable rates, given by (63), of the proposed scheme and the benchmark schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We considered both 𝑑0 = 200 m and 𝑑0 = 10 m regions of the HMT with respect to the transmit power of the pilot signals when the number of the pilot signals is fixed to 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' For all the algorithms we use the same number of pilots, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 23, for both the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Our proposed algorithm uses four pilots in each epoch and there are five epochs, which makes the total number of pilots equals to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We require additional three number of pilots to estimate (𝛽0 1, 𝛽0 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We run the simulation for 1000 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We see that in both cases, the proposed Two-Stage Phase-Shifts Estimation Algorithm gives higher rates than other two benchmark schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 20 10 0 10 20 30 40 Power of pilot signals (in dBm) 15 20 25 30 35 40 45 50 Achievable Rate (in bits/symbol) Maximum Rate d0=200 Proposed Algorithm d0=200 Algorithm from [13] d0=200 Maximum Rate d0=10 Proposed Algorithm d0=10 Algorithm from [13] d0=10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 4: Achievable rate vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' the transmit power of the pilot signals (in dBm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Convergence of The Proposed Algorithm We now numerically evaluate the convergence of the upper bound of the proposed algorithm, given by (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We also compare the actual probability, given by (26), that we obtain by simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 5, we show the convergence property of the error probability and its upper bound of the proposed algorithm for increasing values of 𝜖 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1} when the power of the pilot signal is 𝑃 = 10 dBm and 𝑑0 = 200 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We run the simulation for 1000 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We see that for each value of 𝜖, the proposed algorithm converges towards zero as we increase the number of pilots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Moreover, the upper bound of the error probability also converges to the error probability as the number of pilot signals increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 2This setting corresponds to the noise power spectrum density at the HMT is −174 dBm/Hz and signal bandwidth is 200 KHz, assuming the noise figure of each user to be 6 dB [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 0 500 1000 1500 2000 2500 3000 Number of Pilots 10 1 100 Error Probability Proposed Scheme, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='01 Upper Bound, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='01 Proposed Scheme, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='05 Upper Bound, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='05 Proposed Scheme, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 Upper Bound, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 5: Error Probability Bound v/s Number of Pilots for 𝜖 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1} for 𝑃 = 10 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 6, we compare the convergence property of the error probability of the proposed algorithm with respect to 𝜖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='05 and 𝑑0 = 200 m for different levels of power of the pilot signals, 𝑃 = {5,10,20} dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' As we increase the power of the pilot signals, the estimation accuracy of 𝛼1 and 𝛼2 increases and hence the error probability decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' This is so because, as we increase the power of pilot signals the received signals will be less noisy which increases the chances of estimating the 𝛼1 and 𝛼2 more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 0 500 1000 1500 2000 2500 3000 Number of Pilots 10 2 10 1 100 Error Probability Proposed Scheme, Pilot Power = 5 dBm Upper Bound, Pilot Power = 5 dBm Proposed Scheme, Pilot Power = 10 dBm Upper Bound, Pilot Power = 10 dBm Proposed Scheme, Pilot Power = 20 dBm Upper Bound, Pilot Power = 20 dBm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 6: Error Probability Bound v/s Number of Pilots for 𝜖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='05 for 𝑃 = {5,10,20} dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' CONCLUSION We investigated the problem of estimation of the optimal phase-shift at the HMT-assisted wireless communication system in a noisy environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We proposed a learning algorithm to estimate the optimal phase-shifting parameters and showed that the probability that the phase-shifting parameters generated by the proposed algorithm to deviate by more than 𝜖 from the optimal values decay exponentially fast as the number of pilots grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Our proposed algorithm exploited structural properties of the channel gains in the far-field regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 11 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Proof of Proposition 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Let us define the following events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 𝐴𝑛,𝑚 = |𝑋𝑛 − 𝑋𝑚| > 𝜖, 𝐴𝑛 = |𝑋𝑛 − 𝑋| > 𝜖 2, and 𝐴𝑚 = |𝑋𝑚 − 𝑋| > 𝜖 2 By the triangle inequality, we have |𝑋𝑛 − 𝑋𝑚| ≤ |𝑋𝑛 − 𝑋| + |𝑋𝑚 − 𝑋|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (64) Using (64), the event 𝐴𝑛,𝑚 can be written as |𝑋𝑛 − 𝑋𝑚| ≥ 𝜖 =⇒ |𝑋𝑛 − 𝑋| + |𝑋𝑚 − 𝑋| ≥ 𝜖 Therefore, we have 𝐴𝑛,𝑚 ⊂ {|𝑋𝑛 − 𝑋| + |𝑋 − 𝑋𝑚| > 𝜖} ⊂ � |𝑋𝑛 − 𝑋| > 𝜖 2 � |𝑋 − 𝑋𝑚| > 𝜖 2 � (65) Note that for any two events 𝐴 and 𝐵 where 𝐴 ⊂ 𝐵, then P{𝐴} ≤ P{𝐵}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We use this fact in (65), and we get P{|𝑋𝑛 − 𝑋𝑚| > 𝜖} ≤ P � |𝑋𝑛 − 𝑋| > 𝜖 2 � +P � |𝑋𝑚 − 𝑋| > 𝜖 2 � ■ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Proof of Lemma 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' We consider 𝑟(𝛽1, 𝛽2) as given in (3) which comprises of two complex-valued factors √ 𝑃 × 𝐻(𝛽1, 𝛽2) (see (1)) and 𝜁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Write 𝜁 = 𝑛1 + 𝑗𝑛2, where 𝑛1 and 𝑛2 follows 𝑁 � 0, 𝜎2 2 � and are independent, and write √ 𝑃 × 𝐻(𝛽1, 𝛽2) = 𝑎 + 𝑗𝑏, where 𝑎 and 𝑏 are real values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Therefore, 𝑟(𝛽1, 𝛽2) = |𝑦(𝛽1, 𝛽2)|2 = (𝑎 +𝑛1)2 + (𝑏 +𝑛2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (66) Note that 𝑎+𝑛1 𝜎/ √ 2 ∼ 𝑁 � 𝑎 𝜎/ √ 2,1 � and 𝑏+𝑛2 𝜎/ √ 2 ∼ 𝑁 � 𝑏 𝜎/ √ 2,1 � and they are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Therefore, 2 𝜎2 � (𝑎 +𝑛1)2 + (𝑏 +𝑛2)2 � ∼ 𝜒2 2 � 2 𝜎2 � 𝑎2 + 𝑏2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (67) Applying (67) in (66), we get 𝑋 = 2 𝜎2 𝑟(𝛽1, 𝛽2) ∼ 𝜒2 2 (𝜆1) , where 𝜆1 = 2 𝜎2 ��� √ 𝑃 × 𝐻(𝛽1, 𝛽2) ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The second part of the lemma follows from the additive property of non-central Chi-squared distribution of the sum of 𝑛 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' RVs of 𝜒2 2 (𝜆1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' ■ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Proof of Theorem 3 As 𝑋𝑘,∀𝑘 are independent, applying the definition IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1, the moment generating function of 𝑛� 𝑘=1 (𝑋𝑘 − 𝜇𝑘) is given by E � 𝑒 𝑡 𝑛� 𝑘=1 (𝑋𝑘−𝜇𝑘) � ≤ 𝑒 𝜆2 2 𝑛� 𝑘=1 𝜈2 𝑘, ∀|𝑡| < �� � 1 max 𝑘=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=',𝑛𝑏𝑘 �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Since the moment generating functions uniquely determines the distribution, comparing with the definition IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='1, it follows that 𝑛� 𝑘=1 (𝑋𝑘 − 𝜇𝑘) is a sub-exponential (𝜈∗,𝑏∗) random variable, where 𝑏∗ = max 𝑘=1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=',𝑛𝑏𝑘 and 𝜈∗ = � � 𝑛 ∑︁ 𝑘=1 𝜈2 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' To prove the second part of the Theorem we use the following tail bound on a sub-exponential distribution proved in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Proposition 1 ([18] Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Let 𝑋 is sub-exponential random variable with parameters (𝜈,𝑏) and E [𝑋] = 𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Then P{|𝑋 − 𝜇| ≥ 𝑡} ≤ � 2𝑒− 𝑡2 2𝜈2 , if 0 ≤ 𝑡 ≤ 𝜈2 𝑏 2𝑒− 𝑡 2𝑏 , if 𝑡 ≥ 𝜈2 𝑏 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' The claim immediately follows by applying the above result on 𝑍𝑛 := 𝑛� 𝑘=1 (𝑋𝑘 − 𝜇𝑘), which is sub-exponential (𝜈∗,𝑏∗), where 𝑏∗ = max 𝑘=1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=',𝑛𝑏𝑘 and 𝜈∗ = √︂ 𝑛� 𝑘=1 𝜈2 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Proof of Corollary 1 From Theorem 3, �𝑛 𝑘=1(𝑋𝑘 −𝜇𝑘) is sub-exponential (𝜈∗,𝑏∗), where 𝑏∗ = 4 and 𝜈∗ = 2√𝑛(2 + 2𝑎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Using the parameters (𝜈∗,𝑏∗) = (2√𝑛(2+2𝑎),4) in Proposition 1, we get the required upper bound as P ������ 1 𝑛 𝑛 ∑︁ 𝑘=1 (𝑋𝑘 − 𝜇𝑘) ����� ≥ 𝑡 � ≤ �� �� 2𝑒 − 𝑛𝑡2 8(2+2𝑎)2 , 0 ≤ 𝑡 ≤ (2+2𝑎)2 2𝑒− 𝑛𝑡 8 , 𝑡 ≥ (2+2𝑎)2 P ������ 1 𝑛 𝑛 ∑︁ 𝑘=1 (𝑋𝑘 − 𝜇𝑘) ����� ≥ 𝑡 � ≤ 2𝑒 − 𝑛𝑡2 8(2+2𝑎)2 , 𝑡 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Proof of Lemma 3 If 𝑋 ∼ 𝜒2 𝑝(𝑎), then according to [20], the moment-generating function (MGF) of 𝑋 is given by E [exp{𝑡(𝑋 − (𝑝 + 𝑎)}] = 𝑒−𝑡 ( 𝑝+𝑎)E � 𝑒𝑡𝑋� = 𝑒−𝑡 ( 𝑝+𝑎)𝑒 𝑎𝑡 1−2𝑡 (1−2𝑡) 𝑝/2 = 𝑒 2𝑎𝑡2 1−2𝑡 𝑒−𝑝𝑡 (1−2𝑡) 𝑝/2 , for 𝑡 < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (68) By following some calculus, refer [21], [18, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='8], we obtain 𝑒−𝑝𝑡 (1−2𝑡) 𝑝/2 ≤ 𝑒2𝑝𝑡2, for |𝑡| ≤ 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (69) For |𝑡| ≤ 1 4, we have 𝑒 2𝑎𝑡2 1−2𝑡 ≤ 𝑒4𝑎𝑡2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (70) Applying (69) and (70) to (68), we obtain E [exp{𝑡(𝑋 − (𝑝 + 𝑎)}] ≤ 𝑒2( 𝑝+2𝑎)𝑡2, ∀|𝑡| ≤ 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' (71) Therefore, by (71), 𝑋 is Sub-exponential distribution with parameters �2(𝑝 +2𝑎),4�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' REFERENCES [1] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Wan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Gao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Gao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Di Renzo, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Alouini, “Terahertz massive mimo with holographic reconfigurable intelligent surfaces,” IEEE Transactions on Communications, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 12 [2] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Jamali, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Tulino, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Fischer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Müller, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Schober, “Intelligent surface-aided transmitter architectures for millimeter-wave ultra massive mimo systems,” IEEE Open Journal of the Communications Society, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 144–167, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Larsson, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Edfors, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Tufvesson, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Marzetta, “Massive mimo for next generation wireless systems,” IEEE communications magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 52, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 186–195, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [4] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Wu and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Zhang, “Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,” IEEE Transactions on Wireless Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 5394–5409, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Hu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Alexandropoulos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Zappone, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Yuen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Di Renzo, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Debbah, “Holographic mimo surfaces for 6g wireless networks: Opportunities, challenges, and trends,” IEEE Wireless Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 118–125, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Hu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Rusek, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Edfors, “Beyond massive mimo: The potential of data transmission with large intelligent surfaces,” IEEE Transactions on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 66, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 2746–2758, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [7] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Yoo and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Smith, “Holographic metasurface antennas for uplink massive mimo systems,” arXiv preprint arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='12513, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Zhang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Shlezinger, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Guidi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Dardari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Imani, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Eldar, “Beam focusing for near-field multi-user mimo communications,” IEEE Transactions on Wireless Communications, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [9] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Dai and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Wu, “Efficient broadcasting in ad hoc wireless networks using directional antennas,” IEEE Transactions on Parallel and Distributed Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 17, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 335–347, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [10] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Xiao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' He, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Xia, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Xia, “Hierarchical codebook design for beamforming training in millimeter-wave communication,” IEEE Transactions on Wireless Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 15, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 3380–3392, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [11] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Chen and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Qi, “Beam training based on dynamic hierarchical codebook for millimeter wave massive mimo,” IEEE Communications Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 132–135, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [12] Ö.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Demir, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Björnson, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Sanguinetti, “Channel modeling and channel estimation for holographic massive mimo with planar arrays,” IEEE Wireless Communications Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 997–1001, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Ghermezcheshmeh, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Jamali, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Gacanin, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Zlatanov, “Channel estimation for large intelligent surface-based transceiver using a parametric channel model,” arXiv preprint arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content='02874, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Akdeniz, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Samimi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Sun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Rangan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Rappaport, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Erkip, “Millimeter wave channel modeling and cellular capacity evaluation,” IEEE journal on selected areas in communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 1164–1179, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Ellingson, “Path loss in reconfigurable intelligent surface-enabled channels,” in 2021 IEEE 32nd Annual International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' IEEE, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 829–835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [16] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Selvan and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Janaswamy, “Fraunhofer and fresnel distances: Unified derivation for aperture antennas.” IEEE Antennas and Propagation Magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 59, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 12–15, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Najafi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Jamali, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Schober, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Poor, “Physics-based modeling and scalable optimization of large intelligent reflecting surfaces,” IEEE Transactions on Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 2673–2691, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Wainwright, High-dimensional statistics: A non-asymptotic viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Cambridge University Press, 2019, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [19] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Wang and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Zhang, “Joint beam training and positioning for intelligent reflecting surfaces assisted millimeter wave communications,” IEEE Transactions on Wireless Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 20, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 6282–6297, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' El-Sayed, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Sahar, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Yassmen, “Moment generating function of the unbalanced non-central chi-square distribution,” International Journal of Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 8269, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' Ghosh, “Exponential tail bounds for chisquared random variables,” Journal of Statistical Theory and Practice, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 15, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} +page_content=' 1–6, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf'} diff --git a/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf b/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9c099b9ef818351a19892375b709e143db411512 --- /dev/null +++ b/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e400011ee067cd1003f61cf68685774149e9b19ace08a122d62bde10cb8b0f10 +size 221759 diff --git 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100644 index 0000000000000000000000000000000000000000..17383f65b1f19405677ecef7c4c377fcd42a86b9 --- /dev/null +++ b/NNFJT4oBgHgl3EQfzi34/content/tmp_files/2301.11644v1.pdf.txt @@ -0,0 +1,1013 @@ +Realization of valley-spin polarized current via parametric pump in monolayer MoS2 +Kai-Tong Wang,1, 2 Hui Wang,2 Fuming Xu,1, ∗ Yunjin Yu,1 and Yadong Wei1 +1College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China +2School of Physics and Engineering, Henan University of Science and Technology, Luoyang 471023, China +Monolayer MoS2 is a typical valleytronic material with valley-spin locked valence bands. +We +numerically investigate the valley-spin polarized current in monolayer MoS2 via adiabatic electron +pumping. By introducing an exchange field to break the energy degeneracy of monolayer MoS2, the +top of its valence bands is valley-spin polarized and tunable by the exchange field. A device with +spin-up polarized left lead, spin-down polarized right lead, and untuned central region is constructed +through applying different exchange fields in the corresponding regions. Then, equal amount of +pumped currents with opposite valley-spin polarization are simultaneously generated in the left and +right leads when periodically varying two pumping potentials. Numerical results show that the phase +difference between the pumping potentials can change the direction and hence polarization of the +pumped currents. It is found that the pumped current exhibits resonant behavior in the valley-spin +locked energy window, which depends strongly on the system size and is enhanced to resonant current +peaks at certain system lengths. More importantly, the pumped current periodically oscillates as +a function of the system length, which is closely related to the oscillation of transmission. The +effects of other system parameters, such as the pumping amplitude and the static potential, are also +thoroughly discussed. +I. +INTRODUCTION +Valleytronics has attracted enormous attention on ac- +count of its potential for information processing1–16. In +many crystalline materials, there are two or more min- +ima(maxima) at the conduction(valence) band in the mo- +mentum space, known as valleys. +The degenerate but +inequivalent valley states constitute new pseudospin de- +gree of freedom for low energy carriers. Similar to spin- +tronics, the essential of valleytronics is to generate and +manipulate valley polarization to encode and store infor- +mation. Various materials have been explored to real- +ize valley polarization, including silicon17,18, bismuth19, +diamond20,21, carbon nanotube22,23, etc. In particular, +two-dimensional(2D) honeycomb lattice materials such +as graphene or transition metal dichalcogenides (TMDs) +provide a perfect platform to investigate valleytronics. +Compared to graphene, TMDs labeled as MX2 (M = +Mo, W, X = S, Se, Te), also have two well-separated val- +leys in the Brillouin zone2,24. However, due to inversion +symmetry breaking, TMDs are natural gapped semicon- +ductors, which makes TMDs the promising candidates of +valleytronic materials25–31. +As a typical TMDs material, monolayer MoS2 has a +strong spin-orbit coupling(SOC) interaction26,32, which +leads to the locking between valley and spin at the top of +its valance band. The valley-spin locking means that the +valley and spin can be polarized together, and the lifetime +of polarization can be enhanced due to the large spacing +between K and K′ valleys. In the presence of an exchange +field, TMDs exhibit interesting phenomena, such as the +quantum anomalous Hall effect33,34, spin and valley Hall +effects35, and unconventional superconductivity36. +Be- +sides, an exchange field can induce polarized valleys, +which can be inverted by tuning the spin polarization. +Through the ferromagnetic proximity effect37,38 or mag- +netic doping39,40, the exchange field can be introduced +into TMDs materials, which provides an effective way to +manipulate its valley/spin degree of freedom. In exper- +iments, the exchange field for valley splitting has been +realized in Fe-doped41 or Co-doped monolayer MoS2.42 +EuS as a ferromagnetic substrate can efficiently induce +the magnetic exchange field in monolayer TMDs.43,44 +Based on the valley optical selection rules, the opti- +cal pumping of valley polarization has been experimen- +tally realized by circular polarized light in 2D TMDs45–47. +Very recently, the spin-valley coupled dynamics at the +MoS2-MoSe2 interface is experimentally studied using +optical pumping48; photoinduced valley-selective polar- +ization in monolayer WS2 has been realized with cir- +cularly polarized light pumping.49 Besides, The line +defects50, nonmagnetic disorders51, and spatially vary- +ing potentials16 were predicted to achieve the valley po- +larization in monolayer MoS2. In terms of applications, +it is desirable to obtain pure valley polarized current +by electrical methods. +Accordingly, we propose that +quantum parametric pump can drive valley and spin po- +larized currents in monolayer MoS2 through adiabati- +cally varying two gate voltages. The parametric pump +can produce dc current by periodically varying system +parameters, which has been generalized to various 2D +materials52–55. Specially, spin pump has been reported +in several nanostructures56–58, where pure spin current +and zero charge current are obtained. +In this paper, we numerically study the generation and +manipulation of valley-spin polarized currents via adia- +batic pump in monolayer MoS2. +The system setup is +shown in Fig.1. By magnetic doping, an exchange inter- +action is introduced in the left and right leads, which in- +duces locked valley-spin polarization at the top of valence +band as shown in Fig.1(a) and 1(c). When the pumping +potentials periodically change, fully valley-spin polarized +dc currents are driven into the leads. At one moment, +the current with K valley and spin up is pumped into the +arXiv:2301.11644v1 [cond-mat.mes-hall] 27 Jan 2023 + +2 +FIG. 1: Schematics of the band structures of monolayer MoS2 +for (a) with exchange field M, (b) without exchange field, (c) +with exchange field −M. The red and blue valance bands de- +note valley K with spin up and K′ with spin down. (d) The +pump setup based on monolayer MoS2 consisting of left/right +leads and the scattering region, whose band structures are +correspondingly shown in (a) to (c), respectively. The MoS2 +lattice is represented by the simple honeycomb lattice. The +pumping potentials V1 and V2 are added in the scattering re- +gion, adjacent to the leads. As V1 and V2 periodically change, +electric currents with opposite valley-spin polarizations are si- +multaneously pumped into the left and right leads, as shown +by the block arrows. +left lead while the current with opposite valley-spin po- +larization flows into the right lead. The polarized current +exhibits resonant behavior in the valley-spin locked en- +ergy window, which mainly depends on the system size. +With the increasing of the system length, the pumped +currents show periodic oscillation behavior and robust +resonant current peaks can be observed. We also inves- +tigate the influence of other system parameters, includ- +ing the phase difference, the Rashba SOC strength, the +static potential, and the Fermi energy. It is found that +the phase difference and static potentials can invert the +direction and hence polarization of the pumped current. +The paper is organized as follows. In Sec. II, we in- +troduce the Hamiltonian of monolayer MoS2 and the for- +malism of adiabatic parametric pumping. In Sec. III, +numerical results and relevant discussions are presented. +Finally, a brief summary is given in Sec. IV. +II. +MODEL AND FORMALISM +In monolayer MoS2, the low-energy spectrum at K +and K′ valleys consists of three d orbitals of Mo, i.e., +dz2, dx2−y2, dxy. +The relations between these orbitals +and basis wave functions satisfy: |ϕc⟩ = |dz2⟩, |ϕλ +υ⟩ = +(|dx2−y2⟩+iλ|dxy⟩)/ +√ +2, where the subscript c/υ denotes +the conduction/valence band and λ = ±1 corresponds +to different valleys K and K′. Based on above low-lying +states, the effective Hamiltonian of monolayer MoS2 has +the following form26,59 +H0(k) = at(λkxσx + kyσy) + ∆σz − tSOλσz − 1 +2 +τz, (1) +where a and t are the lattice constant and hopping +strength, respectively. σx,y,z and τz represent the Pauli +matrices of basis functions(|ϕc⟩ and |ϕυ⟩) and spin(↑ and +↓). ∆ is the mass term and the last term is the intrinsic +SOC with strength tSO. +We employ the tight-binding model of MoS2, which +treats monolayer MoS2 as a simplified honeycomb lattice. +The lattice includes A and B sublattices, corresponding +to the dz2 orbit and dx2−y2 + iλdxy orbits of Mo, respec- +tively. In the tight-binding approximation, the Hamilto- +nian can be expressed as60,61 +H0 = +� +i +ϵic† +iαciα + t +� + +c† +iαciα + HSO, +(2) +with +HSO = 2itSO +3 +√ +3 +� +≪i,j≫,α,α′ +υijc† +iατz,αα′cjα′, +(3) +where c† +iα(ciα) is the creation(annihilation) operator at +site i with spin α = ±1, ϵi is the on-site energy. HSO +denotes the intrinsic SOC term and the summation over +the second nearest-neighbor sites only involves B sublat- +tice. Besides, υij = +1(−1) if an electron moves from +site j to site i with taking a left(right) turn62. +Based on this model, we consider a monolayer MoS2 +setup, which contains three parts: the central scattering +region, the left and right leads, as shown in Fig.1(d). +By magnetic doping, different valley-spin polarizations +can be induced in the left and right leads due to the +exchange field39,63. The schematic band structures with +or without the exchange field are depicted in Fig.1(a)-(c). +In the presence of Rashba spin-orbit coupling (RSOC), +the Hamiltonians for the scattering region with pumping +potentials and the leads can be written as +HC = H0+ 3itR +4 +� +,α,α′ +(ταα′×dij)zc† +iαcjα′+V (x, y, t), +(4) +HL/R = H0 ± M +� +i,α,α′ +τz,αα′c† +iαciα′, +(5) +where M and tR denote the strengths of the exchange +field and RSOC, respectively. ταα′ = (τx, τy, τz) is the +Pauli matrix for spin, and dij is the lattice vector con- +necting sites i and j. +The potential term V (x, y, t) = +Vs(x, y) + Vt(x, y, t), where Vs = V0 +� +i Πi(x, y) corre- +sponds to the static potential defining the shape of the +pumping region. Vt = Vp +� +i Πi(x, y)cos(ωt + ϕi) is the +periodic pumping potential. V0 and Vp are the ampli- +tudes of Vs and Vt. i = 1, 2 are the indices of the po- +tential and Πi represents the potential profile, which is + +(a) +(b) +(c) +: +spin up +spin down +K +K' +K +K' +K +K +(d) +个,K +*,K' +Lead-L +Vi(t) +Scattering region +V2(t) +Lead-R3 +highlighted in green in Fig.1(d). ϕi is the initial phase of +the pumping potential. +To evaluate the adiabatic valley-spin pump, we need to +calculate the average current flowing into lead β. Con- +sider a slowly varying time-dependent pumping potential +Vt,i, the average current in one period is expressed as64 +Iβ = qω +2π +� T +0 +dt[ dNβ +dVt,1 +dVt,1 +dt ++ dNβ +dVt,2 +dVt,2 +dt ], +(6) +where the period of Vt,i is T = 2π/ω with frequency ω +and β = L/R labels the lead. +The emissivity +dNβ +dVi +is +defined in terms of the scattering matrix Sββ′ as65,66 +dNβ +dVi += +� dE +2π (−∂Ef) +� +β′ +Im∂Sββ′ +∂Vi +S∗ +ββ′, +(7) +with f the Fermi distribution function. Under the adi- +abatic condition, the pumped current is independent of +the pumping frequency ω, hence we set ω = 1 in the +calculation. +In the language of nonequilibrium Green’s functions, +the pumped current is expressed as67–69 +Iβ = − q +2π +� 2π +0 +dt +� +dE(∂Ef)Tr[ΓβGr dVt +dt Ga]. +(8) +Here Gr/Ga is the retarded/advanced Green’s function +of the central scattering region, which is defined as Gr = +Ga,† = [E − HC − � +β Σr +β]−1. HC is the corresponding +Hamiltonian. Σr +β is the retarded self-energy of lead β, +which can be calculated by surface Green’s function70,71. +Γβ = i(Σr +β − Σa +β) denotes the linewidth function. +As shown by block arrows in Fig.1(d), at one moment +of the pumping period, polarized current with K valley +and spin up (pink arrow) is driven into the left lead, and +equal amount current with K′ valley and spin down (blue +arrow) flows in the right lead. Detailed numerical results +are shown in the following section. +We use the short +term, the pumped current, to stand for the valley-spin +polarized currents in the leads. +III. +RESULTS AND DISCUSSION +In the calculations, the on-site energy is ϵi = ±0.83 eV +for A and B sublattices. Other parameters26,60 are set +as t = 1.27 eV, tSO = 0.038 eV, and the lattice constant +a=0.32 nm. Without loss of generality, we set the ex- +change field strength M = 0.06 eV, and eV is taken as +the energy unit throughout the calculation. The periodic +boundary condition (PBC) is considered for monolayer +MoS2, and thus the edge effect is removed. To realize +PBC, the upper and lower edges of a zigzag MoS2 rib- +bon shown in Fig.1(d) are connected with appropriate +hopping interactions, which is also called the cylinder +boundary. +FIG. 2: The dispersion relation of monolayer MoS2 ribbon +with an exchange field: (a) M = 0.06, (b) M = −0.06. Both +valley and spin are polarized together, where ∆E is defined +as the valley-spin locked energy window. +A. +Valley-spin polarized current +The dispersion of monolayer MoS2 with an exchange +field is plotted in Fig.2. From Fig.2(a), it is clear that +the valley K with spin up is polarized at the valence band +top. However, as the exchange field changes, the polar- +ization of both valley and spin is inverted as shown in +Fig.2(b). In the valley-spin locked window ∆E, perfect +polarization can be realized. To investigate the valley- +spin polarized current, we consider only the ∆E energy +range. In Fig.3, We study the dependence of the pumped +current on the phase difference ϕ12 between V1 and V2. +Due to the inverse valley-spin polarization of the left and +right leads, the holes are forbidden to propagate through +the scattering region, so there is no pumped current gen- +erated in this setup at tR = 0. +Introducing RSOC in +the scattering region, it is found that the pumped cur- +rent arises and flows into left or right lead. We attribute +such a dc current to the spin flip process induced by the +RSOC. Importantly, nonzero valley-spin polarized cur- +rents are pumped into different leads, which depends on +the exchange field in leads. Our results show that IL,R is +an odd function about ϕ12, i.e., I(ϕ12) = −I(−ϕ12). The +maximum of the pumped current appears at ϕ12 = ±π/2. +From Fig.3, when the phase difference ϕ12 shifts from +−π to 0, the valley-polarized holes with spin up will be +pumped out of the scattering region and flow into the +left lead. On the contrary, the opposite valley-polarized +holes with spin down will spread into the right lead when +ϕ12 shifts from 0 to π. It means that the direction of the +pumped current can be tuned by the phase difference ϕ12, +then different valley-spin polarized currents are pumped +into different leads. We calculate the pumped current as +a function of the phase difference ϕ12 for different RSOC +strengths tR. Consequently, with the increasing of tR, +the pumped current increases. IL versus tR at ϕ12 = π +2 is +plotted in the inset. The result is understandable: since +the spin-flip efficiency increases when tR is increased, the +spin-up carriers from one lead can be more easily flipped +as spin-down carriers and flow into the other lead, which + +1.5 +(a) +(b) +1 +0.5 +spin up +E(eV) +spin down +0 +-0.5 +K +K' +K +K' +△E +-1 +-1.5 +0 +0.5 +1 +1.5 +2 0 +0.5 +1 +1.5 +2 +k(π/a) +k (π/a) +X4 +-3 +-2 +-1 +0 +1 +2 +3 +-0.003 +-0.002 +-0.001 +0.000 +0.001 +0.002 +0.003 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.000 +0.002 +0.004 +0.006 +0.008 +ϕ12 +Pumped current + tR=0 + tR=0.02 + tR=0.04 + tR=0.05 +Pumped current +tR +FIG. 3: The pumped current IL as a function of the phase +difference ϕ12 between V1 and V2. Inset: the pumped current +versus RSOC strength tR at ϕ12 = π/2. Other parameters: +Ef = −0.74, L = 10a, V0 = 0.1, Vp = 0.05. +-0.76 +-0.75 +-0.74 +-0.73 +-0.72 +-0.71 +-0.70 +0.000 +0.002 +0.004 +0.006 +0.00 +0.02 +0.04 +0.06 +0.08 +-0.006 +-0.003 +0.000 +0.003 +0.006 + IL +(b) +Transmission +Pumped current +Ef +(a) +0.00 +0.03 +0.06 +0.09 + T + +Pumped current +Vp + IL,V0=0.05 + IR,V0=0.05 + IL,V0=0.07 + IR,V0=0.07 + IL,V0=0.1 + IR,V0=0.1 +FIG. 4: +(a) The pumped current and transmission versus +Fermi energy at V0 = 0.1 and Vp = 0.05. (b) The polarized +current IL and IR versus the pumping potential Vp for dif- +ferent static potentials V0 at Ef = −0.74. Other parameters: +ϕ12 = π/2, L = 10a, tR = 0.05. +is required by the conservation of charge. +In Fig.4(a), the pumped current and the transmission +versus Fermi energy Ef are plotted at ϕ12 = +π +2 . +Ob- +viously, a broad transmission peak arises in the locked +window ∆E, and the pumped current exhibits similar +behavior as the transmission. This result indicates that +the transport of polarized holes is dominated by quan- +tum resonance, which originates from quantum interfer- +ence effect. In fact, the resonance assisted transport is a +common property of electron pump66. Besides, an im- +passable interval for the pumped current is generated +as the Fermi energy is away from the resonant peak as +10 +20 +30 +40 +50 +0.00 +0.02 +0.04 +0.06 +10 +20 +30 +40 +50 +0.000 +0.002 +0.004 +0.006 + +(a) +Transmission +46 +34 +22 +10 +(b) +Pumped current +L/a + tR=0.02 + tR=0.05 + tR=0.08 +FIG. 5: (a) The transmission versus the length L of scattering +region for tR = 0.05. (b) The pumped current IL as a function +of system length L for different RSOC strengths at ϕ12 = +π/2. The numbers label the lengths of the scattering region +where current peaks emerge. Other parameters are the same +as Fig.4. +shown in Fig.4(a). +The variation of polarized current +is well correlated to the transmission. The pumped cur- +rent IL,R versus the pumping potential for different static +potentials is plotted in Fig.4(b). Due to the particle con- +servation, the current flowing into the scattering region +must satisfy IL = −IR, which is confirmed through the +symmetric curves of IL and IR. Furthermore, the results +show that the valley-spin polarized current linearly in- +creases with the increasing of pumping potential. For a +relatively small Vp, the static potential V0 can enhance +the magnitude of pumped currents. +B. +Size effect on the pumped current +In the following, we study the influence of the sys- +tem size on the pumped current. Transmission and the +pumped current versus the length of the scattering re- +gion are plotted in Fig.5. +It is interesting that some +robust peaks of the pumped current appear at certain +lengths, but the currents for other lengths are almost +zero. +Moreover, these peaks show periodic oscillation +behavior with the period length 12a. +In Fig.5(a), we +plot the transmission as a function of the length L for +tR = 0.05. +The transmission exhibits a similar be- +havior as the pumped current, where the transmission +peaks also appear at certain system lengths. It is clear +that these transmission peaks correspond exactly to the +pumped current peaks. It is reasonable that the peri- +odic behavior of the pumped current originates from the +spin precession72–74 induced by Rashba SOC. When the +current carriers travel through the central scattering re- + +5 +0.000 +0.002 +0.004 +0.000 +0.001 +0.002 +10 +20 +30 +40 +50 +0.000 +0.001 +0.002 + IL +(c) +(b) + +Pumped current +(a) +0.00 +0.05 +0.10 + T + + +Pumped current +0.00 +0.05 +0.10 + +Transmission +Transmission +Transmission +Pumped current +L/a +0.00 +0.05 +0.10 + +FIG. 6: The pumped current and transmission versus the +length L of the scattering region for different Fermi energies. +(a): Ef = −0.72, (b): Ef = −0.735, (c): Ef = −0.75. Other +parameters are the same as Fig.4. +gion with RSOC interaction, carrier spin keeps precess- +ing and spin flip occurs, which results in the periodic +oscillating behavior of the pumped current. Our calcu- +lation further demonstrates that the width of scattering +region has almost no influence on the periodic behavior +of polarized current as long as Fermi energy lies in the +valley-spin locked energy window. To evaluate the influ- +ence of RSOC, We show in Fig.5(b) the pumped currents +for different RSOC strengths. It is clear that the RSOC +strength has less influence on the resonant period, which +is L = 12a. However, with the increasing of tR, the reso- +nant current peaks become significant. The peak current +value grows larger as tR increases, which is consistent +with the results shown in Fig.3. +Notice that this periodic oscillation behavior of the +pumped current is different from the even-odd conduc- +tance oscillation of carbon-atom chains.75,76 When two +metallic electrodes are attached to a carbon atomic chain, +the electronic structure of the carbon chain is modified, +which leads to the difference in the density of states be- +tween odd- and even-number carbon chains.75,76 This +leads to the even-odd conductance oscillation driven by +dc bias. +However, for the periodic oscillation of the +valley-spin polarized currents, it is due to the spin- +flipping process induced by Rashba SOC and driven by +periodic pumping potentials. +In Fig.6, we plot the pumped current and transmission +versus the system length for different Fermi energies. Ap- +parently, the periodic oscillation behavior of the pumped +current persists as the Fermi energy changes. +The re- +sult shows that, with the increasing of Ef, the number of +peaks increases while the corresponding periodic length +FIG. 7: The pumped current with respect to both the system +length L and the Fermi energy Ef. +decreases. Besides, the magnitude of current peaks will +decrease as the Fermi energy increases. By calculating +the transmission, it can be seen that the behavior of the +pumped current is still consistent with the transmission, +which means the periodic oscillation is a universal phe- +nomenon in the setup. +To exhibit an overall view of the pumped current, +in Fig.7, we provide a two-dimensional diagram of the +valley-spin polarized current as a function of both the +system length L and the Fermi energy Ef. +By vary- +ing L and Ef, we find that there are five curves with +discrete extrema of currents. The periodic dependence +of the pumped current on the system length L is clearly +shown. When Ef approaches the top of valley, the largest +pumped current appears and becomes more sharp as +shown by the orange regions. +Moreover, it is further +confirmed that, multiple resonant peaks of the pumped +current arise under an appropriate system length L. Our +numerical results reveal that it is possible to design a +high-efficiency device setup for generating valley-spin po- +larized currents. +C. +Influence of the static potential +In this section, we focus on the dependence of IL on +the static potential V0. +For this purpose, the system +length and the RSOC strength are fixed at L = 10a and +tR = 0.05. In Fig.8(a), we plot the pumped current as +well as transmission coefficient versus the static potential +V0. With the increasing of the static potential, a current +peak first emerges at V0 = 0.126 labeled by the blue point +γ3. As V0 scans the critical point γ1 at V0 = 0.151, the +direction of IL is reversed. Continuing to increase V0, we +can see a negative current peak. The result shows that +the static potential can also change the direction and +hence the polarization of the pumped current. Besides, +in the vicinity of the critical point γ1, a transmission + +0.008 +50 +0.006 +0.004 +40 +0.002 +a +30 +0 +20 +10. +-0.76 +-0.74 +-0.72 +-0.706 +0.00 +0.06 +0.12 +0.18 +0.24 +-0.002 +0.000 +0.002 +0.004 +-0.750 +-0.745 +-0.740 +-0.735 +-0.730 +0.1 +0.2 +0.3 +0.4 +γ3 +γ2 +Pumped current +V0 + IL + T/50 +(a) +γ1 +(b) +V0&T +Ef + γ1 transition V0 + γ2 maximum T +0.000 +0.002 +0.004 +0.006 +0.008 + Maximum IL + γ3 maximum IL +FIG. 8: (a) The pumped current as well as the transmission +coefficient T as a function of the static potential V0. A factor +of 1/50 is multiplied to T for better illustration. γ1, γ2, γ3 +label the critical points. (b) The maximum of IL, transition +point of V0 and corresponding maximum of T versus the Fermi +energy. Other parameters: Vp = 0.03, ϕ12 = π/2, L = 10a. +peak is clear, which suggests the influence of the static +potential also results from quantum resonance. +In Fig.8(b), The critical point of V0 and the maxima +of both T and IL versus Fermi energy Ef are plotted. +It is found that the critical value of V0 increases linearly +with the increasing of Ef, which indicates the resonant +energy level depends on the static potential. Besides, the +curves of the peak values for transmission and pumped +current grow with the Fermi energy. The variation of the +pumped current with the Fermi energy is consistent with +the results in Fig.6. +In this work, the valley-spin polarized current is gener- +ated by electric pumping in adiabatic regime, which re- +quires two independently varying system parameters. We +emphasize that similar mechanism can also be achieved +with optical pumping, which is in the non-adiabatic +regime due to the high frequency of light wave. In this +case, the light frequency can serve as a pumping param- +eter. Therefore, non-adiabatic parametric pumping us- +ing electric or optical ways will certainly bring in more +physics. +IV. +CONCLUSIONS +In conclusion, we study the valley-spin polarized cur- +rent in monolayer MoS2 ribbon via parametric electron +pump. In the proposed setup, different valley-spin polar- +ized currents can be controlled to flow into different leads +in the valley-spin locked energy window. The phase dif- +ference between the pumping potentials can change the +direction and hence polarization of the pumped current, +where quantum resonance dominates the transport pro- +cess. Furthermore, the size effect on the valley-spin po- +larized current is numerically investigated. As the length +of the scattering region changes, resonant peaks of the +pumped current arise and show periodic oscillation due +to the spin precession. +With the increasing of Fermi +energy, the number of peaks decreases while the peak +height increases within a fixed length range. The depen- +dence of the resonant pumped current on the scattering +region length and the Fermi energy is numerically re- +vealed in a two-dimensional diagram. +It is also found +that the direction of valley-spin polarized currents can +be inverted by the static pumping potential. As a po- +tential valleytronic device, the pump setup proposed in +this work can serve as a valley-spin polarization source, +which can simultaneously generate opposite polarization +in one device. +The polarized signals can be efficiently +manipulated by many system parameters, and significant +resonant enhancement has been demonstrated. +ACKNOWLEDGMENTS +This +work +was +supported +by +the +National +Natural +Science +Foundation +of +China +(Grant +Nos. +12034014 +and +61674052) +and +the +Natu- +ral +Science +Foundation +of +Shenzhen +(Grant +Nos. +20200812092737002, +JCYJ20190808115415679, +and +JCYJ20190808152801642). Hui Wang also acknowledges +supports from the Outstanding Youth Foundation of +Henan Scientific Committee (212300410041) and the +Key Scientific and Technological Projects in Henan +Province (212102210223). +∗ xufuming@szu.edu.cn +1 A. Rycerz, J. Tworzyd�lo, and C. W. J. Beenakker, Nat. +Phys., 2007, 3, 172-175. +2 D. Xiao, W. Yao, and Q. Niu, Phys. Rev. Lett., 2007, 99, +236809. +3 S.-G. Cheng, J. Zhou, H. Jiang, and Q.-F. Sun, New J. +Phys., 2016, 18, 103024. +4 J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, +K. L. Seyler, W. Yao, and X. Xu, Nat. Rev. Mater., 2016, +1, 16055. +5 D. Gunlycke and C. T. White, Phys. Rev. Lett., 2011, 106, +136806. +6 Y. S. Ang, S. A. Yang, C. Zhang, Z. Ma, and L. K. Ang, +Phys. Rev. B, 2017, 96, 245410. +7 X. Chen, Y. Xu, J. Wang, and H. Guo, Phys. Rev. B, 2019, +99, 064302. + +7 +8 Z.-M. Yu, S. Guan, X.-L. Sheng, W. Gao, and S. A. Yang, +Phys. Rev. Lett., 2020, 124, 037701. +9 L. Zhang, K. Gong, J. Chen, L. Liu, Y. Zhu, D. Xiao, and +H. Guo, Phys. Rev. B, 2014, 90, 195428. +10 Z. Yu, F. Xu, and J. Wang, Carbon, 2016, 99, 451. +11 L. Zhang, Z. Yu, F. Xu, and J. Wang, Carbon, 2018, 126, +183. +12 Z. P. Niu, New J. Phys., 2018 20, 103021. +13 J. Pawlowski, D. Zebrowski, and S. Bednarek, Phys. Rev. +B, 2018, 97, 155412. +14 S.-G. Cheng, H. Liu, H. Jiang, Q.-F. Sun, and X. C. Xie, +Phys. Rev. Lett., 2018, 121, 156801. +15 C. Lei, Y. Ma, T. Zhang, X. Xu, B. Huang, and Y. Dai, +New J. Phys., 2020 22, 033002. +16 D. Gut, M. Prokop, D. Sticlet, and M. P. Nowak, Phys. +Rev. B, 2020, 101, 085425. +17 K. Takashina, Y. Ono, A. Fujiwara, Y. Takahashi, and +Y. Hirayama, Phys. Rev. Lett., 2006 96, 236801. +18 D. Culcer, A. L. Saraiva, B. Koiller, X. Hu, and S. D. +Sarma, Phys. Rev. Lett., 2012, 108, 126804. +19 Z. Zhu, A. Collaudin, B. Fauque, W. Kang, and K. Behnia, +Nat. Phys., 2012, 8, 89. +20 C. E. Nebel, Nat. Mater., 2013, 12, 690. +21 J. Isberg, M. Gabrysch, J. Hammersberg, S. Majdi, K. K. +Kovi, and D. J. Twitchen, Nat. Mater., 2013, 12, 760. +22 F. Xu, Z. Yu, Y. Ren, B. Wang, Y. Wei, and Z. Qiao, New +J. Phys., 2016 18, 113011. +23 A. Palyi and G. Burkard, Phys. Rev. Lett., 2011, 106, +086801. +24 X. Xu, W. Yao, D. Xiao, and T. F. Heinz, Nat. Phys., 2014 +10, 343. +25 T. Cai, S. A. Yang, X. Li, F. Zhang, J. Shi, W. Yao, and +Q. Niu, Phys. Rev. B, 2013, 88, 115140. +26 D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys. +Rev. Lett., 2012, 108, 196802. +27 P. E. Faria Junior, K. Zollner, T. Wo´zniak, M. Kurpas, +M. Gmitra, and J. Fabian, New J. Phys., 2022 24, 083004. +28 Y. G. Semenov and K. W. Kim, Phys. Rev. B, 2016, 93, +041414(R). +29 D. Oliveira, L. V. Lelovsky, M. A. G. Soler, and F. Qu, +Phys. Rev. B, 2018, 97, 125309. +30 H. Tornatzky, A. M. Kaulitz, and J. Maultzsch, Phys. Rev. +Lett., 2018, 121, 167401. +31 K. F. Mak, K. L. Mcgill, J. Park, and P. L. Mceuen, Sci- +ence, 2014, 344, 1489. +32 Z. Y. Zhu, Y. C. Cheng, and U. Schwingenschl¨ogl, Phys. +Rev. B, 2011, 84, 153402. +33 R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, and +Z. Fang, Science, 2010, 329, 61. +34 J. Zhou, Q. Sun, and P. Jena, Phys. Rev. Lett., 2017, 119, +046403. +35 H. Da, Q. Song, P. Dong, H. Ye, and X. Yan, J. Appl. +Phys., 2020, 127, 023903. +36 M. A. Rahimi, A. G. Moghaddam, C. Dykstra, M. Gover- +nale, and U. Z¨ulicke, Phys. Rev. B, 2017, 95, 104515. +37 L.-Y. Gan, Q. Zhang, Y. Cheng, and U. Schwingenschl¨ogl, +Phys. Rev. B, 2013, 88, 235310. +38 S. Teng, X. Mao, Z. Liu, Y. Liu, X. Xu, L. Li, X. Xie, +S. Fan, G. Zhou, J. Li, and J. Li, New J. Phys., 2020 22, +103061. +39 Y. C. Cheng, Q. Y. Zhang, and U. Schwingenschl¨ogl, Phys. +Rev. B, 2014, 89, 155429. +40 J. Zhou and P. Jena, J. Phys. Chem. Lett, 2017, 8, 5764- +5770. +41 Q. Li, X. Zhao, L. Deng, Z. Shi, S. Liu, Q. Wei, L. Zhang, +Y. Cheng, L. Zhang, H. Lu, W. Gao, W. Huang, C. W. Qiu, +G. Xiang, S. J. Pennycook, Q. Xiong, K. P. Loh, and B. +Peng, ACS Nano 2020, 14, 4636. +42 J. Zhou, J. Lin, H. Sims, C. Jiang, C. Cong, J. A. Brehm, +Z. Zhang, L. Niu, Y. Chen, Y. Zhou, Y. Wang, F. Liu, +C. Zhu, T Yu, K. Suenaga, R. Mishra, S. T. Pantelides, +Z.-G. Zhu, W. Gao, Z. Liu, and W. Zhou, Adv. Mater. +2020, 32, 1906536. +43 C. Zhao, T. Norden, P. Zhang, P. Zhao, Y. Cheng, +F. Sun, J. P. Parry, P. Taheri, J. Wang, Y. Yang, +T. Scrace, K. Kang, S. Yang, G.-x. Miao, R. Sabirianov, +G. Kioseoglou, W. Huang, A. Petrou, and H. Zeng, Nat. +Nanotechnol. 2017, 12, 757. +44 T. Norden, C. Zhao, P. Zhang, R. Sabirianov, A. Petrou, +and H. Zeng, Nat. Commun. 2019, 10, 4163. +45 K. F. Mak, K. He, J. Shan, and T. F. Heinz, Nat. Nan- +otechnol., 2012, 7, 494-498. +46 H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, Nat. Nan- +otechnol., 2012, 7, 490-493. +47 K. F. Mak, D. Xiao, and J. Shan, Nat. Photonics 2018, +12, 451. +48 A. Kumar, D. Yagodkin, N. Stetzuhn, S. Kovalchuk, +A. Melnikov, P. Elliott, S. Sharma, C. Gahl, and K. I. +Bolotin, Nano Lett. 2021, 21, 7123. +49 S. Guddala, Y. Kawaguchi, F. Komissarenko, S. Kir- +iushechkina, A. Vakulenko, K. Chen, A. Al`u, V. M. Menon, +and A. B. Khanikaev, Nat. Commun. 2021, 12, 3746. +50 A. Pulkin and O. V. Yazyev, Phys. Rev. B, 2016, 93, +041419(R). +51 X.-T. An, J. Xiao, M. W.-Y. Tu, H. Yu, V. I. Fal’ko, and +W. Yao, Phys. Rev. Lett., 2017, 118, 096602. +52 V. I. Talyanskii, D. S. Novikov, B. D. Simons, and L. S. +Levitov, Phys. Rev. Lett. 2001, 87, 276802. +53 Q. Zhang, K. S. Chan, and Z. Lin, Appl. Phys. Lett., 2011, +98, 032106. +54 M.-J. Wang, J. Wang, and J.-F. Liu, New J. Phys., 2020 +22, 013042. +55 K.-T. Wang, F. Xu, B. Wang, Y. Yu, and Y. Wei, Front. +Phys., 2022, 17, 43501. +56 Y. Xing, B. Wang, Y. Wei, B. Wang, and J. Wang, Phys. +Rev. B, 2004, 70, 245324. +57 W. Y. Deng, W. Luo, H. Geng, M. N. Chen, L. Sheng, and +D. Y. Xing, New J. Phys., 2015, 17, 103018. +58 M. N. Chen, L. Sheng, R. Shen, D. N. Sheng, and D. Y. +Xing, Phys. Rev. B, 2015 91, 125117. +59 H. Ochoa and R. Rold´an, Phys. Rev. B, 2013, 87, 245421. +60 J. Klinovaja and D. Loss, Phys. Rev. B, 2013, 88, 075404. +61 I. Kleftogiannis, C.-S. Tang, and S.-J. Cheng, J. Phys.: +Condens. Matter, 2015, 27, 205302. +62 C. L. Kane and E. J. Mele, Phys. Rev. Lett., 2005, 95, +226801. +63 C. Bai, Y. Yang, and L. Bai, J. Phys.: Condens. Matter, +2018, 30, 495801. +64 P. W. Brouwer, Phys. Rev. B, 1998, 58, R10135. +65 M. B¨uttiker, H. Thomas, and A. Prˆetre, Z. Phys. B Con- +dens. Matter, 1994, 94, 133. +66 Y. Wei, J. Wang, and H. Guo, Phys. Rev. B, 2000, 62, +9947. +67 B. Wang, J. Wang, and H. Guo, Phys. Rev. B, 2003 68, +155326. +68 F. Xu, Y. Xing, and J. Wang, Phys. Rev. B, 2011, 84, +245323. +69 Y. Yu, Y. Zhou, L. Wan, B. Wang, F. Xu, Y. Wei, and + +8 +J. Wang, Nanotechnology, 2016, 27, 185202. +70 D. H. Lee and J. D. Joannopoulos, Phys. Rev. B, 1981, 23, +4988. +71 D. H. Lee and J. D. Joannopoulos, Phys. Rev. B, 1981, 23, +4997. +72 J.-S. Yang, X.-G. He, S.-H. Chen, and C.-R. Chang, Phys. +Rev. B, 2008, 78, 085312. +73 D. Culcer, C. Lechner, and R. Winkler, Phys. Rev. Lett., +2006, 97, 106601. +74 T.-W. Chen and H.-C. Hsu, New J. Phys., 2021, 23, +073017. +75 N. D. Lang and P. Avouris, Phys. Rev. Lett., 1998, 81, +3515. +76 N. D. Lang and P. Avouris, Phys. Rev. Lett., 2000, 84, +358. + diff --git a/NNFJT4oBgHgl3EQfzi34/content/tmp_files/load_file.txt b/NNFJT4oBgHgl3EQfzi34/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a216da1cb508d4bbdd3c3678459bd3c39d58a6df --- /dev/null +++ b/NNFJT4oBgHgl3EQfzi34/content/tmp_files/load_file.txt @@ -0,0 +1,1082 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf,len=1081 +page_content='Realization of valley-spin polarized current via parametric pump in monolayer MoS2 Kai-Tong Wang,1, 2 Hui Wang,2 Fuming Xu,1, ∗ Yunjin Yu,1 and Yadong Wei1 1College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China 2School of Physics and Engineering, Henan University of Science and Technology, Luoyang 471023, China Monolayer MoS2 is a typical valleytronic material with valley-spin locked valence bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' We numerically investigate the valley-spin polarized current in monolayer MoS2 via adiabatic electron pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' By introducing an exchange field to break the energy degeneracy of monolayer MoS2, the top of its valence bands is valley-spin polarized and tunable by the exchange field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' A device with spin-up polarized left lead, spin-down polarized right lead, and untuned central region is constructed through applying different exchange fields in the corresponding regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Then, equal amount of pumped currents with opposite valley-spin polarization are simultaneously generated in the left and right leads when periodically varying two pumping potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Numerical results show that the phase difference between the pumping potentials can change the direction and hence polarization of the pumped currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' It is found that the pumped current exhibits resonant behavior in the valley-spin locked energy window, which depends strongly on the system size and is enhanced to resonant current peaks at certain system lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' More importantly, the pumped current periodically oscillates as a function of the system length, which is closely related to the oscillation of transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The effects of other system parameters, such as the pumping amplitude and the static potential, are also thoroughly discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' INTRODUCTION Valleytronics has attracted enormous attention on ac- count of its potential for information processing1–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In many crystalline materials, there are two or more min- ima(maxima) at the conduction(valence) band in the mo- mentum space, known as valleys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The degenerate but inequivalent valley states constitute new pseudospin de- gree of freedom for low energy carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Similar to spin- tronics, the essential of valleytronics is to generate and manipulate valley polarization to encode and store infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Various materials have been explored to real- ize valley polarization, including silicon17,18, bismuth19, diamond20,21, carbon nanotube22,23, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In particular, two-dimensional(2D) honeycomb lattice materials such as graphene or transition metal dichalcogenides (TMDs) provide a perfect platform to investigate valleytronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Compared to graphene, TMDs labeled as MX2 (M = Mo, W, X = S, Se, Te), also have two well-separated val- leys in the Brillouin zone2,24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' However, due to inversion symmetry breaking, TMDs are natural gapped semicon- ductors, which makes TMDs the promising candidates of valleytronic materials25–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' As a typical TMDs material, monolayer MoS2 has a strong spin-orbit coupling(SOC) interaction26,32, which leads to the locking between valley and spin at the top of its valance band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The valley-spin locking means that the valley and spin can be polarized together, and the lifetime of polarization can be enhanced due to the large spacing between K and K′ valleys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In the presence of an exchange field, TMDs exhibit interesting phenomena, such as the quantum anomalous Hall effect33,34, spin and valley Hall effects35, and unconventional superconductivity36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Be- sides, an exchange field can induce polarized valleys, which can be inverted by tuning the spin polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Through the ferromagnetic proximity effect37,38 or mag- netic doping39,40, the exchange field can be introduced into TMDs materials, which provides an effective way to manipulate its valley/spin degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In exper- iments, the exchange field for valley splitting has been realized in Fe-doped41 or Co-doped monolayer MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='42 EuS as a ferromagnetic substrate can efficiently induce the magnetic exchange field in monolayer TMDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='43,44 Based on the valley optical selection rules, the opti- cal pumping of valley polarization has been experimen- tally realized by circular polarized light in 2D TMDs45–47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Very recently, the spin-valley coupled dynamics at the MoS2-MoSe2 interface is experimentally studied using optical pumping48;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' photoinduced valley-selective polar- ization in monolayer WS2 has been realized with cir- cularly polarized light pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='49 Besides, The line defects50, nonmagnetic disorders51, and spatially vary- ing potentials16 were predicted to achieve the valley po- larization in monolayer MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In terms of applications, it is desirable to obtain pure valley polarized current by electrical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Accordingly, we propose that quantum parametric pump can drive valley and spin po- larized currents in monolayer MoS2 through adiabati- cally varying two gate voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The parametric pump can produce dc current by periodically varying system parameters, which has been generalized to various 2D materials52–55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Specially, spin pump has been reported in several nanostructures56–58, where pure spin current and zero charge current are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In this paper, we numerically study the generation and manipulation of valley-spin polarized currents via adia- batic pump in monolayer MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The system setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' By magnetic doping, an exchange inter- action is introduced in the left and right leads, which in- duces locked valley-spin polarization at the top of valence band as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='1(a) and 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' When the pumping potentials periodically change, fully valley-spin polarized dc currents are driven into the leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' At one moment, the current with K valley and spin up is pumped into the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='11644v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='mes-hall] 27 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 1: Schematics of the band structures of monolayer MoS2 for (a) with exchange field M, (b) without exchange field, (c) with exchange field −M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The red and blue valance bands de- note valley K with spin up and K′ with spin down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' (d) The pump setup based on monolayer MoS2 consisting of left/right leads and the scattering region, whose band structures are correspondingly shown in (a) to (c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The MoS2 lattice is represented by the simple honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The pumping potentials V1 and V2 are added in the scattering re- gion, adjacent to the leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' As V1 and V2 periodically change, electric currents with opposite valley-spin polarizations are si- multaneously pumped into the left and right leads, as shown by the block arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' left lead while the current with opposite valley-spin po- larization flows into the right lead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The polarized current exhibits resonant behavior in the valley-spin locked en- ergy window, which mainly depends on the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' With the increasing of the system length, the pumped currents show periodic oscillation behavior and robust resonant current peaks can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' We also inves- tigate the influence of other system parameters, includ- ing the phase difference, the Rashba SOC strength, the static potential, and the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' It is found that the phase difference and static potentials can invert the direction and hence polarization of the pumped current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' II, we in- troduce the Hamiltonian of monolayer MoS2 and the for- malism of adiabatic parametric pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' III, numerical results and relevant discussions are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Finally, a brief summary is given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' MODEL AND FORMALISM In monolayer MoS2, the low-energy spectrum at K and K′ valleys consists of three d orbitals of Mo, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', dz2, dx2−y2, dxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The relations between these orbitals and basis wave functions satisfy: |ϕc⟩ = |dz2⟩, |ϕλ υ⟩ = (|dx2−y2⟩+iλ|dxy⟩)/ √ 2, where the subscript c/υ denotes the conduction/valence band and λ = ±1 corresponds to different valleys K and K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Based on above low-lying states, the effective Hamiltonian of monolayer MoS2 has the following form26,59 H0(k) = at(λkxσx + kyσy) + ∆σz − tSOλσz − 1 2 τz, (1) where a and t are the lattice constant and hopping strength, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' σx,y,z and τz represent the Pauli matrices of basis functions(|ϕc⟩ and |ϕυ⟩) and spin(↑ and ↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' ∆ is the mass term and the last term is the intrinsic SOC with strength tSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' We employ the tight-binding model of MoS2, which treats monolayer MoS2 as a simplified honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The lattice includes A and B sublattices, corresponding to the dz2 orbit and dx2−y2 + iλdxy orbits of Mo, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In the tight-binding approximation, the Hamilto- nian can be expressed as60,61 H0 = � i ϵic† iαciα + t � c† iαciα + HSO, (2) with HSO = 2itSO 3 √ 3 � ≪i,j≫,α,α′ υijc† iατz,αα′cjα′, (3) where c† iα(ciα) is the creation(annihilation) operator at site i with spin α = ±1, ϵi is the on-site energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' HSO denotes the intrinsic SOC term and the summation over the second nearest-neighbor sites only involves B sublat- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Besides, υij = +1(−1) if an electron moves from site j to site i with taking a left(right) turn62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Based on this model, we consider a monolayer MoS2 setup, which contains three parts: the central scattering region, the left and right leads, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' By magnetic doping, different valley-spin polarizations can be induced in the left and right leads due to the exchange field39,63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The schematic band structures with or without the exchange field are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='1(a)-(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In the presence of Rashba spin-orbit coupling (RSOC), the Hamiltonians for the scattering region with pumping potentials and the leads can be written as HC = H0+ 3itR 4 � ,α,α′ (ταα′×dij)zc† iαcjα′+V (x, y, t), (4) HL/R = H0 ± M � i,α,α′ τz,αα′c† iαciα′, (5) where M and tR denote the strengths of the exchange field and RSOC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' ταα′ = (τx, τy, τz) is the Pauli matrix for spin, and dij is the lattice vector con- necting sites i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The potential term V (x, y, t) = Vs(x, y) + Vt(x, y, t), where Vs = V0 � i Πi(x, y) corre- sponds to the static potential defining the shape of the pumping region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Vt = Vp � i Πi(x, y)cos(ωt + ϕi) is the periodic pumping potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' V0 and Vp are the ampli- tudes of Vs and Vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=" i = 1, 2 are the indices of the po- tential and Πi represents the potential profile, which is (a) (b) (c) : spin up spin down K K' K K' K K (d) 个,K ,K' Lead-L Vi(t) Scattering region V2(t) Lead-R3 highlighted in green in Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' ϕi is the initial phase of the pumping potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' To evaluate the adiabatic valley-spin pump, we need to calculate the average current flowing into lead β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Con- sider a slowly varying time-dependent pumping potential Vt,i, the average current in one period is expressed as64 Iβ = qω 2π � T 0 dt[ dNβ dVt,1 dVt,1 dt + dNβ dVt,2 dVt,2 dt ], (6) where the period of Vt,i is T = 2π/ω with frequency ω and β = L/R labels the lead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The emissivity dNβ dVi is defined in terms of the scattering matrix Sββ′ as65,66 dNβ dVi = � dE 2π (−∂Ef) � β′ Im∂Sββ′ ∂Vi S∗ ββ′, (7) with f the Fermi distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Under the adi- abatic condition, the pumped current is independent of the pumping frequency ω, hence we set ω = 1 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In the language of nonequilibrium Green’s functions, the pumped current is expressed as67–69 Iβ = − q 2π � 2π 0 dt � dE(∂Ef)Tr[ΓβGr dVt dt Ga].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' (8) Here Gr/Ga is the retarded/advanced Green’s function of the central scattering region, which is defined as Gr = Ga,† = [E − HC − � β Σr β]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' HC is the corresponding Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Σr β is the retarded self-energy of lead β, which can be calculated by surface Green’s function70,71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Γβ = i(Σr β − Σa β) denotes the linewidth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' As shown by block arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='1(d), at one moment of the pumping period, polarized current with K valley and spin up (pink arrow) is driven into the left lead, and equal amount current with K′ valley and spin down (blue arrow) flows in the right lead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Detailed numerical results are shown in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' We use the short term, the pumped current, to stand for the valley-spin polarized currents in the leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' RESULTS AND DISCUSSION In the calculations, the on-site energy is ϵi = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='83 eV for A and B sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Other parameters26,60 are set as t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='27 eV, tSO = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='038 eV, and the lattice constant a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='32 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Without loss of generality, we set the ex- change field strength M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='06 eV, and eV is taken as the energy unit throughout the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The periodic boundary condition (PBC) is considered for monolayer MoS2, and thus the edge effect is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' To realize PBC, the upper and lower edges of a zigzag MoS2 rib- bon shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='1(d) are connected with appropriate hopping interactions, which is also called the cylinder boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 2: The dispersion relation of monolayer MoS2 ribbon with an exchange field: (a) M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='06, (b) M = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Both valley and spin are polarized together, where ∆E is defined as the valley-spin locked energy window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Valley-spin polarized current The dispersion of monolayer MoS2 with an exchange field is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='2(a), it is clear that the valley K with spin up is polarized at the valence band top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' However, as the exchange field changes, the polar- ization of both valley and spin is inverted as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In the valley-spin locked window ∆E, perfect polarization can be realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' To investigate the valley- spin polarized current, we consider only the ∆E energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='3, We study the dependence of the pumped current on the phase difference ϕ12 between V1 and V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Due to the inverse valley-spin polarization of the left and right leads, the holes are forbidden to propagate through the scattering region, so there is no pumped current gen- erated in this setup at tR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Introducing RSOC in the scattering region, it is found that the pumped cur- rent arises and flows into left or right lead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' We attribute such a dc current to the spin flip process induced by the RSOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Importantly, nonzero valley-spin polarized cur- rents are pumped into different leads, which depends on the exchange field in leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Our results show that IL,R is an odd function about ϕ12, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', I(ϕ12) = −I(−ϕ12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The maximum of the pumped current appears at ϕ12 = ±π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='3, when the phase difference ϕ12 shifts from −π to 0, the valley-polarized holes with spin up will be pumped out of the scattering region and flow into the left lead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' On the contrary, the opposite valley-polarized holes with spin down will spread into the right lead when ϕ12 shifts from 0 to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' It means that the direction of the pumped current can be tuned by the phase difference ϕ12, then different valley-spin polarized currents are pumped into different leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' We calculate the pumped current as a function of the phase difference ϕ12 for different RSOC strengths tR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Consequently, with the increasing of tR, the pumped current increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' IL versus tR at ϕ12 = π 2 is plotted in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The result is understandable: since the spin-flip efficiency increases when tR is increased, the spin-up carriers from one lead can be more easily flipped as spin-down carriers and flow into the other lead, which 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='5 (a) (b) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='5 spin up E(eV) spin down 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content="5 K K' K K' △E 1 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='5 2 k(π/a) k (π/a) X4 3 2 1 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='008 ϕ12 Pumped current tR=0 tR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='02 tR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='04 tR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='05 Pumped current tR FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 3: The pumped current IL as a function of the phase difference ϕ12 between V1 and V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Inset: the pumped current versus RSOC strength tR at ϕ12 = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Other parameters: Ef = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='74, L = 10a, V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='1, Vp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='006 IL (b) Transmission Pumped current Ef (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='09 T Pumped current Vp IL,V0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='05 IR,V0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='05 IL,V0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='07 IR,V0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='07 IL,V0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='1 IR,V0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 4: (a) The pumped current and transmission versus Fermi energy at V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='1 and Vp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' (b) The polarized current IL and IR versus the pumping potential Vp for dif- ferent static potentials V0 at Ef = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Other parameters: ϕ12 = π/2, L = 10a, tR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' is required by the conservation of charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='4(a), the pumped current and the transmission versus Fermi energy Ef are plotted at ϕ12 = π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Ob- viously, a broad transmission peak arises in the locked window ∆E, and the pumped current exhibits similar behavior as the transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' This result indicates that the transport of polarized holes is dominated by quan- tum resonance, which originates from quantum interfer- ence effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In fact, the resonance assisted transport is a common property of electron pump66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Besides, an im- passable interval for the pumped current is generated as the Fermi energy is away from the resonant peak as 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='06 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='006 (a) Transmission 46 34 22 10 (b) Pumped current L/a tR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='02 tR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='05 tR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='08 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 5: (a) The transmission versus the length L of scattering region for tR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' (b) The pumped current IL as a function of system length L for different RSOC strengths at ϕ12 = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The numbers label the lengths of the scattering region where current peaks emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Other parameters are the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The variation of polarized current is well correlated to the transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The pumped cur- rent IL,R versus the pumping potential for different static potentials is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Due to the particle con- servation, the current flowing into the scattering region must satisfy IL = −IR, which is confirmed through the symmetric curves of IL and IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Furthermore, the results show that the valley-spin polarized current linearly in- creases with the increasing of pumping potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' For a relatively small Vp, the static potential V0 can enhance the magnitude of pumped currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Size effect on the pumped current In the following, we study the influence of the sys- tem size on the pumped current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Transmission and the pumped current versus the length of the scattering re- gion are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' It is interesting that some robust peaks of the pumped current appear at certain lengths, but the currents for other lengths are almost zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Moreover, these peaks show periodic oscillation behavior with the period length 12a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='5(a), we plot the transmission as a function of the length L for tR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The transmission exhibits a similar be- havior as the pumped current, where the transmission peaks also appear at certain system lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' It is clear that these transmission peaks correspond exactly to the pumped current peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' It is reasonable that the peri- odic behavior of the pumped current originates from the spin precession72–74 induced by Rashba SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' When the current carriers travel through the central scattering re- 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='002 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='002 IL (c) (b) Pumped current (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='10 T Pumped current 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='10 Transmission Transmission Transmission Pumped current L/a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 6: The pumped current and transmission versus the length L of the scattering region for different Fermi energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' (a): Ef = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='72, (b): Ef = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='735, (c): Ef = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Other parameters are the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' gion with RSOC interaction, carrier spin keeps precess- ing and spin flip occurs, which results in the periodic oscillating behavior of the pumped current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Our calcu- lation further demonstrates that the width of scattering region has almost no influence on the periodic behavior of polarized current as long as Fermi energy lies in the valley-spin locked energy window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' To evaluate the influ- ence of RSOC, We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='5(b) the pumped currents for different RSOC strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' It is clear that the RSOC strength has less influence on the resonant period, which is L = 12a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' However, with the increasing of tR, the reso- nant current peaks become significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The peak current value grows larger as tR increases, which is consistent with the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Notice that this periodic oscillation behavior of the pumped current is different from the even-odd conduc- tance oscillation of carbon-atom chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='75,76 When two metallic electrodes are attached to a carbon atomic chain, the electronic structure of the carbon chain is modified, which leads to the difference in the density of states be- tween odd- and even-number carbon chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='75,76 This leads to the even-odd conductance oscillation driven by dc bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' However, for the periodic oscillation of the valley-spin polarized currents, it is due to the spin- flipping process induced by Rashba SOC and driven by periodic pumping potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='6, we plot the pumped current and transmission versus the system length for different Fermi energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Ap- parently, the periodic oscillation behavior of the pumped current persists as the Fermi energy changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The re- sult shows that, with the increasing of Ef, the number of peaks increases while the corresponding periodic length FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 7: The pumped current with respect to both the system length L and the Fermi energy Ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Besides, the magnitude of current peaks will decrease as the Fermi energy increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' By calculating the transmission, it can be seen that the behavior of the pumped current is still consistent with the transmission, which means the periodic oscillation is a universal phe- nomenon in the setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' To exhibit an overall view of the pumped current, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='7, we provide a two-dimensional diagram of the valley-spin polarized current as a function of both the system length L and the Fermi energy Ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' By vary- ing L and Ef, we find that there are five curves with discrete extrema of currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The periodic dependence of the pumped current on the system length L is clearly shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' When Ef approaches the top of valley, the largest pumped current appears and becomes more sharp as shown by the orange regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Moreover, it is further confirmed that, multiple resonant peaks of the pumped current arise under an appropriate system length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Our numerical results reveal that it is possible to design a high-efficiency device setup for generating valley-spin po- larized currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Influence of the static potential In this section, we focus on the dependence of IL on the static potential V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' For this purpose, the system length and the RSOC strength are fixed at L = 10a and tR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='8(a), we plot the pumped current as well as transmission coefficient versus the static potential V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' With the increasing of the static potential, a current peak first emerges at V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='126 labeled by the blue point γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' As V0 scans the critical point γ1 at V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='151, the direction of IL is reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Continuing to increase V0, we can see a negative current peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The result shows that the static potential can also change the direction and hence the polarization of the pumped current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Besides, in the vicinity of the critical point γ1, a transmission 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='008 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='004 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='002 a 30 0 20 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='706 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='740 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='735 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='730 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='4 γ3 γ2 Pumped current V0 IL T/50 (a) γ1 (b) V0&T Ef γ1 transition V0 γ2 maximum T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='008 Maximum IL γ3 maximum IL FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 8: (a) The pumped current as well as the transmission coefficient T as a function of the static potential V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' A factor of 1/50 is multiplied to T for better illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' γ1, γ2, γ3 label the critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' (b) The maximum of IL, transition point of V0 and corresponding maximum of T versus the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Other parameters: Vp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='03, ϕ12 = π/2, L = 10a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' peak is clear, which suggests the influence of the static potential also results from quantum resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='8(b), The critical point of V0 and the maxima of both T and IL versus Fermi energy Ef are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' It is found that the critical value of V0 increases linearly with the increasing of Ef, which indicates the resonant energy level depends on the static potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Besides, the curves of the peak values for transmission and pumped current grow with the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The variation of the pumped current with the Fermi energy is consistent with the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In this work, the valley-spin polarized current is gener- ated by electric pumping in adiabatic regime, which re- quires two independently varying system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' We emphasize that similar mechanism can also be achieved with optical pumping, which is in the non-adiabatic regime due to the high frequency of light wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In this case, the light frequency can serve as a pumping param- eter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Therefore, non-adiabatic parametric pumping us- ing electric or optical ways will certainly bring in more physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' CONCLUSIONS In conclusion, we study the valley-spin polarized cur- rent in monolayer MoS2 ribbon via parametric electron pump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' In the proposed setup, different valley-spin polar- ized currents can be controlled to flow into different leads in the valley-spin locked energy window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The phase dif- ference between the pumping potentials can change the direction and hence polarization of the pumped current, where quantum resonance dominates the transport pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Furthermore, the size effect on the valley-spin po- larized current is numerically investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' As the length of the scattering region changes, resonant peaks of the pumped current arise and show periodic oscillation due to the spin precession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' With the increasing of Fermi energy, the number of peaks decreases while the peak height increases within a fixed length range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The depen- dence of the resonant pumped current on the scattering region length and the Fermi energy is numerically re- vealed in a two-dimensional diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' It is also found that the direction of valley-spin polarized currents can be inverted by the static pumping potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' As a po- tential valleytronic device, the pump setup proposed in this work can serve as a valley-spin polarization source, which can simultaneously generate opposite polarization in one device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' The polarized signals can be efficiently manipulated by many system parameters, and significant resonant enhancement has been demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 12034014 and 61674052) and the Natu- ral Science Foundation of Shenzhen (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 20200812092737002, JCYJ20190808115415679, and JCYJ20190808152801642).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Hui Wang also acknowledges supports from the Outstanding Youth Foundation of Henan Scientific Committee (212300410041) and the Key Scientific and Technological Projects in Henan Province (212102210223).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' ∗ xufuming@szu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='cn 1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rycerz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Tworzyd�lo, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Beenakker, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2007, 3, 172-175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xiao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yao, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Niu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2007, 99, 236809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 3 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Cheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Jiang, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Sun, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2016, 18, 103024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Schaibley, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Clark, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rivera, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Ross, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Seyler, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yao, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xu, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2016, 1, 16055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 5 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Gunlycke and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' White, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2011, 106, 136806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 6 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Ang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Ma, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Ang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2017, 96, 245410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 7 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Guo, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2019, 99, 064302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 7 8 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Guan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Sheng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Gao, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2020, 124, 037701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 9 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Gong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xiao, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Guo, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2014, 90, 195428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 10 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wang, Carbon, 2016, 99, 451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 11 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wang, Carbon, 2018, 126, 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 12 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Niu, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2018 20, 103021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 13 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Pawlowski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zebrowski, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Bednarek, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2018, 97, 155412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 14 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Cheng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Jiang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Sun, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xie, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2018, 121, 156801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 15 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Ma, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Huang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Dai, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2020 22, 033002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 16 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Gut, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Prokop, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Sticlet, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Nowak, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2020, 101, 085425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 17 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Takashina, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Ono, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Fujiwara, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Takahashi, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Hirayama, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2006 96, 236801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 18 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Culcer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Saraiva, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Koiller, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Hu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Sarma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2012, 108, 126804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 19 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Collaudin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Fauque, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Kang, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Behnia, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2012, 8, 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 20 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Nebel, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2013, 12, 690.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 21 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Isberg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Gabrysch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Hammersberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Majdi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Kovi, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Twitchen, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2013, 12, 760.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 22 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Ren, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wei, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Qiao, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2016 18, 113011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 23 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Palyi and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Burkard, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2011, 106, 086801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 24 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xiao, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Heinz, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} 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+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Shi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yao, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Niu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2013, 88, 115140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 26 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xiao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Feng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xu, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2012, 108, 196802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 27 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Faria Junior, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zollner, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wo´zniak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Kurpas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Gmitra, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Fabian, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2022 24, 083004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 28 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Semenov and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Kim, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2016, 93, 041414(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 29 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Oliveira, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lelovsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Soler, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Qu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2018, 97, 125309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 30 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Tornatzky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Kaulitz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Maultzsch, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2018, 121, 167401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 31 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Mak, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Mcgill, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Park, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Mceuen, Sci- ence, 2014, 344, 1489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 32 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Cheng, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Schwingenschl¨ogl, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2011, 84, 153402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 33 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Dai, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Fang, Science, 2010, 329, 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 34 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhou, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Sun, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Jena, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2017, 119, 046403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 35 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Da, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Song, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Dong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Ye, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2020, 127, 023903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 36 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rahimi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Moghaddam, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Dykstra, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Gover- nale, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Z¨ulicke, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2017, 95, 104515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 37 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Gan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Cheng, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Schwingenschl¨ogl, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2013, 88, 235310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 38 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Teng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Mao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Fan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Li, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Li, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2020 22, 103061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 39 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Cheng, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhang, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Schwingenschl¨ogl, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2014, 89, 155429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 40 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhou and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Jena, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett, 2017, 8, 5764- 5770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 41 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Deng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Shi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wei, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Cheng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Gao, W.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Pantelides, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Gao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Liu, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhou, Adv.' metadata={'source': 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Kioseoglou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Huang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Petrou, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zeng, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 2017, 12, 757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 44 T.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 2019, 10, 4163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 45 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Mak, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' He, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Shan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Heinz, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Nan- otechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2012, 7, 494-498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 46 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zeng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Dai, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Mak, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xiao, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Shan, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Photonics 2018, 12, 451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 48 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Kumar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yagodkin, 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+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Bolotin, Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 2021, 21, 7123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 49 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Guddala, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Kawaguchi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Komissarenko, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Kir- iushechkina, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Vakulenko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Al`u, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Menon, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Khanikaev, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 2021, 12, 3746.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 50 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Pulkin and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yazyev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2016, 93, 041419(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 51 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' An, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xiao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Tu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yu, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Fal’ko, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2017, 118, 096602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 52 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Talyanskii, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Novikov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Simons, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Levitov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 2001, 87, 276802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 53 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Chan, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lin, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2011, 98, 032106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 54 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Liu, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2020 22, 013042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 55 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wei, Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2022, 17, 43501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 56 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xing, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wei, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2004, 70, 245324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 57 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Deng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Luo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Geng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Sheng, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xing, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2015, 17, 103018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 58 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Sheng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Shen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Sheng, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xing, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2015 91, 125117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 59 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Ochoa and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rold´an, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2013, 87, 245421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 60 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Klinovaja and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Loss, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2013, 88, 075404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 61 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Kleftogiannis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Tang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Cheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' : Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Matter, 2015, 27, 205302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 62 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Kane and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Mele, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2005, 95, 226801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 63 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Bai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yang, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Bai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' : Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Matter, 2018, 30, 495801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 64 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Brouwer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 1998, 58, R10135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 65 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B¨uttiker, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Thomas, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Prˆetre, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B Con- dens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Matter, 1994, 94, 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 66 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wei, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Guo, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2000, 62, 9947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 67 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Guo, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2003 68, 155326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 68 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xing, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2011, 84, 245323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 69 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Zhou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wei, and 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Wang, Nanotechnology, 2016, 27, 185202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 70 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lee and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Joannopoulos, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 1981, 23, 4988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 71 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lee and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Joannopoulos, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 1981, 23, 4997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 72 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' He, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Chen, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Chang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' B, 2008, 78, 085312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 73 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Culcer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lechner, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Winkler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2006, 97, 106601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 74 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Chen and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Hsu, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2021, 23, 073017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 75 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lang and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Avouris, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 1998, 81, 3515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' 76 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lang and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Avouris, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} +page_content=', 2000, 84, 358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf'} diff --git a/OdAzT4oBgHgl3EQfzf5C/content/tmp_files/2301.01769v1.pdf.txt b/OdAzT4oBgHgl3EQfzf5C/content/tmp_files/2301.01769v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..25d8319f9e74e67f233bfb01b825fbaf80552464 --- /dev/null +++ b/OdAzT4oBgHgl3EQfzf5C/content/tmp_files/2301.01769v1.pdf.txt @@ -0,0 +1,1348 @@ +Comprehensive analysis of gene expression profiles to radiation exposure reveals +molecular signatures of low-dose radiation response +Xihaier Luo∗, Sean McCorkle∗, Gilchan Park∗, Vanessa L´opez-Marrero∗, Shinjae Yoo∗, +Edward R. Dougherty†, Xiaoning Qian∗†, Francis J. Alexander∗, Byung-Jun Yoon∗† +∗ Computational Science Initiative, Brookhaven National Laboratory, Upton, NY +† Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX +Abstract—There are various sources of ionizing radiation ex- +posure, where medical exposure for radiation therapy or diag- +nosis is the most common human-made source. Understanding +how gene expression is modulated after ionizing radiation +exposure and investigating the presence of any dose-dependent +gene expression patterns have broad implications for health +risks from radiotherapy, medical radiation diagnostic proce- +dures, as well as other environmental exposure. In this paper, +we perform a comprehensive pathway-based analysis of gene +expression profiles in response to low-dose radiation exposure, +in order to examine the potential mechanism of gene regulation +underlying such responses. To accomplish this goal, we employ +a statistical framework to determine whether a specific group +of genes belonging to a known pathway display coordinated +expression patterns that are modulated in a manner consistent +with the radiation level. Findings in our study suggest that +there exist complex yet consistent signatures that reflect the +molecular response to radiation exposure, which differ between +low-dose and high-dose radiation. +Index Terms—Gene expression analysis, radiation biology, low- +dose radiation response, pathway analysis. +1. Introduction +Environmental threats constitute a major factor in deter- +mining a person’s susceptibility to disease. With the progress +of industrialization and modernization, radiation exposure +has become one of the most serious environmental threats +in today’s world. Mounting evidence suggests that ionizing +radiation is linked to the development of thyroid cancers, +multiple myeloma, and myeloid leukemia in children and +adults [1]. It is well documented that the biological effects of +ionizing radiation on mammalian cells are closely related to +radiation doses and dose rates. In general, low-dose radiation +exposure is far more common than high-dose radiation ex- +posure because low-dose radiation can come from a variety +of sources, including natural sources, cosmic rays, nuclear +power, and various types of radioactive waste. However, +in contrast to the more well-defined effects of high-dose +radiation exposure, the biological effects and consequences +of low-dose radiation and mixed exposures remain poorly +understood [2], [3]. +Historically, the health risks associated with low-dose +ionizing radiation exposure have been estimated by extrap- +olating from available high-dose radiation exposure data. +However, the majority of the data come from experiments +that used extremely high, even supra-lethal, doses. Extrapo- +lating the results of such studies to physiologically relevant +doses can thus be difficult [4]. Furthermore, an increasing +number of studies show that the biological reactions to +high and low doses of radiation are qualitatively distinct, +necessitating a direct examination of low-dose responses to +better understand potential risks [5]. +Genome-wide expression assays using microarrays or +RNA sequencing can provide snapshots of transcriptional +activities in a biological sample, hence studying the gene +expression profiles under low doses of ionizing radiation +can provide novel insights into the biological reactions to +such radiation exposure. In fact, mining gene expression +profiles has proven useful in understanding pathophysiolog- +ical mechanisms, diagnosis and prognosis of complex dis- +eases, and deciding on treatment plans. Several studies have +demonstrated the effectiveness of using gene expression +profiles for traditionally challenging problems, for instance, +discriminating between different subtypes of a complex +disease, such as cancer [6], [7]. Despite these successful +applications, quantification and interpretation at the genetic +level of the impact from radiation exposure on the risk of +developing such diseases are still challenging. Especially, +the small sample size of typical clinical data, on the other +hand, frequently impedes meaningful analysis, making pat- +tern discovery, disease marker identification, risk prediction, +reproducibility, and validation extremely difficult [8], [9]. +Adjusting for multiple hypothesis testing is another critical +issue for all microarray analysis methods. The similarities +of such signatures across different sample types have not +been demonstrated to be strong enough to conclude that +they represent a universal biological mechanism shared by +different sample types [10]–[12]. +In recent years, scientists have gained a better under- +standing of the transcriptional response in cells to radiation +exposure [13]. When cells are exposed to ionizing radiation, +multiple signal transduction pathways are activated, mak- +ing pathway activity a potentially powerful and informa- +tive approach for determining disease states. Furthermore, +pathways, the most well-documented protein interactions, +arXiv:2301.01769v1 [q-bio.GN] 3 Jan 2023 + +are known to closely reflect functional relationships related +to molecular biological activities such as metabolic, sig- +naling, protein interaction, and gene regulation processes. +A growing body of research indicates that tasks such as +class distinction based on differences in pathway activity +can be more stable than distinction based solely on genes. +For example, [14] incorporated pathway information into +expression-based disease diagnosis and proposed a classifi- +cation method based on pathway activities inferred for each +patient. Later in [15], pathway activity patterns are used to +describe a classification scheme for human breast cancer +and to reveal complexity in intrinsic breast cancer subtypes. +The probabilistic inference of differential pathway activity +across different classes (e.g., disease states or phenotypes) +using probabilistic graphical models [16] was shown to +identify molecular signatures that can be used as robust +and reproducible disease markers. The marker identification +method in [16] was further extended in [17], where a novel +algorithm for discovering robust and effective subnetwork +markers in a human protein-protein interaction network that +can accurately predict cancer prognosis and simultaneously +discover multiple synergistic subnetwork markers. It should +be noted that at the heart of these pathway-based analyses +is determining the activity of a given pathway based on the +expression levels of the constituent genes. +The primary goal of this paper is to perform a compre- +hensive pathway-based analysis of gene expression profiles +to investigate the differential time and dose effects, primarily +in low-dose experiments, in order to uncover molecular +signatures of low-dose radiation response. Towards this goal, +we adopt the probabilistic pathway activity inference scheme +in [16], where the pathway activity level is estimated from +gene expression data via the use of a simple probabilistic +graphical model. More specifically, the scheme estimates the +log-likelihood ratio between different classes (e.g., differ- +ent levels of radiation exposure) based on the expression +level of each member gene. The log-likelihood ratios of +the member genes in a given pathway are then aggregated +for probabilistic inference of differential pathway activity. +Through this analysis, we identify the most significantly +differentially activated pathways in response to low-dose +radiation. These pathways are investigated to determine +the presence of consistent dose-dependent gene expression +patterns. Our cross-validation experiments demonstrate that +the proposed method can generate reliable and consistent +pathway analysis results even with limited data. +2. Data +2.1. Low-dose radiation gene expression data +The goal of the current study is to identify poten- +tial molecular signatures underlying the biological response +to low-dose ionizing radiation exposure through pathway- +based analysis of gene expression profiles. For this purpose, +we conducted a thorough literature search and preliminary +analysis to identify human gene expression data suitable +for studying the low-dose radiation response. The gene +Dose Level +Number of Samples +0 Gy +18 +0.005 Gy +16 +0.01 Gy +18 +0.025 Gy +18 +0.05 Gy +17 +0.1 Gy +18 +0.5 Gy +16 +TABLE 1. DESCRIPTION OF THE GENE EXPRESSION DATASET +GSE43151 THAT WAS USED TO INVESTIGATE THE MOLECULAR +SIGNATURES OF LOW-DOSE RADIATION RESPONSE IN THIS STUDY. +expression dataset GSE431511 was identified to be the most +suitable for our study, in terms of sample size and the range +of radiation levels that were considered. Overall, GSE43151 +contains gene expression measurements from 121 blood +samples, where five healthy male donors provided 400 mL +venous peripheral blood samples each [18]. A complete +blood count was performed on each whole blood sample +using an ADVIA Hematology System (Bayer HealthCare). +The standard lymphocyte proportion of 16-45 percent was +met by all samples. Heparin at a final concentration of 34 +U ml−1 was added to whole blood samples. The blood +was then diluted 1:10 with Iscove’s Modified Dulbecco’s +Medium (IMDM, Life Technologies). Finally, blood samples +were incubated overnight at 37 Cina 5% CO2 concentration. +For the ex vivo irradiation, whole blood exposures were +performed at the ICO-4000 facility (Fontenay-aux-Roses, +France) with a Co source at a low dose rate (50 mGy +min−1). Exposures were carried out independently on each +donor’s blood sample. The kerma rate was calculated us- +ing a Physikalisch-Technische Werkst¨atten (PTW) ionization +chamber that was irradiated under the same conditions as the +samples. Doses of 5, 10, 25, 50, 100, and 500 mGy were +tested (See Table. 1), as well as sham irradiated conditions. +Following ex vivo irradiation, blood samples were incubated +at 37 degrees Celsius for 150, 300, 450, and 600 minutes +in a 5% CO2 atmosphere. +A density medium was used to collect CD4+ T lym- +phocytes for cell sorting. Following that, total RNA was +extracted from CD4+ T lymphocytes using RNeasy Mini +columns from the RNeasy Mini Kit (Qiagen) as directed by +the manufacturer. For all RNA samples, the RIN (RNA in- +tegrity number) was calculated for assigning integrity values +to RNA measurements. For gene expression assays, all RIN +values were greater than the recommended value of 7. +Before performing the pathway analysis based on the +GSE43151 gene expression dataset, all 121 samples in the +dataset were normalized, filtered, and analyzed using GAGE +in R software [19]. Following the filtering step, a total of +10,875 probes were chosen, where the basic filtering criteria +consisted of removing a probe when it was undetected in at +least 75% of the replicates considered. +1. https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE43151 + +(D.1) Overall algorithm - pseudo code +(D.2) Rank pathways +Rank pathways using computed t-scores +for pathway in KEGG database +for gene in selected pathway +compute the active score +compute the t-score +• +hsa00010 +• +hsa00020 +• +hsa00030 +• +hsa00040 +• +… +• +hsa00400 +• +hsa00560 +• +hsa04907 +• +hsa00790 +• +… +Task 1: low-dose Task 2: high-dose +(C.1) Label samples +(C.2) Build conditional distributions +(C.3) Estimate activity score +Task 1: zero-dose vs low-dose Task 2: zero-dose vs high-dose +. . . +• +Zero-dose +• +Low-dose +• +High-dose +Compute log-likelihood ratio by Equation 1 +• Task 1 +• Task 2 +ACHicbVDLSsNAFJ3UV62vqEs3g0VoNzXRom6EghsFxXsA9oQJpNJO3SiTMTpYR+iBt/xY0LRdy4EPwbp20W2nrgXg7n3MvMPV7MqFSW9W3kFhaXlfyq4W19Y3NLXN7pyl5IjBpYM64aHtIEkY +j0lBUMdKOBUGhx0jLG1yM/dY9EZLy6FYNY+KEqBfRgGKktOSax9duyvjDCJ5DxnuwFJRg9y5BftbL8BDOSmXLFoVawI4T+yMFEGumt+dn2Ok5BECjMkZce2YuWkSCiKGRkVuokMcID1CMdTSMUEumk+NG8EArPgy40BUpOF/b6QolHIYenoyRKovZ72x+J/XSVRw5qQ0ihNFIjx9KEgYVByOk4I+FQrNtQEYUH1XyHuI4Gw0nkWdAj27MnzpHlUsU8q1ZtqsXaVxZEHe2AflIANTkENXI6aAMHsEzeAVvxpPxYrwbH9PRnJHt7I/ML5+AJo8npc=Llow = log(f( +)/f( +)) +ACHXicbVDLSsNAFJ3UV62vqEs3g0VoNzWRom6EghsFxXsA9oQJpNJOnQyiTMToYT+iBt/xY0LRVy4Ef/GaZuFth64l8M59zJzj5cwKpVlfRuFpeWV1bXiemljc2t7x9zda8s4FZi0cMxi0fWQJIx +y0lJUMdJNBEGRx0jHG15O/M4DEZLG/E6NEuJEKOQ0oBgpLblm/cbNBjQcjOEFZHEIK0EF9u9T5Oe9Co/hvFR1zbJVs6aAi8TOSRnkaLrmZ9+PcRoRrjBDUvZsK1FOhoSimJFxqZ9KkiA8RCHpacpRKSTa8bwyOt+DCIhS6u4FT9vZGhSMpR5OnJCKmBnPcm4n9eL1XBuZNRnqSKcDx7KEgZVDGcRAV9KghWbKQJwoLqv0I8QAJhpQMt6RDs+ZMXSfukZp/W6rf1cuM6j6MIDsAhqAbnIEGuAJN0AIYPIJn8ArejCfjxXg3PmajBSPf2Qd/YHz9AEXwnu8=Lhigh = log(f( +)/f( +)) +(B.1) GSE Database +(B.2) Identify the low-dose radiation data set +(B.3) Sample classification +Zero radiation samples +Low-dose radiation samples +ERR127303 +ERR127302 +ERR127305 +ERR127304 +ERR127309 +ERR127307 +ERR127306 +ERR127308 +26472 +2029 +51582 +6418 +51377 +11146 +5147 +11261 +304 +10628 +336 +308 +10935 +5511 +145376 +5037 +57805 +8525 +6588 +341 +22853 +27344 +116154 +221476 +142679 +710 +7035 +1026 +9749 +27329 +153218 +51050 +5611 +8434 +5274 +25913 +331 +2646 +54577 +94274 +5627 +301 +161742 +9479 +2873 +1032 +9491 +284352 +5570 +9858 +7349 +23145 +27290 +7076 +1028 +57761 +7079 +124790 +89932 +−2 +0 +2 +Value +Color Key +High-dose radiation samples +(D.3) Expert interpretation +(A.1) KEGG Pathway Database +(A.2) Extract the pathway information +(A.3) Gene list +gene 1 +gene 2 +gene k +. . . +A. Build a gene list from a selected pathway from the KEGG database +B. Identify low-dose radiation data set from GSE database +C. Probabilistic inference of pathway activity +D. Rank pathways based on their discriminative powers +Figure 1. Overview of the pathway-based analysis of gene expression profiles in response to low-dose radiation exposure. +2.2. Pathway database +We used the KEGG (Kyoto Encyclopedia of Genes +and Genomes) database to obtain a reliable set of known +biological pathways [20]. KEGG is a collection of manually +drawn pathway maps for understanding high-level functions +and utilities of the biological system. The genomic infor- +mation is maintained in the GENES database, which is a +collection of gene catalogs for all fully sequenced genomes +and some partially sequenced genomes with current annota- +tions of gene functions. The PATHWAY database’s higher- +order functional information is augmented with a collection +of ortholog group tables for information about conserved +subpathways, which are frequently encoded by positionally +related genes on the chromosome and are especially valuable +in predicting gene functions. In our case, we identified 343 +pathways relevant to the gene expression dataset GSE43151 +from the available 548 KEGG pathway maps by discarding +the pathways that did not contain any gene whose measure- +ment was included in GSE43151. +3. Methods +In this section, we describe the technical details of the +pathway-based gene expression data analysis procedure that +was used to detect potential molecular signatures underlying +low-dose radiation response. Figure 1 provides an overview +of the overall procedure. +3.1. Pathway activity inference +To perform the pathway analysis, we first identified the +genes whose measurements were included in the gene ex- +pression dataset GSE43151 for the pathways of our interest. + +P53 SIGNALING PATHWAY +Target genes +Cyclin D +CDK416 +Response +G1 arest +p21 +Cyclin E +(sustaine d) +-irradlia +143-3- +CDK2 +Cell cyc le arrest +UV +/Rerrima +Genotoxic +Cyelin E +ATM +CHK2 +G2 arrest +Cell cyc le +Cellular se rnescence +drugs + DNA damage +Gadd45 +Cdc2 +(sustained) +Nutrition +ATR +CHK1 +B99 +deprivation +Hypoxia +Heaticold . +Fas +shock +Nitric oxide +DRS +CASP8 +PIDD +Bil +A poptosis +Stress signak +Noxa +PUIMAP53AIP +tBid +Cytc +Jncogene +Bcl-xL +Ras, BCR-ABL) +7Sival +-CASP9 +CASP3 +SCYL1EPI +Bc12 +Araf-1 ++p +ROS + PIGs +P14ARF +MDM2 +r53 +DNA +IVitoc hordrior +ScotinPERPPAG608Siah +Apoptosis +AAIFM2 +MDMX +IGF-BF3 +HIGF +Cell cy le +PAIBAI-1KAIGDAiFTSP1Maspin +Irhibition ofangiogene sis +ar retastasis +P48p53R2Gadd45Sestins +DNA re pair and +PTENTSC2IGF-BP3 +TS AF6 +Exosorre rrediated +secretion +MDM2Cop-1PIRH2CyelinGSiah-1WiplANp73 +p53 regative feedback +041156/4/20 +(c) Kanehisa LaboratoriesFor every pathway, member genes that were missing in the +given dataset were removed from the gene set. Consider a +pathway G that consist of n genes {gk}n +k=1 whose mea- +surements were available in the dataset. In the context of +binary classification, we assume that the expression level of +gene gk (k = 1, 2, . . . , n) has a phenotype-dependent dis- +tribution. Let us denote the conditional probability density +function (PDF) of gene gk expression level under phenotype +1 as f 1 +k(x) and the conditional PDF under phenotype 2 +as f 2 +k(x) with x representing the expression level of gene +gk. In our case, we classify radiation exposures into three +categories: zero-dose, low-dose, and high-dose. We compare +low-dose and high-dose samples separately to zero-dose +samples, which means that if zero-dose samples are treated +as phenotype 1, either low-dose or high-dose samples will +be treated as phenotype 2. +After examining different probability distribution mod- +els, we assumed that both f 1 +k(x) and f 2 +k(x) are Guassian in +this study. Having these conditional PDFs, we can calculate +the log-likelihood ratio (LLR) between the two phenotypes +at a given expression level x of gene gk as follows +Lk(x) = log[f 1 +k(x)/f 2 +k(x)] +(1) +For any given gene gk in the pathway G, the associated log- +likelihood ratio Lk(x) in (1) indicates which phenotype is +more likely based on the expression level x of gene gk. By +combining the evidence–in the form of LLR–from all the +member genes in the pathway, we can assess the overall +activity level of the pathway at hand to infer which of the +two phenotypes the collective expression pattern of its mem- +ber genes points to and how significantly so, as discussed +in [16]. More specifically, provided with a set {xj,k}m +j=1 of +m samples (i.e., gene expression measurements) for each +gene gk, we first calculated activity levels {Sj}m +j=1 defined +as +Sj = +n +� +k=1 +Lk(xj,k) +(2) +The activity level Sj in (2) incorporates information from +every gene in the pathway of interest and can be used to +predict the phenotype (class label) based on the overall +activation level of the given pathway in sample j. +Note that to calculate the log-likelihood ratio Lk(x) in +(1), we must first estimate the conditional PDF f c +k(x) for +each phenotype c ∈ {1, 2}. We assume that the expression +of gene gk under the phenotype c follows a Gaussian dis- +tribution with a mean of µc +k and a standard deviation of σc +k. +These parameters were calculated using all of the available +samples that correspond to the phenotype c. After that, the +estimated conditional PDFs can be utilized to compute the +log-likelihood ratios. In practice, we often have insufficient +training data to estimate the PDFs of f 1 +k(x) and f 2 +k(x) +with confidence. As a result, the computation of the log- +likelihood ratio may be sensitive to relatively small changes +in the gene expression levels. To alleviate this issue, we +normalized the data as recommended in [16]. Namely, Lk(x) +was normalized to obtain �Lk(x) as follows +�Lk(x) = +Lk(x) − E[Lk(x)] +� +E[(Lk(x) − E[Lk(x)])2] +. +(3) +While the use of (1) and (2) without normalization for infer- +ring the pathway activity level would be equivalent to using +a Naive Bayes model (NBM) for classifying the phenotype +(class label) given the expression profile of the member +genes that belong to a given pathway, this normalization step +in (3) makes the pathway activity scoring scheme diverge +from the traditional NBM. +3.2. Pathways as potential markers for discriminat- +ing low-dose response from high-dose response +To examine the ability of a pathway to discriminate +between two phenotypes, we computed the t-test statistics +scores using the activity levels Sj for all member genes (as +defined in (2)) and averaged the absolute value of the t-test +scores to compute an aggregated differential activity score. +The aggregated score–which we refer to as the pathway +activity score–was then used as an indicator of the pathway’s +discriminative power [21]. It should be noted that low-dose +and high-dose samples were analyzed separately to detect +most strongly differentially activated pathways under each +radiation exposure level. We had three types of samples: +zero radiation, low-dose radiation (0.005 Gy to 0.1 Gy), +and high-dose radiation (0.5 Gy). Despite the fact that +different low-dose levels of ionizing radiation have been +tested, we treated all dose levels between 0.005 Gy and 0.1 +Gy as the same type (i.e., low-dose radiation). Based on this +categorization, we ranked all relevant KEGG pathways to +based on the strongest differential pathway activity between +zero-dose against low-dose radiations, and separately, based +on zero-dose against high-dose radiations. This is illustrated +in Fig. 1(C). +4. Results +4.1. Pathway analysis results +To begin, we evaluated all relevant pathways in the +KEGG database and ranked the pathways based on their +discriminative power following the procedures elaborated +in Sec. 3 and illustrated in Fig. 1. In particular, we ranked +the pathways based on their discriminative power, assessed +based on the aggregated differential activity score obtained +by averaging the absolute value of the t-test scores of the +member genes in a given pathway [21] and estimating the +p-value. +Fig. 2(a) shows the top five pathways that have been +identified as being the most deferentially activated in the +presence of low-dose radiation. +The top pathway was associated with Natural killer cell +mediated cytotoxicity, focusing on natural killer cells, which +are innate immune system lymphocytes involved in early + +AB6nicbVDL +SgNBEOyNrxhfUY9eBoMQL2FXgn +oMevEY0TwgWcLspDcZMju7zMw +KIeQTvHhQxKtf5M2/cZLsQRML +Goqbrq7gkRwbVz328mtrW9sbu +W3Czu7e/sHxcOjpo5TxbDBYhG +rdkA1Ci6xYbgR2E4U0igQ2ApGt +zO/9YRK81g+mnGCfkQHkoecUW +OlhzI97xVLbsWdg6wSLyMlyFDv +Fb+6/ZilEUrDBNW647mJ8SdUG +c4ETgvdVGNC2YgOsGOpBFqfz +I/dUrOrNInYaxsSUPm6u+JCY20 +HkeB7YyoGeplbyb+53VSE17E +y6T1KBki0VhKoiJyexv0ucKmRF +jSyhT3N5K2JAqyoxNp2BD8JZf +XiXNi4p3WaneV0u1myOPJzAKZ +TBgyuowR3UoQEMBvAMr/DmCOf +FeXc+Fq05J5s5hj9wPn8Ai5mN +Uw=(a) +AB6nicbVDL +SgNBEOyNrxhfUY9eBoMQL2FXgn +oMevEY0TwgWcLspDcZMju7zMw +KIeQTvHhQxKtf5M2/cZLsQRML +Goqbrq7gkRwbVz328mtrW9sbu +W3Czu7e/sHxcOjpo5TxbDBYhG +rdkA1Ci6xYbgR2E4U0igQ2ApGt +zO/9YRK81g+mnGCfkQHkoecUW +Olh3Jw3iuW3Io7B1klXkZKkKHe +K351+zFLI5SGCap1x3MT40+oM +pwJnBa6qcaEshEdYMdSPU/m +R+6pScWaVPwljZkobM1d8TExp +PY4C2xlRM9TL3kz8z+ukJrz2J +1wmqUHJFovCVBATk9nfpM8VMiP +GlCmuL2VsCFVlBmbTsG4C2/ +vEqaFxXvslK9r5ZqN1kceTiBUy +iDB1dQgzuoQwMYDOAZXuHNEc6 +L8+58LFpzTjZzDH/gfP4AjR6N +VA=(b) +Figure 2. Ranking of most differentially activated pathways and their +discriminative power in terms of the pathway activity score. (a) Top +differentially activated pathways under low-dose radiation exposure. The +aggregated t-test scores reflect the discriminative power of the pathways +for discriminating between zero-dose and low-dose samples. (b) Top differ- +entially activated pathways for high-dose radiation exposure (zero-dose vs +high-dose). Comparison between (a) and (b) show a significant difference +between the list of top pathways that are differentially activated under low- +dose radiation and those under high-dose radiation. +defenses against both allogeneic and autologous cells. Many +studies have been conducted to investigate the direct effects +of low-dose ionizing radiation (LDIR) on natural killer cells +and the potential mechanism [22], [23]. The results of the +experiments showed that a simplified strategy based on +LDIR leads to effective expansion and increased activity of +natural killer cells, providing a novel approach for adoptive +cellular immunotherapy. +The second pathway is related to Adherens junction (AJ), +which is the most common type of intercellular adhesion. AJ +initiates and maintains cell adhesion while also controlling +the actin cytoskeleton. In [24], three types of junctional +proteins were chosen for immunohistochemical labeling, +and experimental results showed that not only high, but +also low and moderate doses of cranial irradiation increase +cerebral vessel permeability in mice. In-vitro studies showed +that irradiation alters junctional morphology, reduces cell +number, and causes senescence in brain endothelial cells. +Another study [25] discovered that gamma-radiation, even +at low doses, rapidly disrupts tight junctions, adherens +junctions, and the actin cytoskeleton, resulting in barrier +dysfunction in the mouse colon in vivo. Radiation-induced +epithelial junction disruption and barrier dysfunction are +mediated by oxidative stress, which can be mitigated by +NAC supplementation prior to IR. +Another pathway linked to Sphingolipid metabolism was +also highly ranked. Sphingolipids, a type of membrane +lipid, are bioactive molecules that play a variety of roles +in fundamental cellular processes such as cell division, +differentiation, and cell death. Many studies on the effect +of sphingolipids on cancer treatment have been conducted. +Microbeam radiation can induce radiosensitivity in elements +within the cytoplasm, according to [26]. The effect could be +inhibited by agents that disrupt the formation of lipid rafts +(filipin), demonstrating once again that membranes could be +a target of ionizing radiation. The authors of [27] concluded +that, while other pathways are activated to induce radiation +or chemoresistance, sphingolipids play a significant role. +The JAK-STAT signaling pathway and Glycosphin- +golipid biosynthesis have also been revealed to be very +important in the study of radiation effects. For example, +erythropoietin (EPO), which was originally identified as an +erythrocyte growth factor, is now used to treat anemia and +fatigue in cancer patients receiving radiation therapy and +chemotherapy. The study in [28] demonstrated previously +unknown EPO-mediated HNSCC cell invasion via the Janus +kinase (JAK)-signal transducer and activator of transcription +(STAT) signaling pathway. On the other hand, the findings in +[29] suggest that glycosphingolipid biosynthesis on the cell +surface contributed to the activation of ionizing radiation- +induced apoptosis via ceramide production. The functional +importance of this pathway to eradicating cancer cells with +ionizing radiation has been proven, with sphingolipid break- +down activated as a mechanism of ceramide formation after +cell irradiation. +In a similar manner, Fig. 2(b) shows the top five path- +ways that have been identified as being most differentially +activated in the presence of high-dose radiation. The genes +found in the identified pathways are closely related to the +radiotherapy regimen. Graft-versus-host disease (GVHD), +for example, is a fatal complication of allogeneic hematopoi- +etic stem cell transplantation in which immunocompetent +donor T cells attack genetically diverse host cells. Many +clinical studies have found a link between GVHD severity +and radiation dose, with more severe GVHD after condi- +tioning regimens that included radiation therapy compared +to those that only included chemotherapy [30], [31]. Another +example is allograft rejection. By definition, the recipient’s +alloimmune response to nonself antigens expressed by donor +tissues causes allograft rejection. According to research, +the complex pathophysiology involves host tissue damage +caused by the conditioning regimen (chemotherapy and/or +irradiation) [32]. After nonmyeloablative conditioning with +low-dose irradiation, the use of recombinant fusion protein +promotes mixed lymphoid chimerism. +Interestingly, we can see that there is relatively small +overlap between the set of pathways there were most re- +sponsive to low-dose radiation exposure and those that were +responsive to high-dose radiation exposure. For example, as +shown in Fig. 2, only one pathway (i.e., Natural killer cell +mediated cytotoxicity) was among the top 5 differentially +activate pathways under both low-dose and high-dose ra- +diation. However, we can see more pathways in common +as we go down the list further. For example, when we +compare the top ten pathways that are the most responsive + +Natural killer cell mediated cytotoxicity +Pathway Name +Adherens junction +Sphingolipid metabolism +JAK-STAT signaling pathway +Glycosphingolipid biosynthesis +2.5 +12.5 +0.0 +5.0 +7.5 +10.0 +Aggregated t-test scoreNatural killer cell mediated cytotoxicity +Graft-versus-host disease +Viral myocarditis +Allograft rejection +Autoimmune thyroid disease +2.5 +12.5 +0.0 +5.0 +7.5 +10.0 +Aggregated t-test scoreto low-dose and high-dose radiation exposure, we find four +common pathways: Natural killer cell mediated cytotoxic- +ity, Adherens junction, Glycosphingolipid biosynthesis, and +Antigen processing and presentation. +4.2. Differential dose effect on radiation responsive +pathways +Next, we investigated the differential dose effects on the +top pathways that were most responsive to either low-dose +or high-dose radiation exposure. As noted earlier in Sec. 3.1, +the probabilistic pathway activity inference scheme [16], +which we adopted in this current study, is equivalent to using +a simple probabilistic graphical model (PGM)–namely, a +NBM–when we use (2) for calculating the pathway activity +score based on the LLRs of the member genes belonging +to the pathway. We wanted to find out whether this PGM +constructed to detect the presence of low-dose (or high-dose) +radiation exposure yields consistent activity inference results +as the radiation dose level changes. +Figure 3 shows the inference result based on the PGM +trained to discriminate between zero-dose and low-dose +samples. The y-axis shows the aggregated LLRs and the +x-axis corresponds to the radiation dose level. For each +dose level, the dots show the distribution of the pathway +activity scores for all samples radiated at the given dose +level. The results are shown for the top five pathways that +were found to be most responsive to low-dose radiation. +As we can see in Fig. 3, all low-dose responsive pathways +yielded similar trends, where the inferred differential activity +levels generally decreased as the radiation exposure level +increased. These results imply that these pathways, and the +gene expression profiles of the members therein, may reflect +potential molecular signatures underlying the biological re- +sponse to low-dose radiation exposure. +We carried out a similar analysis based on the top five +high-dose radiation response pathways that were identified +in our study. The analysis results are summarized in Fig. 4. +As before, the y-axis shows the pathway activity score +obtained by aggregating the LLRs of the member genes in +the pathway at hand. It should however be noted that, in this +case, the LLR is obtained by comparing the likelihood ratios +between zero-dose response and high-dose response. The +resulting PGM is therefore trained to discriminate between +zero-dose samples and high-dose samples. Interestingly, ex- +cept for the first pathway (i.e., Natural killer cell mediated +cytotoxicity), which was the top-ranked pathway in both +low-dose as well as high-dose differential activity analysis +(see Fig. 2), the pathway activity levels did not change +significantly as the dose level increased. Considering that +the pathway activity scores reflect the presence of potential +molecular signatures of high-dose radiation response, this +may imply that these top pathways that were responsive +to high-dose radiation exposure might not be substantially +perturbed when the radiation dose level is relatively low. +4.3. Reproducibility of the identified pathways +We conducted cross-validation experiments to assess the +reproducibility of pathway analysis results and the signifi- +cance of the identified pathways. To begin the experiment, +we randomly selected 70% of zero-dose, low-dose, and +high-dose samples, and we repeated this process ten times, +taking into account the total size of our dataset. The top- +ranked pathways identified by the algorithm are depicted +in Fig. 5. Because the different sample selection introduces +randomness, we first counted the show-up cases of pathways +from the top ten most activated pathways. Then, we ranked +our cross-validation results based on the total number of +counts (shown in blue color). We also computed the mean +and standard deviation of the aggregated t-test scores for +each pathway (shown in red color). The cross-validation +experiments for low-dose radiation responsive pathways are +shown in see Fig. 5(a). As we can see, Fig. 5(a) demonstrates +the consistency of the identified pathways when compared +to the results originally obtained using the whole dataset +(see Fig. 2 for comparison). Pathways Natural killer cell +mediated cytotoxicity and JAK-STAT signaling pathway, for +example, have been identified as being highly related to low- +dose radiation response. We suspect that the difference is +due to the radiation dose level. As previously discussed, +we discovered a direct relationship between dose level and +activation. Such differences are expected in a mixed and +random combination of different dose levels. +Noticeably, such consistency was not observed in the +high-dose experiments shown in Fig. 5(b). In many top- +ranked pathways, as shown in Fig. 4, there is a weak dis- +tinction between high-dose samples. The last column, which +represents the distribution of the calculated aggregated t-test +scores of high-dose samples, in particular, shows a narrow- +band distribution (See Fig. 4(b), (d), and (e)). Because the +calculated statistical scores are so close, when randomness +is introduced into data sampling, the cross-validation results +in Fig. 5(b) appear more random. To validate this, we +expanded our ranked pathway list to the top 30 pathways +and found a larger number of overlapping pathways between +the experiments using full dataset and the cross-validation +experiments using only 70% of the dataset. In this case, the +average ranking of the pathways Natural killer cell mediated +cytotoxicity and Allograft rejection, for example, were 17th +and 22nd, respectively. It should be noted that the radiation +dose level that we categorized as “high-dose” in this study is +still relatively low. We expect that gene expression analysis +of samples that underwent higher-dose radiation exposure +may result in more consistent pathway identification results +with clear molecular signatures. +Finally, we also investigated the assumption regarding +the conditional distribution of the gene expression values. +We used the one-sample Kolmogorov-Smirnov (KS) test to +determine the goodness of fit. The test compares the under- +lying distribution F(x) of a sample to a given distribution +G(x), which in our case is a Gaussian distribution. The +null hypothesis holds that the two distributions are identical, +with F(x) = G(x) for all x; the alternative holds that + +AB6nicbVDLSgNBEOyN +rxhfUY9eBoMQL2FXgnoMevEY0TwgWcLspDcZMju7zMwKIeQT +vHhQxKtf5M2/cZLsQRMLGoqbrq7gkRwbVz328mtrW9sbuW3 +Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApG +tzO/9YRK81g+mnGCfkQHkoecUWOlhzI97xVLbsWdg6wSLyMl +yFDvFb+6/ZilEUrDBNW647mJ8SdUGc4ETgvdVGNC2YgOsGOp +pBFqfzI/dUrOrNInYaxsSUPm6u+JCY20HkeB7YyoGeplbyb+ +53VSE17Ey6T1KBki0VhKoiJyexv0ucKmRFjSyhT3N5K2JAq +yoxNp2BD8JZfXiXNi4p3WaneV0u1myOPJzAKZTBgyuowR3U +oQEMBvAMr/DmCOfFeXc+Fq05J5s5hj9wPn8Ai5mNUw=(a) +AB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXgnoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoq +brq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoec +UWOlh3Jw3iuW3Io7B1klXkZKkKHeK351+zFLI5SGCap1x3MT40+oMpwJnBa6qcaEshEdYMdSPU/mR+6pScWaVPwljZkobM +1d8TExpPY4C2xlRM9TL3kz8z+ukJrz2J1wmqUHJFovCVBATk9nfpM8VMiPGlCmuL2VsCFVlBmbTsG4C2/vEqaFxXvslK9 +r5ZqN1kceTiBUyiDB1dQgzuoQwMYDOAZXuHNEc6L8+58LFpzTjZzDH/gfP4AjR6NVA=(b) +AB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXgnoMevEY +0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoecUWOlhzI7xVLbsWdg6wSLyMlyFDvFb+6/ZilEUrDBNW647mJ +8SdUGc4ETgvdVGNC2YgOsGOpBFqfzI/dUrOrNInYaxsSUPm6u+JCY20HkeB7YyoGeplbyb+53VSE17Ey6T1KBki0VhKoiJyexv0ucKmRFjSyhT3N5K2JAqyoxNp2BD8JZfXiXNi4p3WaneV0u1myOPJzAKZTBgyuowR3UoQEMBvAMr/DmCOfFeXc+Fq05 +J5s5hj9wPn8AjqONVQ=(c) +AB6nicbVBNSwMxEJ3U +r1q/qh69BItQL2VXinosevFY0X5Au5RsNtuGZrNLkhXK0p/g +xYMiXv1F3vw3pu0etPXBwO9GWbm+Yng2jONyqsrW9sbhW3 +Szu7e/sH5cOjto5TRVmLxiJWXZ9oJrhkLcONYN1EMRL5gnX8 +8e3M7zwxpXksH80kYV5EhpKHnBJjpYdqcD4oV5yaMwdeJW5O +KpCjOSh/9YOYphGThgqidc91EuNlRBlOBZuW+qlmCaFjMmQ9 +SyWJmPay+alTfGaVAIexsiUNnqu/JzISaT2JfNsZETPSy95M +/M/rpSa89jIuk9QwSReLwlRgE+PZ3zjgilEjJpYQqri9FdMR +UYQam07JhuAuv7xK2hc197JWv69XGjd5HEU4gVOogtX0IA7 +aEILKAzhGV7hDQn0gt7Rx6K1gPKZY/gD9PkDkCiNVg=(d) +AB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXgnoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoq +brq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoec +UWOlhzKe94olt+LOQVaJl5ESZKj3il/dfszSCKVhgmrd8dzE+BOqDGcCp4VuqjGhbEQH2LFU0gi1P5mfOiVnVumTMFa2pCFz +9fEhEZaj6PAdkbUDPWyNxP/8zqpCa/9CZdJalCyxaIwFcTEZPY36XOFzIixJZQpbm8lbEgVZcamU7AheMsvr5LmRcW7rFTv +q6XaTRZHk7gFMrgwRXU4A7q0AGA3iGV3hzhPivDsfi9ack80cwx84nz+RrY1X(e) +Figure 3. The pathway activity level measured in terms of the aggregated log-likelihood ratios (LLRs) in response to different levels of radiation exposure. +Dose-dependent activity level is shown for the top five pathways that were most differentially activated under low-dose radiation exposure. (a) Natural +killer cell mediated cytotoxicity (b) Adherens junction (c) Sphingolipid metabolism (d) JAK-STAT signaling pathway (e) Glycosphingolipid biosynthesis. +All plots in (a)–(e) for the top low-dose response pathways display similar trends, where the differential activity levels reflecting the presence of potential +molecular signatures of low-dose radiation response decrease as the radiation dose level increases. +AB6nicbVDLSgNBEOyN +rxhfUY9eBoMQL2FXgnoMevEY0TwgWcLspDcZMju7zMwKIeQT +vHhQxKtf5M2/cZLsQRMLGoqbrq7gkRwbVz328mtrW9sbuW3 +Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApG +tzO/9YRK81g+mnGCfkQHkoecUWOlhzI97xVLbsWdg6wSLyMl +yFDvFb+6/ZilEUrDBNW647mJ8SdUGc4ETgvdVGNC2YgOsGOp +pBFqfzI/dUrOrNInYaxsSUPm6u+JCY20HkeB7YyoGeplbyb+ +53VSE17Ey6T1KBki0VhKoiJyexv0ucKmRFjSyhT3N5K2JAq +yoxNp2BD8JZfXiXNi4p3WaneV0u1myOPJzAKZTBgyuowR3U +oQEMBvAMr/DmCOfFeXc+Fq05J5s5hj9wPn8Ai5mNUw=(a) +AB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXgnoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoq +brq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoec +UWOlh3Jw3iuW3Io7B1klXkZKkKHeK351+zFLI5SGCap1x3MT40+oMpwJnBa6qcaEshEdYMdSPU/mR+6pScWaVPwljZkobM +1d8TExpPY4C2xlRM9TL3kz8z+ukJrz2J1wmqUHJFovCVBATk9nfpM8VMiPGlCmuL2VsCFVlBmbTsG4C2/vEqaFxXvslK9 +r5ZqN1kceTiBUyiDB1dQgzuoQwMYDOAZXuHNEc6L8+58LFpzTjZzDH/gfP4AjR6NVA=(b) +AB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXgnoMevEY +0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoecUWOlhzI7xVLbsWdg6wSLyMlyFDvFb+6/ZilEUrDBNW647mJ +8SdUGc4ETgvdVGNC2YgOsGOpBFqfzI/dUrOrNInYaxsSUPm6u+JCY20HkeB7YyoGeplbyb+53VSE17Ey6T1KBki0VhKoiJyexv0ucKmRFjSyhT3N5K2JAqyoxNp2BD8JZfXiXNi4p3WaneV0u1myOPJzAKZTBgyuowR3UoQEMBvAMr/DmCOfFeXc+Fq05 +J5s5hj9wPn8AjqONVQ=(c) +AB6nicbVBNSwMxEJ3U +r1q/qh69BItQL2VXinosevFY0X5Au5RsNtuGZrNLkhXK0p/g +xYMiXv1F3vw3pu0etPXBwO9GWbm+Yng2jONyqsrW9sbhW3 +Szu7e/sH5cOjto5TRVmLxiJWXZ9oJrhkLcONYN1EMRL5gnX8 +8e3M7zwxpXksH80kYV5EhpKHnBJjpYdqcD4oV5yaMwdeJW5O +KpCjOSh/9YOYphGThgqidc91EuNlRBlOBZuW+qlmCaFjMmQ9 +SyWJmPay+alTfGaVAIexsiUNnqu/JzISaT2JfNsZETPSy95M +/M/rpSa89jIuk9QwSReLwlRgE+PZ3zjgilEjJpYQqri9FdMR +UYQam07JhuAuv7xK2hc197JWv69XGjd5HEU4gVOogtX0IA7 +aEILKAzhGV7hDQn0gt7Rx6K1gPKZY/gD9PkDkCiNVg=(d) +AB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXgnoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoq +brq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoec +UWOlhzKe94olt+LOQVaJl5ESZKj3il/dfszSCKVhgmrd8dzE+BOqDGcCp4VuqjGhbEQH2LFU0gi1P5mfOiVnVumTMFa2pCFz +9fEhEZaj6PAdkbUDPWyNxP/8zqpCa/9CZdJalCyxaIwFcTEZPY36XOFzIixJZQpbm8lbEgVZcamU7AheMsvr5LmRcW7rFTv +q6XaTRZHk7gFMrgwRXU4A7q0AGA3iGV3hzhPivDsfi9ack80cwx84nz+RrY1X(e) +Figure 4. The pathway activity level measured in terms of the aggregated log-likelihood ratios (LLRs) in response to different levels of radiation exposure. +As before, dose-dependent activity level is shown for the top five pathways that were most differentially activated under high-dose radiation exposure. (a) +Natural killer cell mediated cytotoxicity (b) Graft-versus-host disease (c) Viral myocarditis (d) Allograft rejection (e) Autoimmune thyroid disease. Except +for the top pathway in (a), the differential activity levels reflecting the presence of potential molecular signatures of high-dose radiation response do not +significantly change as the radiation dose level increases. This implies that the pathways that are responsive to high-dose radiation exposure may not be +substantially perturbed under relatively lower-dose radiation exposure. +they are not. We classify the samples as having a Gaussian +distribution if the P-value is greater than 0.05; otherwise, +they have a non-Gaussian distribution. Figure 6 depicts +the computed results, which show that 70.45 percent of +the low-dose samples and 89.63 percent of the high-dose +samples adhere to the Gaussian assumption. This indicates +that during the pathway analysis, it is appropriate to assume +that the conditional distribution of the gene expression data +is Gaussian. + +1.8 +1.6 +1.4 +1.2 +1.0 +0.8 +0 +0.005 +0.01 +0.025 +0.05 +0.1 +0.5 +Dose level1.6 +1.4 +1.2 +1.0 +0.8 +0 +0.005 +0.01 +0.025 +0.05 +0.1 +0.5 +Dose level1.3 +1.2 +1.1 +1.0 +0.9 +0.8 +0.7 +0 +0.005 +0.01 +0.025 +0.05 +0.1 +0.5 +Dose level1.4 +1.2 +1.0 +0.8 +0 +0.005 +0.01 +0.025 +0.05 +0.1 +0.5 +Dose level1.4 +1.2 +1.0 +0.8 +0 +0.005 +0.01 +0.025 +0.05 +0.1 +0.5 +Dose level1.8 +1.6 +1.4 +1.2 +1.0 +0.8 +0 +0.005 +0.01 +0.025 +0.05 +0.1 +0.5 +Dose level4 +2 +0 +2 +0 +0.005 +0.01 +0.025 +0.05 +0.1 +0.5 +Dose level2.0 +1.5 +1.0 +0 +0.005 +0.01 +0.025 +0.05 +0.1 +0.5 +Dose level2 +0 +2 +0 +0.005 +0.01 +0.025 +0.05 +0.1 +0.5 +Dose level2 +1 +0 +-2 +0 +0.005 +0.01 +0.025 +0.05 +0.1 +0.5 +Dose levelAB6nicbVDLSgNBEO +yNrxhfUY9eBoMQL2FXgnoMevEY0T +wgWcLspDcZMju7zMwKIeQTvHhQxK +tf5M2/cZLsQRMLGoqbrq7gkRwbV +z328mtrW9sbuW3Czu7e/sHxcOjpo +5TxbDBYhGrdkA1Ci6xYbgR2E4U0i +gQ2ApGtzO/9YRK81g+mnGCfkQHko +ecUWOlhzI97xVLbsWdg6wSLyMlyF +DvFb+6/ZilEUrDBNW647mJ8SdUGc +4ETgvdVGNC2YgOsGOpBFqfzI/dU +rOrNInYaxsSUPm6u+JCY20HkeB7Y +yoGeplbyb+53VSE17Ey6T1KBki0 +VhKoiJyexv0ucKmRFjSyhT3N5K2J +AqyoxNp2BD8JZfXiXNi4p3WaneV0 +u1myOPJzAKZTBgyuowR3UoQEMBv +AMr/DmCOfFeXc+Fq05J5s5hj9wPn +8Ai5mNUw=(a) +AB6nicbVDLSgNBEO +yNrxhfUY9eBoMQL2FXgnoMevEY0T +wgWcLspDcZMju7zMwKIeQTvHhQxK +tf5M2/cZLsQRMLGoqbrq7gkRwbV +z328mtrW9sbuW3Czu7e/sHxcOjpo +5TxbDBYhGrdkA1Ci6xYbgR2E4U0i +gQ2ApGtzO/9YRK81g+mnGCfkQHko +ecUWOlh3Jw3iuW3Io7B1klXkZKkK +HeK351+zFLI5SGCap1x3MT40+oMp +wJnBa6qcaEshEdYMdSPU/mR+6p +ScWaVPwljZkobM1d8TExpPY4C2x +lRM9TL3kz8z+ukJrz2J1wmqUHJFo +vCVBATk9nfpM8VMiPGlCmuL2VsC +FVlBmbTsG4C2/vEqaFxXvslK9r5 +ZqN1kceTiBUyiDB1dQgzuoQwMYDO +AZXuHNEc6L8+58LFpzTjZzDH/gfP +4AjR6NVA=(b) +Figure 5. Cross validation results of the top ranked pathways. (a) Cross- +validation results for pathways most responsive to low-dose radiation. +(b) Cross-validation results for pathways most responsive to high-dose +radiation. +Figure 6. Kolmogorov-Smirnov (KS) test results. We checked the normality +of the gene expression values in low-dose and high-dose samples using the +KS test. Results indicate that the Gaussian assumption holds in most cases. +5. Conclusion +The current study aimed to unveil molecular signatures +of biological responses exposed to low or very low doses +of ionizing radiation through pathway-based analysis of +genome-wide expression profiles. Gene expression patterns +under the radiation exposure at six different dose levels +ranging from 5 mGy to 500 mGy were investigated, where +the measurements in the original study [18] were made using +blood samples obtained from five different donors during +five independent irradiation sessions. Our investigation was +conducted at the pathway level, as pathway-based gene +expression analysis is known to yield more robust and repro- +ducible results and as it may shed light on potential molecu- +lar mechanisms underlying low-dose radiation response. To +determine the differential activity level of a given pathway +under different levels of radiation exposure, a probabilistic +pathway activity inference scheme was adopted that aggre- +gates the log-likelihood ratios (LLRs) of the member genes +in a given pathway to infer its differential activity. This +allows robust detection of pathways, whose member genes +display possibly subtle yet consistent coordinated expression +patterns in response to low-dose radiation exposure. We +searched through the KEGG database to prioritize pathways +based on their differential activity levels modulated by low- +dose radiation exposure. Our analysis identified the top +pathways that may be associated with low-dose radiation re- +sponse. Findings in this study reflect the complicated nature +of the biological response to low-dose ionizing radiation, +as well as the fact that low-dose exposures affect many +different gene pathways that are not significantly altered +after higher doses of radiotherapy. +One limitation of the current study is the small sample +size of the analyzed dataset (GSE43151). While it has been +challenging to find large-scale human gene expression data +under low-dose radiation exposure, should such data be +available in the future, their analysis would shed further light +onto the unique molecular signatures of low-dose radiation +response. Furthermore, the pathway activity level inference +scheme in (2) makes specific modeling assumptions, upon +which the derived results depend. In fact, the adopted +scheme [16] assumes that the gene expression levels of +the member genes in a given pathway are conditionally +independent given the class label (e.g., presence/absence of +radiation exposure as was considered in the current study) +and follow Gaussian distributions. Although we carried out +some preliminary validation of this modeling assumption +(e.g., see Fig. 6), it would be also worth validating the +pathway analysis results using other methods [33], [34], +which may be potentially pursued in our future studies. +Acknowledgements +This work is supported by the U.S. Department of +Energy, Office of Science, RadBio program under Award +KP1601011/FWP CC121. +References +[1] +N. R. Council et al., “Health risks from exposure to low levels of +ionizing radiation: Beir vii phase 2,” 2006. +[2] +W. M. Bonner, “Low-dose radiation: thresholds, bystander effects, +and adaptive responses,” Proceedings of the National Academy of +Sciences, vol. 100, no. 9, pp. 4973–4975, 2003. +[3] +L. Mullenders, M. Atkinson, H. Paretzke, L. Sabatier, and S. Bouf- +fler, “Assessing cancer risks of low-dose radiation,” Nature Reviews +Cancer, vol. 9, no. 8, pp. 596–604, 2009. +[4] +D. J. Brenner, R. Doll, D. T. Goodhead, E. J. Hall, C. E. Land, +J. B. Little, J. H. Lubin, D. L. Preston, R. J. Preston, J. S. Puskin +et al., “Cancer risks attributable to low doses of ionizing radiation: +assessing what we really know,” Proceedings of the National Academy +of Sciences, vol. 100, no. 24, pp. 13 761–13 766, 2003. +[5] +M. Tubiana, L. E. Feinendegen, C. Yang, and J. M. Kaminski, “The +linear no-threshold relationship is inconsistent with radiation biologic +and experimental data,” Radiology, vol. 251, no. 1, pp. 13–22, 2009. + +100 +Gaussian +Non-Gaussian +80 +Percentage +60 +40 +20 +0 +Low dose +High doseNatural killer cell mediated cytotoxicity + Name +JAK-STAT signaling pathway +Pathway +Glycosphingolipid biosynthesis - lacto and neolacto series +T cell receptor signaling pathway +Human T-cell leukemia virus 1 infection +0 +10 +0 +10 +Counts +Aggregated t-test scoreGlycosphingolipid biosynthesis - lacto and neolacto series +T cell receptor signaling pathway +Cellular senescence +Human T-cell leukemia virus 1 infection +JAK-STAT signaling pathway +0 +5 +0 +10 +Counts +Aggregated t-test score[6] +T. R. Golub, D. K. Slonim, P. Tamayo, C. Huard, M. Gaasenbeek, +J. P. Mesirov, H. Coller, M. L. Loh, J. R. Downing, M. A. Caligiuri +et al., “Molecular classification of cancer: class discovery and class +prediction by gene expression monitoring,” science, vol. 286, no. +5439, pp. 531–537, 1999. +[7] +A. A. Alizadeh, M. B. Eisen, R. E. Davis, C. Ma, I. S. Lossos, +A. Rosenwald, J. C. Boldrick, H. Sabet, T. Tran, X. Yu et al., “Distinct +types of diffuse large b-cell lymphoma identified by gene expression +profiling,” Nature, vol. 403, no. 6769, pp. 503–511, 2000. +[8] +T. Ideker, V. Thorsson, A. F. Siegel, and L. E. Hood, “Testing for +differentially-expressed genes by maximum-likelihood analysis of +microarray data,” Journal of computational biology, vol. 7, no. 6, +pp. 805–817, 2000. +[9] +P. Baldi and A. D. Long, “A bayesian framework for the analysis of +microarray expression data: regularized t-test and statistical inferences +of gene changes,” Bioinformatics, vol. 17, no. 6, pp. 509–519, 2001. +[10] U. M. Braga-Neto and E. R. Dougherty, “Is cross-validation valid +for small-sample microarray classification?” Bioinformatics, vol. 20, +no. 3, pp. 374–380, 2004. +[11] S. Michiels, S. Koscielny, and C. Hill, “Prediction of cancer outcome +with microarrays: a multiple random validation strategy,” The Lancet, +vol. 365, no. 9458, pp. 488–492, 2005. +[12] L. Ein-Dor, O. Zuk, and E. Domany, “Thousands of samples are +needed to generate a robust gene list for predicting outcome in +cancer,” Proceedings of the National Academy of Sciences, vol. 103, +no. 15, pp. 5923–5928, 2006. +[13] P. Khatri, M. Sirota, and A. J. Butte, “Ten years of pathway analysis: +current approaches and outstanding challenges,” PLoS computational +biology, vol. 8, no. 2, p. e1002375, 2012. +[14] E. Lee, H.-Y. Chuang, J.-W. Kim, T. Ideker, and D. Lee, “Inferring +pathway activity toward precise disease classification,” PLoS compu- +tational biology, vol. 4, no. 11, p. e1000217, 2008. +[15] M. L. Gatza, J. E. Lucas, W. T. Barry, J. W. Kim, Q. Wang, M. D. +Crawford, M. B. Datto, M. Kelley, B. Mathey-Prevot, A. Potti et al., +“A pathway-based classification of human breast cancer,” Proceedings +of the National Academy of Sciences, vol. 107, no. 15, pp. 6994–6999, +2010. +[16] J. Su, B.-J. Yoon, and E. R. Dougherty, “Accurate and reliable cancer +classification based on probabilistic inference of pathway activity,” +PLOS ONE, vol. 4, no. 12, p. e8161, 2009. +[17] N. Khunlertgit and B.-J. Yoon, “Incorporating topological information +for predicting robust cancer subnetwork markers in human protein- +protein interaction network,” in BMC bioinformatics, vol. 17, no. 13. +Springer, 2016, pp. 143–152. +[18] I. Nosel, A. Vaurijoux, J.-F. Barquinero, and G. Gruel, “Characteriza- +tion of gene expression profiles at low and very low doses of ionizing +radiation,” DNA repair, vol. 12, no. 7, pp. 508–517, 2013. +[19] W. Luo, M. S. Friedman, K. Shedden, K. D. Hankenson, and P. J. +Woolf, “Gage: generally applicable gene set enrichment for pathway +analysis,” BMC bioinformatics, vol. 10, no. 1, pp. 1–17, 2009. +[20] M. Kanehisa and S. Goto, “Kegg: kyoto encyclopedia of genes and +genomes,” Nucleic acids research, vol. 28, no. 1, pp. 27–30, 2000. +[21] L. Tian, S. A. Greenberg, S. W. Kong, J. Altschuler, I. S. Kohane, and +P. J. Park, “Discovering statistically significant pathways in expression +profiling studies,” Proceedings of the National Academy of Sciences, +vol. 102, no. 38, pp. 13 544–13 549, 2005. +[22] G. Yang, Q. Kong, G. Wang, H. Jin, L. Zhou, D. Yu, C. Niu, +W. Han, W. Li, and J. Cui, “Low-dose ionizing radiation induces +direct activation of natural killer cells and provides a novel approach +for adoptive cellular immunotherapy,” Cancer Biotherapy and Radio- +pharmaceuticals, vol. 29, no. 10, pp. 428–434, 2014. +[23] J.-U. Jeong, T. N. T. Uong, W.-K. Chung, T.-K. Nam, S.-J. Ahn, J.-Y. +Song, S.-K. Kim, D.-J. Shin, E. Cho, K. W. Kim et al., “Effect of +irradiation-induced intercellular adhesion molecule-1 expression on +natural killer cell-mediated cytotoxicity toward human cancer cells,” +Cytotherapy, vol. 20, no. 5, pp. 715–727, 2018. +[24] N. S´andor, F. R. Walter, A. Bocsik, P. S´antha, B. Schilling-T´oth, +V. L´ener, Z. Varga, Z. Kah´an, M. A. Deli, G. S´afr´any et al., “Low +dose cranial irradiation-induced cerebrovascular damage is reversible +in mice,” PloS one, vol. 9, no. 11, p. e112397, 2014. +[25] P. K. Shukla, R. Gangwar, B. Manda, A. S. Meena, N. Yadav, +E. Szabo, A. Balogh, S. C. Lee, G. Tigyi, and R. Rao, “Rapid +disruption of intestinal epithelial tight junction and barrier dysfunction +by ionizing radiation in mouse colon in vivo: protection by n-acetyl-l- +cysteine,” American Journal of Physiology-Gastrointestinal and Liver +Physiology, vol. 310, no. 9, pp. G705–G715, 2016. +[26] C. Shao, M. Folkard, B. D. Michael, and K. M. Prise, “Targeted +cytoplasmic irradiation induces bystander responses,” Proceedings of +the National Academy of Sciences, vol. 101, no. 37, pp. 13 495– +13 500, 2004. +[27] J.-P. Truman, M. Garc´ıa-Barros, L. M. Obeid, and Y. A. Hannun, +“Evolving concepts in cancer therapy through targeting sphingolipid +metabolism,” Biochimica et Biophysica Acta (BBA)-Molecular and +Cell Biology of Lipids, vol. 1841, no. 8, pp. 1174–1188, 2014. +[28] S. Y. Lai, E. E. Childs, S. Xi, F. M. Coppelli, W. E. Gooding, A. Wells, +R. L. Ferris, and J. R. Grandis, “Erythropoietin-mediated activation +of jak-stat signaling contributes to cellular invasion in head and neck +squamous cell carcinoma,” Oncogene, vol. 24, no. 27, pp. 4442–4449, +2005. +[29] M. Aureli, V. Murdica, N. Loberto, M. Samarani, A. Prinetti, R. Bassi, +and S. Sonnino, “Exploring the link between ceramide and ionizing +radiation,” Glycoconjugate journal, vol. 31, no. 6, pp. 449–459, 2014. +[30] G. R. Hill and J. L. Ferrara, “The primacy of the gastrointestinal tract +as a target organ of acute graft-versus-host disease: rationale for the +use of cytokine shields in allogeneic bone marrow transplantation,” +Blood, The Journal of the American Society of Hematology, vol. 95, +no. 9, pp. 2754–2759, 2000. +[31] B. R. Blazar, W. J. Murphy, and M. Abedi, “Advances in graft- +versus-host disease biology and therapy,” Nature reviews immunology, +vol. 12, no. 6, pp. 443–458, 2012. +[32] M. R. Van den Brink and S. J. Burakoff, “Cytolytic pathways in +haematopoietic stem-cell transplantation,” Nature Reviews Immunol- +ogy, vol. 2, no. 4, pp. 273–281, 2002. +[33] J. Han, X. Shi, Y. Zhang, Y. Xu, Y. Jiang, C. Zhang, L. Feng, H. Yang, +D. Shang, Z. Sun et al., “Esea: discovering the dysregulated pathways +based on edge set enrichment analysis,” Scientific reports, vol. 5, +no. 1, pp. 1–15, 2015. +[34] M. Rydenfelt, B. Klinger, M. Kl¨unemann, and N. Bl¨uthgen, “Speed2: +inferring upstream pathway activity from differential gene expres- +sion,” Nucleic acids research, vol. 48, no. W1, pp. W307–W312, +2020. + diff --git a/OdAzT4oBgHgl3EQfzf5C/content/tmp_files/load_file.txt b/OdAzT4oBgHgl3EQfzf5C/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f15d3cb2b1c7fc2e5db80ff00989487871743af3 --- /dev/null +++ b/OdAzT4oBgHgl3EQfzf5C/content/tmp_files/load_file.txt @@ -0,0 +1,1127 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf,len=1126 +page_content='Comprehensive analysis of gene expression profiles to radiation exposure reveals molecular signatures of low-dose radiation response Xihaier Luo∗, Sean McCorkle∗, Gilchan Park∗, Vanessa L´opez-Marrero∗, Shinjae Yoo∗, Edward R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Dougherty†, Xiaoning Qian∗†, Francis J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Alexander∗, Byung-Jun Yoon∗† ∗ Computational Science Initiative, Brookhaven National Laboratory, Upton, NY † Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX Abstract—There are various sources of ionizing radiation ex- posure, where medical exposure for radiation therapy or diag- nosis is the most common human-made source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Understanding how gene expression is modulated after ionizing radiation exposure and investigating the presence of any dose-dependent gene expression patterns have broad implications for health risks from radiotherapy, medical radiation diagnostic proce- dures, as well as other environmental exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' In this paper, we perform a comprehensive pathway-based analysis of gene expression profiles in response to low-dose radiation exposure, in order to examine the potential mechanism of gene regulation underlying such responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' To accomplish this goal, we employ a statistical framework to determine whether a specific group of genes belonging to a known pathway display coordinated expression patterns that are modulated in a manner consistent with the radiation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Findings in our study suggest that there exist complex yet consistent signatures that reflect the molecular response to radiation exposure, which differ between low-dose and high-dose radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Index Terms—Gene expression analysis, radiation biology, low- dose radiation response, pathway analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Introduction Environmental threats constitute a major factor in deter- mining a person’s susceptibility to disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' With the progress of industrialization and modernization, radiation exposure has become one of the most serious environmental threats in today’s world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Mounting evidence suggests that ionizing radiation is linked to the development of thyroid cancers, multiple myeloma, and myeloid leukemia in children and adults [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' It is well documented that the biological effects of ionizing radiation on mammalian cells are closely related to radiation doses and dose rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' In general, low-dose radiation exposure is far more common than high-dose radiation ex- posure because low-dose radiation can come from a variety of sources, including natural sources, cosmic rays, nuclear power, and various types of radioactive waste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' However, in contrast to the more well-defined effects of high-dose radiation exposure, the biological effects and consequences of low-dose radiation and mixed exposures remain poorly understood [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Historically, the health risks associated with low-dose ionizing radiation exposure have been estimated by extrap- olating from available high-dose radiation exposure data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' However, the majority of the data come from experiments that used extremely high, even supra-lethal, doses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Extrapo- lating the results of such studies to physiologically relevant doses can thus be difficult [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Furthermore, an increasing number of studies show that the biological reactions to high and low doses of radiation are qualitatively distinct, necessitating a direct examination of low-dose responses to better understand potential risks [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Genome-wide expression assays using microarrays or RNA sequencing can provide snapshots of transcriptional activities in a biological sample, hence studying the gene expression profiles under low doses of ionizing radiation can provide novel insights into the biological reactions to such radiation exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' In fact, mining gene expression profiles has proven useful in understanding pathophysiolog- ical mechanisms, diagnosis and prognosis of complex dis- eases, and deciding on treatment plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Several studies have demonstrated the effectiveness of using gene expression profiles for traditionally challenging problems, for instance, discriminating between different subtypes of a complex disease, such as cancer [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Despite these successful applications, quantification and interpretation at the genetic level of the impact from radiation exposure on the risk of developing such diseases are still challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Especially, the small sample size of typical clinical data, on the other hand, frequently impedes meaningful analysis, making pat- tern discovery, disease marker identification, risk prediction, reproducibility, and validation extremely difficult [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Adjusting for multiple hypothesis testing is another critical issue for all microarray analysis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' The similarities of such signatures across different sample types have not been demonstrated to be strong enough to conclude that they represent a universal biological mechanism shared by different sample types [10]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' In recent years, scientists have gained a better under- standing of the transcriptional response in cells to radiation exposure [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' When cells are exposed to ionizing radiation, multiple signal transduction pathways are activated, mak- ing pathway activity a potentially powerful and informa- tive approach for determining disease states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Furthermore, pathways, the most well-documented protein interactions, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='01769v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='GN] 3 Jan 2023 are known to closely reflect functional relationships related to molecular biological activities such as metabolic, sig- naling, protein interaction, and gene regulation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' A growing body of research indicates that tasks such as class distinction based on differences in pathway activity can be more stable than distinction based solely on genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' For example, [14] incorporated pathway information into expression-based disease diagnosis and proposed a classifi- cation method based on pathway activities inferred for each patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Later in [15], pathway activity patterns are used to describe a classification scheme for human breast cancer and to reveal complexity in intrinsic breast cancer subtypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' The probabilistic inference of differential pathway activity across different classes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=', disease states or phenotypes) using probabilistic graphical models [16] was shown to identify molecular signatures that can be used as robust and reproducible disease markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' The marker identification method in [16] was further extended in [17], where a novel algorithm for discovering robust and effective subnetwork markers in a human protein-protein interaction network that can accurately predict cancer prognosis and simultaneously discover multiple synergistic subnetwork markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' It should be noted that at the heart of these pathway-based analyses is determining the activity of a given pathway based on the expression levels of the constituent genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' The primary goal of this paper is to perform a compre- hensive pathway-based analysis of gene expression profiles to investigate the differential time and dose effects, primarily in low-dose experiments, in order to uncover molecular signatures of low-dose radiation response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Towards this goal, we adopt the probabilistic pathway activity inference scheme in [16], where the pathway activity level is estimated from gene expression data via the use of a simple probabilistic graphical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' More specifically, the scheme estimates the log-likelihood ratio between different classes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=', differ- ent levels of radiation exposure) based on the expression level of each member gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' The log-likelihood ratios of the member genes in a given pathway are then aggregated for probabilistic inference of differential pathway activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Through this analysis, we identify the most significantly differentially activated pathways in response to low-dose radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' These pathways are investigated to determine the presence of consistent dose-dependent gene expression patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Our cross-validation experiments demonstrate that the proposed method can generate reliable and consistent pathway analysis results even with limited data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Data 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Low-dose radiation gene expression data The goal of the current study is to identify poten- tial molecular signatures underlying the biological response to low-dose ionizing radiation exposure through pathway- based analysis of gene expression profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' For this purpose, we conducted a thorough literature search and preliminary analysis to identify human gene expression data suitable for studying the low-dose radiation response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' The gene Dose Level Number of Samples 0 Gy 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='005 Gy 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='01 Gy 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='025 Gy 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='05 Gy 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='1 Gy 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='5 Gy 16 TABLE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' DESCRIPTION OF THE GENE EXPRESSION DATASET GSE43151 THAT WAS USED TO INVESTIGATE THE MOLECULAR SIGNATURES OF LOW-DOSE RADIATION RESPONSE IN THIS STUDY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' expression dataset GSE431511 was identified to be the most suitable for our study, in terms of sample size and the range of radiation levels that were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Overall, GSE43151 contains gene expression measurements from 121 blood samples, where five healthy male donors provided 400 mL venous peripheral blood samples each [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' A complete blood count was performed on each whole blood sample using an ADVIA Hematology System (Bayer HealthCare).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' The standard lymphocyte proportion of 16-45 percent was met by all samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Heparin at a final concentration of 34 U ml−1 was added to whole blood samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' The blood was then diluted 1:10 with Iscove’s Modified Dulbecco’s Medium (IMDM, Life Technologies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Finally, blood samples were incubated overnight at 37 Cina 5% CO2 concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' For the ex vivo irradiation, whole blood exposures were performed at the ICO-4000 facility (Fontenay-aux-Roses, France) with a Co source at a low dose rate (50 mGy min−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Exposures were carried out independently on each donor’s blood sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' The kerma rate was calculated us- ing a Physikalisch-Technische Werkst¨atten (PTW) ionization chamber that was irradiated under the same conditions as the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Doses of 5, 10, 25, 50, 100, and 500 mGy were tested (See Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' 1), as well as sham irradiated conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Following ex vivo irradiation, blood samples were incubated at 37 degrees Celsius for 150, 300, 450, and 600 minutes in a 5% CO2 atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' A density medium was used to collect CD4+ T lym- phocytes for cell sorting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Following that, total RNA was extracted from CD4+ T lymphocytes using RNeasy Mini columns from the RNeasy Mini Kit (Qiagen) as directed by the manufacturer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' For all RNA samples, the RIN (RNA in- tegrity number) was calculated for assigning integrity values to RNA measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' For gene expression assays, all RIN values were greater than the recommended value of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Before performing the pathway analysis based on the GSE43151 gene expression dataset, all 121 samples in the dataset were normalized, filtered, and analyzed using GAGE in R software [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Following the filtering step, a total of 10,875 probes were chosen, where the basic filtering criteria consisted of removing a probe when it was undetected in at least 75% of the replicates considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' https://www.' metadata={'source': 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code (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='2) Rank pathways Rank pathways using computed t-scores for pathway in KEGG database for gene in selected pathway compute the active score compute the t-score hsa00010 hsa00020 hsa00030 hsa00040 … hsa00400 hsa00560 hsa04907 hsa00790 … Task 1: low-dose Task 2: high-dose (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='1) Label samples (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='2) Build conditional distributions (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='3) Estimate activity score Task 1: zero-dose vs low-dose Task 2: zero-dose vs high-dose .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='Value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='Color Key ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='High-dose radiation samples ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='(D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='3) Expert interpretation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='1) KEGG Pathway Database (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='2) Extract the pathway information (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='3) Gene list gene 1 gene 2 gene k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Build a gene list from a selected pathway from the KEGG database B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Identify low-dose radiation data set from GSE database C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Probabilistic inference of pathway activity D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Rank pathways based on their discriminative powers Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Overview of the pathway-based analysis of gene expression profiles in response to low-dose radiation exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Pathway database We used the KEGG (Kyoto Encyclopedia of Genes and Genomes) database to obtain a reliable set of known biological pathways [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' KEGG is a collection of manually drawn pathway maps for understanding high-level functions and utilities of the biological system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' The genomic infor- mation is maintained in the GENES database, which is a collection of gene catalogs for all fully sequenced genomes and some partially sequenced genomes with current annota- tions of gene functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' The PATHWAY database’s higher- order functional information is augmented with a collection of ortholog group tables for information about conserved subpathways, which are frequently encoded by positionally related genes on the chromosome and are especially valuable in predicting gene functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' In our case, we identified 343 pathways relevant to the gene expression dataset GSE43151 from the available 548 KEGG pathway maps by discarding the pathways that did not contain any gene whose measure- ment was included in GSE43151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Methods In this section, we describe the technical details of the pathway-based gene expression data analysis procedure that was used to detect potential molecular signatures underlying low-dose radiation response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Figure 1 provides an overview of the overall procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Pathway activity inference To perform the pathway analysis, we first identified the genes whose measurements were included in the gene ex- pression dataset GSE43151 for the pathways of our interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' P53 SIGNALING PATHWAY Target genes Cyclin D CDK416 Response G1 arest p21 Cyclin E (sustaine d) irradlia 143-3- CDK2 Cell cyc le arrest UV /Rerrima Genotoxic Cyelin E ATM CHK2 G2 arrest Cell cyc le Cellular se rnescence drugs DNA damage Gadd45 Cdc2 (sustained) Nutrition ATR CHK1 B99 deprivation Hypoxia Heaticold .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Fas shock Nitric oxide DRS CASP8 PIDD Bil A poptosis Stress signak Noxa PUIMAP53AIP tBid Cytc Jncogene Bcl-xL Ras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' BCR-ABL) 7Sival CASP9 CASP3 SCYL1EPI Bc12 Araf-1 +p ROS PIGs P14ARF MDM2 r53 DNA IVitoc hordrior ScotinPERPPAG608Siah Apoptosis AAIFM2 MDMX IGF-BF3 HIGF Cell cy le PAIBAI-1KAIGDAiFTSP1Maspin Irhibition ofangiogene sis ar retastasis P48p53R2Gadd45Sestins DNA re pair and PTENTSC2IGF-BP3 TS AF6 Exosorre rrediated secretion MDM2Cop-1PIRH2CyelinGSiah-1WiplANp73 p53 regative feedback 041156/4/20 (c) Kanehisa LaboratoriesFor every pathway,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' member genes that were missing in the given dataset were removed from the gene set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Consider a pathway G that consist of n genes {gk}n k=1 whose mea- surements were available in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' In the context of binary classification, we assume that the expression level of gene gk (k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' , n) has a phenotype-dependent dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Let us denote the conditional probability density function (PDF) of gene gk expression level under phenotype 1 as f 1 k(x) and the conditional PDF under phenotype 2 as f 2 k(x) with x representing the expression level of gene gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' In our case, we classify radiation exposures into three categories: zero-dose, low-dose, and high-dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' We compare low-dose and high-dose samples separately to zero-dose samples, which means that if zero-dose samples are treated as phenotype 1, either low-dose or high-dose samples will be treated as phenotype 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' After examining different probability distribution mod- els, we assumed that both f 1 k(x) and f 2 k(x) are Guassian in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Having these conditional PDFs, we can calculate the log-likelihood ratio (LLR) between the two phenotypes at a given expression level x of gene gk as follows Lk(x) = log[f 1 k(x)/f 2 k(x)] (1) For any given gene gk in the pathway G, the associated log- likelihood ratio Lk(x) in (1) indicates which phenotype is more likely based on the expression level x of gene gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' By combining the evidence–in the form of LLR–from all the member genes in the pathway, we can assess the overall activity level of the pathway at hand to infer which of the two phenotypes the collective expression pattern of its mem- ber genes points to and how significantly so, as discussed in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' More specifically, provided with a set {xj,k}m j=1 of m samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=', gene expression measurements) for each gene gk, we first calculated activity levels {Sj}m j=1 defined as Sj = n � k=1 Lk(xj,k) (2) The activity level Sj in (2) incorporates information from every gene in the pathway of interest and can be used to predict the phenotype (class label) based on the overall activation level of the given pathway in sample j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Note that to calculate the log-likelihood ratio Lk(x) in (1), we must first estimate the conditional PDF f c k(x) for each phenotype c ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' We assume that the expression of gene gk under the phenotype c follows a Gaussian dis- tribution with a mean of µc k and a standard deviation of σc k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' These parameters were calculated using all of the available samples that correspond to the phenotype c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' After that, the estimated conditional PDFs can be utilized to compute the log-likelihood ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' In practice, we often have insufficient training data to estimate the PDFs of f 1 k(x) and f 2 k(x) with confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' As a result, the computation of the log- likelihood ratio may be sensitive to relatively small changes in the gene expression levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' To alleviate this issue, we normalized the data as recommended in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Namely, Lk(x) was normalized to obtain �Lk(x) as follows �Lk(x) = Lk(x) − E[Lk(x)] � E[(Lk(x) − E[Lk(x)])2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' (3) While the use of (1) and (2) without normalization for infer- ring the pathway activity level would be equivalent to using a Naive Bayes model (NBM) for classifying the phenotype (class label) given the expression profile of the member genes that belong to a given pathway, this normalization step in (3) makes the pathway activity scoring scheme diverge from the traditional NBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Pathways as potential markers for discriminat- ing low-dose response from high-dose response To examine the ability of a pathway to discriminate between two phenotypes, we computed the t-test statistics scores using the activity levels Sj for all member genes (as defined in (2)) and averaged the absolute value of the t-test scores to compute an aggregated differential activity score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' The aggregated score–which we refer to as the pathway activity score–was then used as an indicator of the pathway’s discriminative power [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' It should be noted that low-dose and high-dose samples were analyzed separately to detect most strongly differentially activated pathways under each radiation exposure level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' We had three types of samples: zero radiation, low-dose radiation (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='005 Gy to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='1 Gy), and high-dose radiation (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='5 Gy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Despite the fact that different low-dose levels of ionizing radiation have been tested, we treated all dose levels between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='005 Gy and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='1 Gy as the same type (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=', low-dose radiation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Based on this categorization, we ranked all relevant KEGG pathways to based on the strongest differential pathway activity between zero-dose against low-dose radiations, and separately, based on zero-dose against high-dose radiations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' 1(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Pathway analysis results To begin, we evaluated all relevant pathways in the KEGG database and ranked the pathways based on their discriminative power following the procedures elaborated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' 3 and illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' In particular, we ranked the pathways based on their discriminative power, assessed based on the aggregated differential activity score obtained by averaging the absolute value of the t-test scores of the member genes in a given pathway [21] and estimating the p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' 2(a) shows the top five pathways that have been identified as being the most deferentially activated in the presence of low-dose radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' The top pathway was associated with Natural killer cell mediated cytotoxicity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' focusing on natural killer cells,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' which ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content='are innate immune system lymphocytes involved in early ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf'} +page_content=' 0 or l1(Dfn) > 1, QCC(fn ◦AND2) ∈ Ω( +� +nl0(Dfn)+l1(Dfn)+log log n) +holds. +• Otherwise (If fn satisfies l0(Dfn) = 0 and l1(Dfn) ≤ 1), QCC(fn ◦ AND2) ∈ Θ(1) holds. +In the proof of Proposition 1, the fooling set argument, a standard technique in communication complexity, +plays a fundamental role. +Proof technique +Let us now explain the main idea for the desired protocol used in Theorem 2 and Theorem 3. +To create the desired protocol for SYM ◦ AND2, we first decompose the symmetric function SYM(x) = D(|x|) into +the two symmetric functions SYM0(x) := D0(|x|) and SYM1(x) := D1(|x|) as follows: +D0(m) := +� +D(m) +if m ≤ l0(D) +0 +otherwise +, +D1(m) = +� +D(m) +if m > n − l1(D) +0 +otherwise +. +Note that the function D takes a constant value on the interval [l0(D), n − l1(D)]. As discussed in Section 5, it +turns out that computing SYM0 ◦ AND2 and SYM1 ◦ AND2 separately is enough to compute the entire function +SYM ◦ AND2. Therefore, we only need to design two distinct protocols: one protocol for SYM0 ◦ AND2 and the +other protocol for SYM1 ◦ AND2. We now explain how to design the two protocols. +• To compute SYM0 ◦ AND2, we simply use our first result. This uses O( +� +nl0(D)) qubits of communication +since Q(SYM0) = O( +� +nl0(D)) holds [26, 27]. +• To compute SYM1 ◦ AND2, Alice and Bob directly compute the number of elements in the set {i ≤ n | +AND2(xi, yi) = 1} under the condition2 min{|x|, |y|} ≥ n − l0(D). By taking the negation on the inputs, this +problem is reduced to the computation of the number of elements in the set {i ≤ n | xi = 0 or yi = 0} under +the condition min{|x|, |y|} ≤ l0(D). In fact, this problem and related problems have been analyzed in several +works [28, 29, 30, 31] and it is shown in [29] that O(l0(D)) classical communication is sufficient when shared +randomness is allowed (and the additional O(log log n) bits of communication3 are required to convert the +shared randomness into private randomness). +Combining the above protocols, we create the desired protocol for SYM ◦ AND2 with O( +� +nl0(Df) + l1(Df)) com- +munication. One thing which should be noted is that as seen in the above protocol, what Alice and Bob needed +to share beforehand is shared randomness, not shared entanglement. This means that we in fact show the upper +bound O( +� +nl0(Df) + l1(Df)) in a weaker communication model where shared randomness is allowed but shared +entanglement is not allowed. +1.4 +Organization of the paper +In Section 2, we list several notations and facts used in this paper. In Section 3, we generalize the protocol for +Set-Disjointness [11] and create a useful protocol which is used for our main results. In Section 4, we treat the first +result and show Theorem 1. In Section 5, we treat the second result and show Theorem 2 and Theorem 3. +2If the condition does not hold, SYM1 ◦ AND2(x, y) must be zero. +Alice and Bob check this condition with only two bits of +communication. +3In this case, min{|x|, |y|} ≥ n − l0(D) holds and therefore Newman’s theorem tells us that O(log log #{x | |x| ≥ n − l0(D)}) bits +simulates the shared randomness. As shown in Section 5, the additional bits required are in fact bounded by O(log log n). +4 + +2 +Preliminaries +For any function f, we denote the quantum communication complexity of zero-error protocols, the bounded-error +quantum communication complexity (with error ≤ 1/3) without shared entanglement, the bounded-error quantum +communication complexity (with error ≤ 1/3) with shared entanglement of a function f by QCCE(f), QCC(f) and +QCC∗(f) respectively. Trivially, it holds that QCC∗(f) ≤ QCC(f) ≤ QCCE(f). We also denote the bounded- +error query complexity of a function f by Q(f). For a n-bit string x, we denote the bitwise negation of x by +¬x = (¬x1, . . . , ¬xn). +Symmetric function +Here we list several important facts about symmetric functions. For any symmetric function +f, f can be represented as f(x) = Df(|x|) using some function Df : {0, 1, . . ., n} → {0, 1}. Denoting +l0(Df) += +max +� +l | 1 ≤ l ≤ n/2 and Df(l) ̸= Df(l − 1) +� +, +l1(Df) += +max +� +n − l | n/2 ≤ l < n and Df(l) ̸= Df(l + 1) +� +, +prior works [26, 27] show that the query complexity Q(f) of a symmetric function f is characterized as Q(f) = +Θ( +� +n(l0(Df) + l1(Df))). +3 +Communication cost for finding elements +This section is devoted to show Proposition 2, which is the quantum communication version of [11, Theorem 5.16]. +Proposition 2. There is a protocol FIND-MOREk using O(� n +k QCCE(G)) qubits and using shared randomness +which satisfies the following: +• The protocol outputs a coordinate i ∈ [n] such that G(Xi, Yi) = 1 w.p. ≥ 99/100 when there exist at least k +such coordinates. +• The protocol answers “there is no such coordinate” w.p. 1 when there is no such coordinate. +• The protocol does not use any shared entanglement. +The proof is given in Section 3.2. +3.1 +A key lemma +To show Proposition 2, we first show the following lemma: +Lemma 1. For γ ∈ N, there is a protocol FIND-EXACTγ using O( +� +n +γ QCCE(G)) qubits and shared randomness +which satisfies the followings: +• The protocol outputs a coordinate i ∈ [n] such that G(Xi, Yi) = 1 w.p. ≥ 99/100 when there exist exactly k +such coordinates for some k satisfying 3k/2 < γ < 3k. +• The protocol answers “there is no such coordinate” w.p. 1 when there is no such coordinate. +• The protocol does not use any shared entanglement. +In the proof of Lemma 1, we use Lemma 2 which is a modified protocol of the one given in [11, Section 7]. See +Appendix A for the modification. +Lemma 2. There is a protocol FIND-ONE with O(√nQCCE(G)) cost which satisfies the followings: +• The protocol outputs the coordinate i ∈ [n] such that G(Xi, Yi) = 1 w.p. ≥ 99/100 when such i exists. +• The protocol answers “there is no such coordinate” w.p. 1 when there is no such coordinate. +5 + +• The protocol does not use any shared entanglement. +Proof of Lemma 1. We first divide the set {1, . . . , n} into n/γ subsets Aj = {(j − 1)γ + 1, . . . , jγ} (1 ≤ j ≤ n/γ), +each containing γ sub-inputs. Using shared randomness, Alice and Bob pick the set of coordinates {i1, . . . , in/γ} ⊂ +[n] where each ij is chosen uniformly at random from the set Aj. +Alice and Bob then perform the protocol +FIND-ONE pretending the inputs are (Xi1, . . . , Xin/γ) for Alice and (Yi1, . . . , Yin/γ) for Bob. +Since FIND-ONE +requires O(√nQCCE(G)) qubits of communication for the input length n, this protocol with the input length n/γ +requires O( +� +n +γ QCCE(G)) qubits of communication. +We now analyze the correct probability of this protocol, following the technique used in [11, Lemma 5.15]. +Assume there exist exactly k coordinates satisfying G(Xi, Yi) = 1 and 3k/2 < γ < 3k holds. Suppose i0 satisfies +G(Xi0, Yi0) = 1. Then the coordinate i0 is chosen as the shared randomness w.p. 1/γ. Given that i0 is chosen, one +of other coordinates i′ satisfying G(Xi′, Yi′) = 1 is chosen w.p. 0 if i0, i′ are in the same subset Aj(1 ≤ j ≤ n/γ) and +w.p. 1/γ if i0 and i′ are in two different subsets. Therefore, the probability of “the coordinate i0 alone is chosen” +is at least +1 +γ +� +1 − k − 1 +γ +� +≥ 1 +γ +� +1 − k +γ +� +. +Considering the events “the coordinate i0 is chosen” are mutually disjoint, we see that the probability of “exactly +one such coordinate is chosen” is at least k/γ − (k/γ)2. Since 3k/2 < γ < 3k holds, we observe that the probability +is at least 2/9. This shows the event “at least one element is chosen” occurs w.p. ≥ 2/9. +Therefore, by the property of FIND-ONE, our new protocol satisfies the followings: +• The protocol outputs the coordinate i ∈ [n] such that G(Xi, Yi) = 1 w.p. Ω(1) when there exist exactly k +such coordinates for some k satisfying 3k/2 < γ < 3k. +• The protocol answers “there is no such coordinate” w.p. 1 when there is no such coordinate. +• The protocol does not use any shared entanglement. +To amplify the success probability Ω(1) to 99/100, Alice and Bob perform this above protocol recursively while +at each repetition checking if the output iout satisfies G(Xiout, Yiout) = 1. This repetition uses only some constant +overhead on the communication cost and hence we obtain the desired statement. +3.2 +Proof of Proposition 2 +Using the protocol FIND-EXACTγ, we show Proposition 2 as follows. +Proof of Proposition 2. The protocol FIND-MOREk is executed as follows: +(1) For j = 0 to log2(n/k), Alice and Bob perform FIND-EXACTγj where γj = 2jk. +(2) As shared randomness, Alice and Bob pick one coordinate i uniformly at random from the set [n] and check if +G(Xi, Yi) = 1. This is repeated for O(1) times. +We first analyze the communication cost of this protocol. The first step requires +log2(n/k) +� +j=0 +O +�� n +2jk QCCE(G) +� += O +��n +k QCCE(G) +� log2(n/k) +� +j=0 +1 +2j/2 = O +��n +k QCCE(G) +� +qubits of communication. The second step requires O(QCCE(G)) qubits of communication. Therefore, in total, +O +�� n +k QCCE(G) +� +qubits are used in this protocol. +Next we analyze the correct probability of this protocol. Let k∗ ≥ k be the number of coordinates satisfying +G(Xi, Yi) = 1. If k∗ ≤ n/3, then there exists j satisfying 3k∗/2 < γj < 3k∗. Therefore, FIND-EXACTγj finds the +desired coordinate w.p. ≥ 99/100. On the other hand, if k∗ > n/3, the second step finds the desired coordinate +w.p. 1/3. Then O(1) repetitions increase the success probability to 99/100. +6 + +4 +Communication protocol for symmetric functions +In [24, Theorem 22 and Theorem 25], the following theorem has been shown (with a slightly different expression): +Theorem ([24, Theorem 22 and Theorem 25]). Suppose FIND-MOREk uses m EPR-pairs as shared entanglement +and arbitrarily much shared randomness. Then for any symmetric function f : {0, 1}n → {0, 1} and any two- +party function G : {0, 1}j × {0, 1}k → {0, 1}, there is a protocol with O(Q(f)QCCE(G)) qubits which satisfies the +followings: +• The protocol successfully computes f ◦ G with probability ≥ 99/100. +• The protocol uses m · O(l0(Df) + l1(Df)) EPR-pairs as shared entanglement. +• The protocol uses O(log n) bits of shared randomness. +As is shown in Proposition 2, our modified protocol FIND-MOREk does not use any shared entanglement. +Therefore, we set m = 0 in the statement above and obtain the following theorem. (Note that O(log n) bits of +shared randomness are included in a part of communication since the O(log n) bits are negligible compared to +Q(f) ≥ Ω(√n) when f is not trivial.) +Theorem 1. For any symmetric function f : {0, 1}n → {0, 1} and any two-party function G : {0, 1}j × {0, 1}k → +{0, 1}, +QCC(f ◦ G) ∈ O(Q(f)QCCE(G)). +5 +Tight upper bound for symmetric functions +In this section, we show the following two theorems: +Theorem 2. For any symmetric function SYMn : {0, 1}n → {0, 1}, QCC∗(SYMn ◦ AND2) ∈ O( +� +nl0(D) + l1(D) +holds. +Theorem 3. For any symmetric function SYMn : {0, 1}n → {0, 1}, QCC(SYMn ◦ AND2) ∈ O( +� +nl0(D) + l1(D) + +log log n) holds. +To show these theorems, we use the following protocol that is a modification of the protocol given in [29, +Theorem 3.1]. For completeness, we describe the modification in Appendix B. +Proposition 3. Suppose the inputs x, y ∈ {0, 1}n satisfy max{|x|, |y|} ≤ k. There is a public coin classical protocol4 +with O(k) bits of communication which computes the set {i|xi = yi = 1} ⊂ [n] w.p. 99/100. +Following the technique used in [10, Section 4], we prove Theorem 2 and Theorem 3 as follows: +Proof of Theorem 2 and Theorem 3. Let us first describe some important facts based on the arguments in [10, 25]. +For any symmetric function fn, the corresponding function Dfn is constant on the interval [l0(Dfn), n − l1(Dfn)]. +Without loss of generality, assume Dfn takes 0 on the interval. (If Dfn takes 1 on the interval, we take the negation +of Dfn.) Defining D0 and D1 : {0, . . . , n} → {0, 1} as +D0(m) = +� +Dfn(m) +if m ≤ l0(Dfn) +0 +otherwise +, D1(m) = +� +Dfn(m) +if m > n − l1(Dfn) +0 +otherwise +, +Dfn = D0 ∨ D1 holds. +Therefore, by defining f 0 +n(x) := D0(|x|) and f 1 +n(x) := D1(|x|), we get fn ◦ AND2 = +(f 0 +n ◦ AND2)∨(f 1 +n ◦ AND2). This means, computing f 0 +n ◦ AND2 and f 1 +n ◦ AND2 separately is sufficient to compute the +entire function fn◦AND2. As another important fact needed for our explanation, we note that the query complexity +of f 0 +n equals to O( +� +nl0(Dfn)) which is proven in [26]. +From now on, we describe two protocols: one protocol for the computation of f 0 +n and the other one for the +computation of f 1 +n. +4Note that this protocol may use many amount of shared randomness. +7 + +• Protocol for f 0 +n: We simply apply the protocol of Theorem 1 with G = AND2 (note that f 0 +n is a symmetric +function). This protocol uses O( +� +nl0(Dfn)) qubits because Q(f 1 +n) = Θ( +� +nl0(Dfn)) holds. +• Protocol for f 1 +n: First, Bob sends Alice one bit: 1 if |¬y| ≤ l1(Dfn) and 0 otherwise. If Alice receives 1 +and |¬x| ≤ l1(Dfn) holds, they perform the protocol of Proposition 3 with the inputs ¬x and ¬y. Otherwise, +min{|x|, |y|} < n−l0(Dfn) holds and therefore f 0 +n ◦AND2(x, y) must be zero by the definition of D1. After the +execution of the protocol of Proposition 3, Alice and Bob know the set {i ≤ n | xi = yi = 0}. Next, Alice sends +|¬x| and Bob sends |¬y| using log l0(Dfn) communication, and they finally compute #{i ≤ n | xi = yi = 1} +as #{i ≤ n | xi = yi = 1} = n + #{i ≤ n | xi = yi = 0} − |¬x| − |¬y|. This protocol uses O(l1(Dfn)) +communication bits. +We then evaluate the cost for public coins. Even though the execution of this protocol may require much shared +randomness, Newman’s theorem [4] ensures that O(log log |S|) bits are sufficient when the inputs x, y belong +to a set S. Since |¬x|, |¬y| ≤ l1(Dfn) holds when executed and using the fact #{x ∈ {0, 1}n | |¬x| ≤ k} ≤ nk, +we conclude that O(log(log nl1(Dfn ))) = O(log l1(Dfn) + log log n) bits of shared randomness are sufficient. +Moreover, since O(log l1(Dfn)) bits of shared randomness are negligible compared to O(l1(Dfn)) bits in +communication and therefore included as a part of communication with no additional communication cost, +we only need to use O(log log n) bits as a shared randomness. +Combining these two protocols, we get the desired protocol with O( +� +nl0(Dfn)+l1(Dfn)) cost which uses O(log log n) +public coins. This shows QCC∗(fn ◦ AND2) ∈ O( +� +nl0(Dfn) + l1(Dfn)) and QCC(fn ◦ AND2) ∈ O( +� +nl0(Dfn) + +l1(Dfn) + log log n) by Alice sending O(log log n) random bits instead of the shared randomness. +By combining the arguments we showed so far, we obtain the tight bound QCC∗(fn ◦ AND2) ∈ Θ( +� +nl0(Dfn) + +l1(Dfn)) on the communication model with shared entanglement. On the model without shared entanglement, our +bound QCC(fn ◦ AND2) ∈ O( +� +nl0(Dfn) + l1(Dfn) + log log n) still have the additive log log n difference from the +lower bound. We next show this upper bound is indeed optimal by using a standard technique, the fooling set +argument. +Proposition 1. For any non-trivial symmetric function fn : {0, 1}n → {0, 1}, +• if the function fn satisfies l0(Dfn) > 0 or l1(Dfn) > 1, QCC(fn ◦AND2) ∈ Ω( +� +nl0(Dfn)+l1(Dfn)+log log n) +holds. +• Otherwise (i.e., if fn satisfies l0(Dfn) = 0 and l1(Dfn) ≤ 1), QCC(fn ◦ AND2) ∈ Θ(1) holds. +Proof. Let us first prove that QCC(fn ◦ AND2) ∈ Θ(1) holds when l0(Dfn) = 0 and l1(Dfn) ≤ 1 hold. In this case, +there are only two types of the functions: fn = ANDn or fn = ¬ANDn. In either case of the functions, Alice and +Bob only need to send one single bit expressing whether x = (1, . . . , 1) for Alice (y = (1, . . . , 1) for Bob). Therefore +we obtain QCC(fn ◦ AND2) ∈ Θ(1) since a lower bound QCC(fn ◦ AND2) ∈ Ω(1) is trivial. +The rest is to show QCC(fn ◦ AND2) ∈ Ω( +� +nl0(Dfn) + l1(Dfn) + log log n) holds assuming l0(Dfn) > 0 or +l1(Dfn) > 1. First, we note that the log log n factor becomes negligible comparing to +� +nl0(Dfn) + l1(Dfn) when +l0(Dfn) > 0 holds. This means that the well-known lower bound Ω( +� +nl0(Dfn) + l1(Dfn)) [10] already gives a +tight lower bound. Therefore, we only need to show QCC(fn ◦ AND2) ∈ Ω(l1(Dfn) + log log n) holds assuming +l0(Dfn) = 0. +Moreover, the lower bound QCC∗(fn ◦ AND2) ∈ Ω( +� +nl0(Dfn) + l1(Dfn)) shown in [10] implies +QCC(fn ◦AND2) ∈ Ω(l1(Dfn)). Therefore, it is sufficient to show QCC(fn ◦AND2) ∈ Ω(log log n) when l0(Dfn) = 0 +and l1(Dfn) > 1 hold. +Assuming l0(Dfn) = 0, l1(Dfn) > 1 and Dfn ≡ 0 on [l0(Dfn), n − l1(Dfn)] without loss of generality, we show +QCC(fn ◦ AND2) ∈ Ω(log log n). To show this, we use the fooling set argument [2, 3]. Define +FSn := {(x, y) ∈ {0, 1}n × {0, 1}n | x = y and |¬x| = l1(Dfn) − 1}. +Then we see that for any (x, y) ∈ FSn, fn ◦ AND2(x, y) = 1 and for any (x, y), (x′, y′) ∈ FSn, (x, y) ̸= (x′, y′) implies +fn ◦ AND2(x, y′) = fn ◦ AND2(x′, y) = 0. Therefore, the deterministic communication complexity DCC(fn ◦ AND2) +satisfies +DCC(fn ◦ AND2) ≥ log2 |FSn| +8 + +by the fooling set argument. As shown in [32], it is well-known that QCC(f) ≥ log DCC(f) for any function f. +Therefore, by observing |FSn| = +� +n +l1(Dfn)−1 +� +≥ Ω(n) for l1(Dfn) > 1, we obtain the desired statement QCC(fn ◦ +AND2) ≥ Ω(log log n). +Acknowledgement +The author was partially supported by the MEXT Q-LEAP grant No. JPMXS0120319794. The author would +like to take this opportunity to thank the “Nagoya University Interdisciplinary Frontier Fellowship” supported by +Nagoya University and JST, the establishment of university fellowships towards the creation of science technology +innovation, Grant Number JPMJFS2120. The author also would like to thank Fran¸cois Le Gall for his kindness +and valuable comments and Ronald de Wolf for kind comments on an earlier draft of this paper. +References +[1] Andrew Chi-Chih Yao. Some complexity questions related to distributive computing (preliminary report). In +11th Annual ACM Symposium on Theory of Computing, pages 209–213, 1979. +[2] Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, 1996. +[3] Anup Rao and Amir Yehudayoff. Communication Complexity: and Applications. Cambridge University Press, +2020. +[4] Ilan Newman. Private vs. common random bits in communication complexity. Information Processing Letters, +39(2):67–71, 1991. +[5] Andrew Chi-Chih Yao. Quantum circuit complexity. In 34th Annual Foundations of Computer Science, pages +352–361, 1993. +[6] Richard Cleve and Harry Buhrman. Substituting quantum entanglement for communication. Physical Review +A, 56(2):1201, 1997. +[7] Harry Buhrman, Richard Cleve, and Wim van Dam. Quantum entanglement and communication complexity. +SIAM Journal on Computing, 30(6):1829–1841, 2001. +[8] Gilles Brassard. Quantum communication complexity. Foundations of Physics, 33(11):1593–1616, 2003. +[9] Harry Buhrman, Richard Cleve, Serge Massar, and Ronald de Wolf. Nonlocality and communication complexity. +Reviews of modern physics, 82:665–698, 2010. +[10] Alexander A Razborov. Quantum communication complexity of symmetric predicates. Izvestiya: Mathematics, +67(1):145, 2003. +[11] Scott Aaronson and Andris Ambainis. Quantum search of spatial regions. Theory of Computing, 1(4):47–79, +2005. +[12] Richard Cleve, Wim van Dam, Michael Nielsen, and Alain Tapp. Quantum entanglement and the commu- +nication complexity of the inner product function. +In Selected Papers from the First NASA International +Conference on Quantum Computing and Quantum Communications, QCQC ’98, pages 61–74, 1998. +[13] Harry Buhrman and Ronald de Wolf. Communication complexity lower bounds by polynomials. In 16th Annual +IEEE Conference on Computational Complexity, pages 120–130. IEEE, 2001. +[14] John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. Proposed experiment to test local +hidden-variable theories. Physical Review Letters, 23:880–884, Oct 1969. +9 + +[15] N. David Mermin. Simple unified form for the major no-hidden-variables theorems. Physical Review Letters, +65:3373–3376, Dec 1990. +[16] Dmitry Gavinsky, Julia Kempe, Oded Regev, and Ronald de Wolf. Bounded-error quantum state identification +and exponential separations in communication complexity. In Proceedings of the thirty-eighth annual ACM +symposium on Theory of Computing, pages 594–603, 2006. +[17] Dmitry Gavinsky. On the role of shared entanglement. Quantum Information & Computation, 8(1):82–95, +2008. +[18] Yaoyun Shi and Yufan Zhu. Quantum communication complexity of block-composed functions. Quantum +Information & Computation, 9(5):444–460, 2009. +[19] Troy Lee and Shengyu Zhang. Composition theorems in communication complexity. In Automata, Languages +and Programming, pages 475–489, 2010. +[20] Anil Ada, Arkadev Chattopadhyay, Omar Fawzi, and Phuong Nguyen. The nof multiparty communication +complexity of composed functions. In Automata, Languages, and Programming, pages 13–24, 2012. +[21] Harry Buhrman, Richard Cleve, and Avi Wigderson. Quantum vs. classical communication and computation. +In 30th annual ACM symposium on Theory of computing, pages 63–68, 1998. +[22] Peter Høyer and Ronald de Wolf. Improved quantum communication complexity bounds for disjointness and +equality. In Annual Symposium on Theoretical Aspects of Computer Science, pages 299–310, 2002. +[23] Sourav Chakraborty, Arkadev Chattopadhyay, Nikhil S. Mande, and Manaswi Paraashar. Quantum query- +to-communication simulation needs a logarithmic overhead. In 35th Computational Complexity Conference, +2020. +[24] Sourav Chakraborty, Arkadev Chattopadhyay, Peter Høyer, Nikhil S. Mande, Manaswi Paraashar, and Ronald +de Wolf. Symmetry and quantum query-to-communication simulation. In 39th International Symposium on +Theoretical Aspects of Computer Science, pages 20:1–20:23, 2022. +[25] Alexander A. Sherstov. The pattern matrix method. SIAM Journal on Computing, 40(6):1969–2000, 2011. +[26] Ramamohan Paturi. On the degree of polynomials that approximate symmetric boolean functions (preliminary +version). In 24th annual ACM symposium on Theory of computing, pages 468–474, 1992. +[27] Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds +by polynomials. Journal of the ACM (JACM), 48(4):778–797, 2001. +[28] Wei Huang, Yaoyun Shi, Shengyu Zhang, and Yufan Zhu. The communication complexity of the hamming +distance problem. Information Processing Letters, 99(4):149–153, 2006. +[29] Joshua Brody, Amit Chakrabarti, Ranganath Kondapally, David P. Woodruff, and Grigory Yaroslavtsev. +Beyond set disjointness: The communication complexity of finding the intersection. In ACM Symposium on +Principles of Distributed Computing, PODC ’14, pages 106–113, 2014. +[30] Joshua Brody, Amit Chakrabarti, Ranganath Kondapally, David P. Woodruff, and Grigory Yaroslavtsev. +Certifying equality with limited interaction. Algorithmica, 76(3):796–845, 2016. +[31] Dawei Huang, Seth Pettie, Yixiang Zhang, and Zhijun Zhang. The communication complexity of set intersection +and multiple equality testing. In 31st Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1715–1732, +2020. +[32] Ilan Kremer and Noam Nisan. Quantum communication. Technical report, 1995. +10 + +A +Modification for Lemma 2 +Here we describe how the protocol given in [11, Section 7] is modified to the protocol in Theorem 2. +In [11, +Section 7], the authors proposed a protocol that finds i ∈ [n] such that xi ∧ yi = 1 where Alice is given x ∈ {0, 1}n +and Bob is given y ∈ {0, 1}n. In the protocol, Alice and Bob perform the query +OAND : |i, z⟩A|i⟩B �→ |i, z ⊕ (xi ∧ yi)⟩A|i⟩B +for O(√n) times and other operations which require O(√n) communication. Since the query operation is imple- +mented using 2-qubits of communication, this protocol requires 2O(√n) + O(√n) = O(√n) communication. +Our modification for finding i such that G(Xi, Yi) = 1 is simple. We just replace the query OAND to +OG : |i, z⟩A|i⟩B �→ |i, z ⊕ G(Xi, Yi)⟩A|i⟩B. +This protocol indeed finds the desired coordinate i, which is shown in the same manner as in [11, Section 7]. Let us +analyze the communication cost of this protocol. Since QCCE(G) denotes the exact communication complexity of +G, the operation OG is implemented using 2QCCE(G) qubits. (First QCCE(G) communication is used to compute +G and the second QCCE(G) is used to compute reversely and clear the unwanted registers.) Other operations are +the same as in the original protocol and therefore use O(√n) communication. Considering that the operation OG is +performed for O(√n) times, we see that our modified protocol uses O(√n) + QCCE(G)O(√n) = O(QCCE(G)√n) +qubits of communication. +B +Modification for Proposition 3 +In [29, Theorem 3.1], the authors originally showed the following. +Theorem 4. Suppose the inputs x, y ∈ {0, 1}n satisfy max{|x|, |y|} ≤ k. There exists an O( +√ +k)-round constructive +randomized classical protocol that outputs the set {i | xi = yi = 1} with success probability 1 − 1/poly(k). In the +model of shared randomness the total expected communication is O(k). +To modify this theorem for Proposition 3, we need to take care of the success probability and the expected +communication. To take care of the success probability, we first take a sufficiently large constant k0 such that for +any k ≥ k0, 1/poly(k) ≤ 1/200. If k < k0 holds, the parties perform the protocol in Theorem 4 with the constant +k0. This requires O(k0) expected communication. Otherwise (i.e., when k > k0 holds), the parties perform the +protocol in Theorem 4 with the constant k, which requires O(k) expected communication. Since k0 is a constant, +the protocol by this modification still requires O(k) expected communication with error ≤ 1/200. +To convert the expected communication to the worst-case communication, we use the Markov’s inequality. +Suppose this protocol requires C · k expected communication. Then the probability of “the communication cost +≥ 200C · k” is less than or equal to 1/200 by the Markov’s inequality. We create the desired protocol by Alice and +Bob aborting communication when its cost gets 200C · k. This modified protocol still have the success probability +≥ 99/100, since the first modification has the error 1/200 and the second modification affects the error at most +1/200. +11 + diff --git a/PNAyT4oBgHgl3EQfg_jD/content/tmp_files/load_file.txt b/PNAyT4oBgHgl3EQfg_jD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bbf428d4252f0faa4a1ff152733e30323fc500fe --- /dev/null +++ b/PNAyT4oBgHgl3EQfg_jD/content/tmp_files/load_file.txt @@ -0,0 +1,484 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf,len=483 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='00370v1 [quant-ph] 1 Jan 2023 Matching upper bounds on symmetric predicates in quantum communication complexity Daiki Suruga January 3, 2023 Abstract In this paper, we focus on the quantum communication complexity of functions of the form f ◦ G = f(G(X1, Y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' , G(Xn, Yn)) where f : {0, 1}n → {0, 1} is a symmetric function, G : {0, 1}j × {0, 1}k → {0, 1} is any function and Alice (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Bob) is given (Xi)i≤n (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' (Yi)i≤n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Recently, Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [STACS 2022] showed that the quantum communication complexity of f ◦ G is O(Q(f)QCCE(G)) when the parties are allowed to use shared entanglement, where Q(f) is the query complexity of f and QCCE(G) is the exact communication complexity of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In this paper, we first show that the same statement holds without shared entanglement, which generalizes their result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Based on the improved result, we next show tight upper bounds on f ◦ AND2 for any symmetric function f (where AND2 : {0, 1} × {0, 1} → {0, 1} denotes the 2-bit AND function) in both models: with shared entanglement and without shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This matches the well-known lower bound by Razborov [Izv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 67(1) 145, 2003] when shared entanglement is allowed and improves Razborov’s bound when shared entanglement is not allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='1 Motivation Communication complexity The model of (classical) communication complexity was originally introduced by Yao [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In this model, there are two players, Alice who receives x ∈ X and Bob who receives y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Their goal is to compute a known function f : X × Y → {0, 1} with as little communication as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Due to this simple structure, lower and upper bounds on communication complexity problems have applications on many other fields such as VLSI design, circuit complexity, data structure, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' (See [2, 3] for good references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=') Communication complexity has been investigated in many prior works since its introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In communication complexity, Set-Disjointness (DISJn(x, y) = ¬ � i≤n(xi∧yi)), Equality (EQn(x, y) = ¬ � i≤n(xi⊕ yi)), and Inner-Product function (IPn(x, y) = � i≤n(xi ∧yi)) are three of the most well-studied functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Denoting the private randomized communication complexity of a function f (with error ≤ 1/3) as CC(f), it has been shown that CC(DISJn) = CC(IPn) = Θ(n) and CC(EQn) = Θ(log n) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Note that if shared randomness between the two parties is allowed, CCpub(DISJn) = CCpub(IPn) = Θ(n) and CCpub(EQn) = Θ(1) hold where CCpub(f) denotes the randomized communication complexity of a function f with error ≤ 1/3 and with shared randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Observing from CC(EQn) ̸= CCpub(EQn), we see that the shared randomness sometimes enables to reduce the communication complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Therefore, we need to carefully treat the effect of the shared randomness when analyzing the communi- cation complexity of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' (Note that if CCpub(f) is strictly larger than O(log n), Newman’s theorem [4] tells us that CCpub(f) = O(CC(f)) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=') In 1993, Yao [5] introduced the model of quantum communication complexity based on the model of classical communication complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The main difference between the classical and quantum model is that Alice and Bob use quantum bits to transmit their information in the quantum model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' As quantum information science has been growing up rapidly, quantum communication complexity has been widely studied [6, 7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In the case of quantum communication complexity, the three functions mentioned above satisfy QCC(DISJn) = Θ(√n) [10, 11], QCC(IPn) = Θ(n) [12] and QCC(EQn) = Θ(log n) [13] , where QCC(f) denotes the private quantum communication complexity of a function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' If Alice and Bob have shared entanglement, QCC∗(DISJn) = Θ(√n) [10, 11], QCC∗(IPn) = Θ(n) [12] 1 and QCC∗(EQn) = Θ(1) [13] hold where QCC∗(f) denotes the quantum communication complexity of the function f when shared entanglement is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Even though the power of entanglement is not significant in these examples, careful treatment of shared entanglement is important since many non-trivial properties of entanglement have been witnessed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=', [14, 15, 16, 17, 9]), including Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [17] that shows Newman’s theorem [4] does not hold in case of shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Composed functions In both classical and quantum communication complexity, many important functions have the form f ◦ G : (X, Y ) �→ f((G(X1, Y1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' , G(Xn, Yn)) ∈ {0, 1} where X = (Xi)i≤n ∈ {0, 1}nj, Y = (Yi)i≤n ∈ {0, 1}nk, f : {0, 1}n → {0, 1} and G : {0, 1}j × {0, 1}k → {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This fact is already observed in the three of the most well-studied functions: Set-Disjointness (¬ORn ◦ AND2), Equality (ANDn ◦ XOR2), and Inner-Product function (XORn ◦ AND2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' As a natural consequence of its importance, functions of this form have been investigated deeply [18, 19, 20] in both classical and quantum communication complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Even though the functions f ◦ G are in general difficult to analyze in detail because of their generality, the analysis may become simpler when G has a simpler form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Let us explain in detail about upper and lower bounds on the quantum communication complexity when G is a simple function such as AND2, XOR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In the case of upper bounds, Buhrman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [21] showed QCC(f ◦ G) = O(Q(f) log n) holds when G ∈ {AND2, XOR2}, where Q(f) denotes the bounded error query complexity of a function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Applying this result, we immediately get QCC(DISJn) = O(√n log n) because Q(ORn) = O(√n) holds by Grover’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This is an important result since it shows that the fundamental function DISJn can be computed more efficiently than in classical scenario (recall CCpub(DISJn) = Θ(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This upper bound QCC(DISJn) = O(√n log n) was later improved by [22] and finally improved to O(√n) by [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [21] gives many important upper bounds for functions f ◦ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' On the other hand, Razborov [10] treated lower bounds of QCC∗(f ◦ G) and showed several tight bounds when f is a symmetric function and G is AND2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' For example, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [10] shows QCC∗(DISJn) = Ω(√n) and QCC∗(IPn) = Ω(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Combining the O(√n) bound [11] and Ω(√n) bound [10] imply QCC(DISJn) = Θ(√n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Our contributions can be understood as a generalization of these works [21, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' As described above, the relation QCC(f ◦ G) = O(Q(f) log n) holds when the function G is either AND2 or XOR2 [21], and this upper bound was then improved to O(√n) by Aaronson and Ambainis [11] when f = ORn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This implies that the log n factor in [21] is not required in the case of Set-Disjointness function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Considering this fact, one may wonder whether the log n overhead is not required for arbitrary function when G ∈ {AND2, XOR2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [23] treated this problem and gave a negative answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' They exhibited a function f that requires Ω(Q(f) log n) communication to compute f ◦XOR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This means that the upper bound O(Q(f) log n) in [21] is tight for generic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Interestingly, their subsequent work [24] generalized the result and proved the log n overhead is not required when f is a symmetric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In this paper, we focus on functions of the form SYM ◦ G where SYM is a symmetric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' As described below in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='2 and Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='3, our first result generalizes the paper [24] and our second result shows a tight lower and upper bound on the quantum communication complexity of such functions SYM ◦ G when G = AND2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='2 First result: On improving the result [24] As mentioned above, the paper [24] showed that the log n factor in O(Q(f) log n) upper bound is not required when we focus on a symmetric function f = SYM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' More precisely, it is shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [24] that there exists a protocol for a function SYM ◦ G with O(Q(SYM)QCCE(G)) qubits of communication (QCCE(G) denotes the exact communication complexity of G) which uses shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Even though the amount of shared entanglement in their protocol is not so large, there are cases when the amount of the entanglement is significantly larger than the communication cost O(Q(SYM)QCCE(G)) as stated in [24, Remark 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Thus, in general the shared entanglement can not be included as a part of the communication in their protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' We improve their result and show that the same statement holds even without any shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' That is, we show the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' For any symmetric function f : {0, 1}n → {0, 1} and any two-party function G : {0, 1}j × {0, 1}k → {0, 1}, QCC(f ◦ G) ∈ O(Q(f)QCCE(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 2 Proof technique In the paper [24], the desired protocol is constructed by employing a new technique called noisy amplitude amplification, which needs a certain amount of entanglement shared between Alice and Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Based on the noisy amplitude amplification technique, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [24] shows the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Theorem ([24, Theorem 21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Suppose Alice (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Bob) is given (Xi)i≤n ∈ {0, 1}jn (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' (Yi)i≤n ∈ {0, 1}kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' There is a protocol which satisfies the followings: The protocol uses O(√nQCCE(G)) qubits of communication and ⌈log n⌉ EPR pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The protocol finds the coordinate i satisfying G(Xi, Yi) = 1 with probability 99/100 when such i exists, and outputs “No” with probability 1 when no such i exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Using this protocol as a subroutine, the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [24] constructed the main protocol for f ◦ G, which inherently requires a certain amount of the entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' On the other hand, in the case of Set-Disjointness, Aaronson and Ambainis [11] showed a protocol with O(√n) qubits of communication which does not use any shared entanglement but does find a coordinate i satisfying xi ∧ yi = 1 with probability 99/100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Based on the construction of the protocol in [11] rather than the noisy amplitude amplification technique used in [24], we successfully construct a generalized version of the above theorem in Proposition 2 which does not require any shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Once we show the generalized version, the rest is shown in a similar manner as in [24], which is described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Thus, we obtain the protocol for SYM ◦ G using O(Q(SYM)QCCE(G)) qubits which does not use any shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='3 Second result: On tight upper bounds for SYM ◦ AND2 In our second result, we focus on tight upper bounds on the quantum communication complexity of SYM ◦ AND2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' We first note here that the paper [24] and our first result already exhibit protocols with O(Q(SYM)) qubits which are more efficient than the protocol in [21] with O(Q(SYM) log n) qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' However, even a protocol with O(Q(SYM)) qubits of communication does not generally give a tight upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' For example, the quantum communication complexity of ANDn ◦ AND2 is O(1) but Q(ANDn) = Θ(√n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Therefore, we need to develop another technique to show a tight upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In this framework, Razborov [10] and Sherstov [25] showed the following strong result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Theorem ([10, 25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Let SYMn : {0, 1}n → {0, 1} be a symmetric function and D : {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' , n} → {0, 1} be a function satisfying1 SYMn(x) = D(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Define l0(D) = max � l | 1 ≤ l ≤ n/2 and D(l) ̸= D(l − 1) � , l1(D) = max � n − l | n/2 ≤ l < n and D(l) ̸= D(l + 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Then we have QCC∗(SYMn◦AND2) ∈ Ω( � nl0(D)+l1(D)) and QCC(SYMn◦AND2) ∈ O({ � nl0(D)+l1(D)} log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This theorem already shows the nearly tight bound QCC∗(SYMn ◦ AND2) = ˜Θ( � nl0(D) + l1(D)) up to a multiplicative log n factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' To show an exact tight upper bound, it is thus sufficient to create a protocol with O( � nl0(D) + l1(D)) qubits of communication by removing the log n factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In this paper, we successfully show that the multiplicative log n factor is not required in the model with shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' That is, we get the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' For any symmetric function SYMn : {0, 1}n → {0, 1}, QCC∗(SYMn ◦ AND2) ∈ O( � nl0(D) + l1(D)) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In the model without shared entanglement, we also show a similar statement, albeit with an additive log log n factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Thus we show Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' For any symmetric function SYMn : {0, 1}n → {0, 1}, QCC(SYMn ◦ AND2) ∈ O( � nl0(D) + l1(D) + log log n) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 1Note that for any symmetric function f, there is a corresponding function D satisfying f(x) = D(|x|) where |x| denotes the Hamming weight of a bit string x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 3 This shows, for the first time, the tight relation QCC∗(SYMn ◦ AND2) = Θ( � nl0(D)+ l1(D)) in the model with shared entanglement, matching the lower bound by [10, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In the model without shared entanglement, however, there is still a log log n gap between the communication cost of our protocol and the lower bound [10, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' To fill this gap, we also show that our protocol without shared entanglement is in fact optimal: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' For any non-trivial symmetric function fn : {0, 1}n → {0, 1}, if the function fn satisfies l0(Dfn) > 0 or l1(Dfn) > 1, QCC(fn ◦AND2) ∈ Ω( � nl0(Dfn)+l1(Dfn)+log log n) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Otherwise (If fn satisfies l0(Dfn) = 0 and l1(Dfn) ≤ 1), QCC(fn ◦ AND2) ∈ Θ(1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In the proof of Proposition 1, the fooling set argument, a standard technique in communication complexity, plays a fundamental role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Proof technique Let us now explain the main idea for the desired protocol used in Theorem 2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' To create the desired protocol for SYM ◦ AND2, we first decompose the symmetric function SYM(x) = D(|x|) into the two symmetric functions SYM0(x) := D0(|x|) and SYM1(x) := D1(|x|) as follows: D0(m) := � D(m) if m ≤ l0(D) 0 otherwise , D1(m) = � D(m) if m > n − l1(D) 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Note that the function D takes a constant value on the interval [l0(D), n − l1(D)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' As discussed in Section 5, it turns out that computing SYM0 ◦ AND2 and SYM1 ◦ AND2 separately is enough to compute the entire function SYM ◦ AND2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Therefore, we only need to design two distinct protocols: one protocol for SYM0 ◦ AND2 and the other protocol for SYM1 ◦ AND2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' We now explain how to design the two protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' To compute SYM0 ◦ AND2, we simply use our first result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This uses O( � nl0(D)) qubits of communication since Q(SYM0) = O( � nl0(D)) holds [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' To compute SYM1 ◦ AND2, Alice and Bob directly compute the number of elements in the set {i ≤ n | AND2(xi, yi) = 1} under the condition2 min{|x|, |y|} ≥ n − l0(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' By taking the negation on the inputs, this problem is reduced to the computation of the number of elements in the set {i ≤ n | xi = 0 or yi = 0} under the condition min{|x|, |y|} ≤ l0(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In fact, this problem and related problems have been analyzed in several works [28, 29, 30, 31] and it is shown in [29] that O(l0(D)) classical communication is sufficient when shared randomness is allowed (and the additional O(log log n) bits of communication3 are required to convert the shared randomness into private randomness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Combining the above protocols, we create the desired protocol for SYM ◦ AND2 with O( � nl0(Df) + l1(Df)) com- munication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' One thing which should be noted is that as seen in the above protocol, what Alice and Bob needed to share beforehand is shared randomness, not shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This means that we in fact show the upper bound O( � nl0(Df) + l1(Df)) in a weaker communication model where shared randomness is allowed but shared entanglement is not allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='4 Organization of the paper In Section 2, we list several notations and facts used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In Section 3, we generalize the protocol for Set-Disjointness [11] and create a useful protocol which is used for our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In Section 4, we treat the first result and show Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In Section 5, we treat the second result and show Theorem 2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 2If the condition does not hold, SYM1 ◦ AND2(x, y) must be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Alice and Bob check this condition with only two bits of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 3In this case, min{|x|, |y|} ≥ n − l0(D) holds and therefore Newman’s theorem tells us that O(log log #{x | |x| ≥ n − l0(D)}) bits simulates the shared randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' As shown in Section 5, the additional bits required are in fact bounded by O(log log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 4 2 Preliminaries For any function f, we denote the quantum communication complexity of zero-error protocols, the bounded-error quantum communication complexity (with error ≤ 1/3) without shared entanglement, the bounded-error quantum communication complexity (with error ≤ 1/3) with shared entanglement of a function f by QCCE(f), QCC(f) and QCC∗(f) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Trivially, it holds that QCC∗(f) ≤ QCC(f) ≤ QCCE(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' We also denote the bounded- error query complexity of a function f by Q(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' For a n-bit string x, we denote the bitwise negation of x by ¬x = (¬x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' , ¬xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Symmetric function Here we list several important facts about symmetric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' For any symmetric function f, f can be represented as f(x) = Df(|x|) using some function Df : {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=', n} → {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Denoting l0(Df) = max � l | 1 ≤ l ≤ n/2 and Df(l) ̸= Df(l − 1) � , l1(Df) = max � n − l | n/2 ≤ l < n and Df(l) ̸= Df(l + 1) � , prior works [26, 27] show that the query complexity Q(f) of a symmetric function f is characterized as Q(f) = Θ( � n(l0(Df) + l1(Df))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 3 Communication cost for finding elements This section is devoted to show Proposition 2, which is the quantum communication version of [11, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' There is a protocol FIND-MOREk using O(� n k QCCE(G)) qubits and using shared randomness which satisfies the following: The protocol outputs a coordinate i ∈ [n] such that G(Xi, Yi) = 1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' ≥ 99/100 when there exist at least k such coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The protocol answers “there is no such coordinate” w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 1 when there is no such coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The protocol does not use any shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The proof is given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='1 A key lemma To show Proposition 2, we first show the following lemma: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' For γ ∈ N, there is a protocol FIND-EXACTγ using O( � n γ QCCE(G)) qubits and shared randomness which satisfies the followings: The protocol outputs a coordinate i ∈ [n] such that G(Xi, Yi) = 1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' ≥ 99/100 when there exist exactly k such coordinates for some k satisfying 3k/2 < γ < 3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The protocol answers “there is no such coordinate” w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 1 when there is no such coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The protocol does not use any shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In the proof of Lemma 1, we use Lemma 2 which is a modified protocol of the one given in [11, Section 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' See Appendix A for the modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' There is a protocol FIND-ONE with O(√nQCCE(G)) cost which satisfies the followings: The protocol outputs the coordinate i ∈ [n] such that G(Xi, Yi) = 1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' ≥ 99/100 when such i exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The protocol answers “there is no such coordinate” w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 1 when there is no such coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 5 The protocol does not use any shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' We first divide the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' , n} into n/γ subsets Aj = {(j − 1)γ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' , jγ} (1 ≤ j ≤ n/γ), each containing γ sub-inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Using shared randomness, Alice and Bob pick the set of coordinates {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' , in/γ} ⊂ [n] where each ij is chosen uniformly at random from the set Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Alice and Bob then perform the protocol FIND-ONE pretending the inputs are (Xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' , Xin/γ) for Alice and (Yi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' , Yin/γ) for Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Since FIND-ONE requires O(√nQCCE(G)) qubits of communication for the input length n, this protocol with the input length n/γ requires O( � n γ QCCE(G)) qubits of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' We now analyze the correct probability of this protocol, following the technique used in [11, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Assume there exist exactly k coordinates satisfying G(Xi, Yi) = 1 and 3k/2 < γ < 3k holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Suppose i0 satisfies G(Xi0, Yi0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Then the coordinate i0 is chosen as the shared randomness w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 1/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Given that i0 is chosen, one of other coordinates i′ satisfying G(Xi′, Yi′) = 1 is chosen w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 0 if i0, i′ are in the same subset Aj(1 ≤ j ≤ n/γ) and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 1/γ if i0 and i′ are in two different subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Therefore, the probability of “the coordinate i0 alone is chosen” is at least 1 γ � 1 − k − 1 γ � ≥ 1 γ � 1 − k γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Considering the events “the coordinate i0 is chosen” are mutually disjoint, we see that the probability of “exactly one such coordinate is chosen” is at least k/γ − (k/γ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Since 3k/2 < γ < 3k holds, we observe that the probability is at least 2/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This shows the event “at least one element is chosen” occurs w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' ≥ 2/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Therefore, by the property of FIND-ONE, our new protocol satisfies the followings: The protocol outputs the coordinate i ∈ [n] such that G(Xi, Yi) = 1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Ω(1) when there exist exactly k such coordinates for some k satisfying 3k/2 < γ < 3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The protocol answers “there is no such coordinate” w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 1 when there is no such coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The protocol does not use any shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' To amplify the success probability Ω(1) to 99/100, Alice and Bob perform this above protocol recursively while at each repetition checking if the output iout satisfies G(Xiout, Yiout) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This repetition uses only some constant overhead on the communication cost and hence we obtain the desired statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='2 Proof of Proposition 2 Using the protocol FIND-EXACTγ, we show Proposition 2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The protocol FIND-MOREk is executed as follows: (1) For j = 0 to log2(n/k), Alice and Bob perform FIND-EXACTγj where γj = 2jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' (2) As shared randomness, Alice and Bob pick one coordinate i uniformly at random from the set [n] and check if G(Xi, Yi) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This is repeated for O(1) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' We first analyze the communication cost of this protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The first step requires log2(n/k) � j=0 O �� n 2jk QCCE(G) � = O ��n k QCCE(G) � log2(n/k) � j=0 1 2j/2 = O ��n k QCCE(G) � qubits of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The second step requires O(QCCE(G)) qubits of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Therefore, in total, O �� n k QCCE(G) � qubits are used in this protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Next we analyze the correct probability of this protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Let k∗ ≥ k be the number of coordinates satisfying G(Xi, Yi) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' If k∗ ≤ n/3, then there exists j satisfying 3k∗/2 < γj < 3k∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Therefore, FIND-EXACTγj finds the desired coordinate w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' ≥ 99/100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' On the other hand, if k∗ > n/3, the second step finds the desired coordinate w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Then O(1) repetitions increase the success probability to 99/100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 6 4 Communication protocol for symmetric functions In [24, Theorem 22 and Theorem 25], the following theorem has been shown (with a slightly different expression): Theorem ([24, Theorem 22 and Theorem 25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Suppose FIND-MOREk uses m EPR-pairs as shared entanglement and arbitrarily much shared randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Then for any symmetric function f : {0, 1}n → {0, 1} and any two- party function G : {0, 1}j × {0, 1}k → {0, 1}, there is a protocol with O(Q(f)QCCE(G)) qubits which satisfies the followings: The protocol successfully computes f ◦ G with probability ≥ 99/100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The protocol uses m · O(l0(Df) + l1(Df)) EPR-pairs as shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The protocol uses O(log n) bits of shared randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' As is shown in Proposition 2, our modified protocol FIND-MOREk does not use any shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Therefore, we set m = 0 in the statement above and obtain the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' (Note that O(log n) bits of shared randomness are included in a part of communication since the O(log n) bits are negligible compared to Q(f) ≥ Ω(√n) when f is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=') Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' For any symmetric function f : {0, 1}n → {0, 1} and any two-party function G : {0, 1}j × {0, 1}k → {0, 1}, QCC(f ◦ G) ∈ O(Q(f)QCCE(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 5 Tight upper bound for symmetric functions In this section, we show the following two theorems: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' For any symmetric function SYMn : {0, 1}n → {0, 1}, QCC∗(SYMn ◦ AND2) ∈ O( � nl0(D) + l1(D) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' For any symmetric function SYMn : {0, 1}n → {0, 1}, QCC(SYMn ◦ AND2) ∈ O( � nl0(D) + l1(D) + log log n) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' To show these theorems, we use the following protocol that is a modification of the protocol given in [29, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' For completeness, we describe the modification in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Suppose the inputs x, y ∈ {0, 1}n satisfy max{|x|, |y|} ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' There is a public coin classical protocol4 with O(k) bits of communication which computes the set {i|xi = yi = 1} ⊂ [n] w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 99/100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Following the technique used in [10, Section 4], we prove Theorem 2 and Theorem 3 as follows: Proof of Theorem 2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Let us first describe some important facts based on the arguments in [10, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' For any symmetric function fn, the corresponding function Dfn is constant on the interval [l0(Dfn), n − l1(Dfn)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Without loss of generality, assume Dfn takes 0 on the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' (If Dfn takes 1 on the interval, we take the negation of Dfn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=') Defining D0 and D1 : {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' , n} → {0, 1} as D0(m) = � Dfn(m) if m ≤ l0(Dfn) 0 otherwise , D1(m) = � Dfn(m) if m > n − l1(Dfn) 0 otherwise , Dfn = D0 ∨ D1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Therefore, by defining f 0 n(x) := D0(|x|) and f 1 n(x) := D1(|x|), we get fn ◦ AND2 = (f 0 n ◦ AND2)∨(f 1 n ◦ AND2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This means, computing f 0 n ◦ AND2 and f 1 n ◦ AND2 separately is sufficient to compute the entire function fn◦AND2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' As another important fact needed for our explanation, we note that the query complexity of f 0 n equals to O( � nl0(Dfn)) which is proven in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' From now on, we describe two protocols: one protocol for the computation of f 0 n and the other one for the computation of f 1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 4Note that this protocol may use many amount of shared randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 7 Protocol for f 0 n: We simply apply the protocol of Theorem 1 with G = AND2 (note that f 0 n is a symmetric function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This protocol uses O( � nl0(Dfn)) qubits because Q(f 1 n) = Θ( � nl0(Dfn)) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Protocol for f 1 n: First, Bob sends Alice one bit: 1 if |¬y| ≤ l1(Dfn) and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' If Alice receives 1 and |¬x| ≤ l1(Dfn) holds, they perform the protocol of Proposition 3 with the inputs ¬x and ¬y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Otherwise, min{|x|, |y|} < n−l0(Dfn) holds and therefore f 0 n ◦AND2(x, y) must be zero by the definition of D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' After the execution of the protocol of Proposition 3, Alice and Bob know the set {i ≤ n | xi = yi = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Next, Alice sends |¬x| and Bob sends |¬y| using log l0(Dfn) communication, and they finally compute #{i ≤ n | xi = yi = 1} as #{i ≤ n | xi = yi = 1} = n + #{i ≤ n | xi = yi = 0} − |¬x| − |¬y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This protocol uses O(l1(Dfn)) communication bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' We then evaluate the cost for public coins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Even though the execution of this protocol may require much shared randomness, Newman’s theorem [4] ensures that O(log log |S|) bits are sufficient when the inputs x, y belong to a set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Since |¬x|, |¬y| ≤ l1(Dfn) holds when executed and using the fact #{x ∈ {0, 1}n | |¬x| ≤ k} ≤ nk, we conclude that O(log(log nl1(Dfn ))) = O(log l1(Dfn) + log log n) bits of shared randomness are sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Moreover, since O(log l1(Dfn)) bits of shared randomness are negligible compared to O(l1(Dfn)) bits in communication and therefore included as a part of communication with no additional communication cost, we only need to use O(log log n) bits as a shared randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Combining these two protocols, we get the desired protocol with O( � nl0(Dfn)+l1(Dfn)) cost which uses O(log log n) public coins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This shows QCC∗(fn ◦ AND2) ∈ O( � nl0(Dfn) + l1(Dfn)) and QCC(fn ◦ AND2) ∈ O( � nl0(Dfn) + l1(Dfn) + log log n) by Alice sending O(log log n) random bits instead of the shared randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' By combining the arguments we showed so far, we obtain the tight bound QCC∗(fn ◦ AND2) ∈ Θ( � nl0(Dfn) + l1(Dfn)) on the communication model with shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' On the model without shared entanglement, our bound QCC(fn ◦ AND2) ∈ O( � nl0(Dfn) + l1(Dfn) + log log n) still have the additive log log n difference from the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' We next show this upper bound is indeed optimal by using a standard technique, the fooling set argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' For any non-trivial symmetric function fn : {0, 1}n → {0, 1}, if the function fn satisfies l0(Dfn) > 0 or l1(Dfn) > 1, QCC(fn ◦AND2) ∈ Ω( � nl0(Dfn)+l1(Dfn)+log log n) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Otherwise (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=', if fn satisfies l0(Dfn) = 0 and l1(Dfn) ≤ 1), QCC(fn ◦ AND2) ∈ Θ(1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Let us first prove that QCC(fn ◦ AND2) ∈ Θ(1) holds when l0(Dfn) = 0 and l1(Dfn) ≤ 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In this case, there are only two types of the functions: fn = ANDn or fn = ¬ANDn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In either case of the functions, Alice and Bob only need to send one single bit expressing whether x = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' , 1) for Alice (y = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' , 1) for Bob).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Therefore we obtain QCC(fn ◦ AND2) ∈ Θ(1) since a lower bound QCC(fn ◦ AND2) ∈ Ω(1) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The rest is to show QCC(fn ◦ AND2) ∈ Ω( � nl0(Dfn) + l1(Dfn) + log log n) holds assuming l0(Dfn) > 0 or l1(Dfn) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' First, we note that the log log n factor becomes negligible comparing to � nl0(Dfn) + l1(Dfn) when l0(Dfn) > 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This means that the well-known lower bound Ω( � nl0(Dfn) + l1(Dfn)) [10] already gives a tight lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Therefore, we only need to show QCC(fn ◦ AND2) ∈ Ω(l1(Dfn) + log log n) holds assuming l0(Dfn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Moreover, the lower bound QCC∗(fn ◦ AND2) ∈ Ω( � nl0(Dfn) + l1(Dfn)) shown in [10] implies QCC(fn ◦AND2) ∈ Ω(l1(Dfn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Therefore, it is sufficient to show QCC(fn ◦AND2) ∈ Ω(log log n) when l0(Dfn) = 0 and l1(Dfn) > 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Assuming l0(Dfn) = 0, l1(Dfn) > 1 and Dfn ≡ 0 on [l0(Dfn), n − l1(Dfn)] without loss of generality, we show QCC(fn ◦ AND2) ∈ Ω(log log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' To show this, we use the fooling set argument [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Define FSn := {(x, y) ∈ {0, 1}n × {0, 1}n | x = y and |¬x| = l1(Dfn) − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Then we see that for any (x, y) ∈ FSn, fn ◦ AND2(x, y) = 1 and for any (x, y), (x′, y′) ∈ FSn, (x, y) ̸= (x′, y′) implies fn ◦ AND2(x, y′) = fn ◦ AND2(x′, y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Therefore, the deterministic communication complexity DCC(fn ◦ AND2) satisfies DCC(fn ◦ AND2) ≥ log2 |FSn| 8 by the fooling set argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' As shown in [32], it is well-known that QCC(f) ≥ log DCC(f) for any function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Therefore, by observing |FSn| = � n l1(Dfn)−1 � ≥ Ω(n) for l1(Dfn) > 1, we obtain the desired statement QCC(fn ◦ AND2) ≥ Ω(log log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Acknowledgement The author was partially supported by the MEXT Q-LEAP grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' JPMXS0120319794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The author would like to take this opportunity to thank the “Nagoya University Interdisciplinary Frontier Fellowship” supported by Nagoya University and JST, the establishment of university fellowships towards the creation of science technology innovation, Grant Number JPMJFS2120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The author also would like to thank Fran¸cois Le Gall for his kindness and valuable comments and Ronald de Wolf for kind comments on an earlier draft of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' References [1] Andrew Chi-Chih Yao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Some complexity questions related to distributive computing (preliminary report).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In 11th Annual ACM Symposium on Theory of Computing, pages 209–213, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [2] Eyal Kushilevitz and Noam Nisan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Communication Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Cambridge University Press, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [3] Anup Rao and Amir Yehudayoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Communication Complexity: and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Cambridge University Press, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [4] Ilan Newman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Private vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' common random bits in communication complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Information Processing Letters, 39(2):67–71, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [5] Andrew Chi-Chih Yao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Quantum circuit complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In 34th Annual Foundations of Computer Science, pages 352–361, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [6] Richard Cleve and Harry Buhrman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Substituting quantum entanglement for communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Physical Review A, 56(2):1201, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [7] Harry Buhrman, Richard Cleve, and Wim van Dam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Quantum entanglement and communication complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' SIAM Journal on Computing, 30(6):1829–1841, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [8] Gilles Brassard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Quantum communication complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Foundations of Physics, 33(11):1593–1616, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [9] Harry Buhrman, Richard Cleve, Serge Massar, and Ronald de Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Nonlocality and communication complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Reviews of modern physics, 82:665–698, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [10] Alexander A Razborov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Quantum communication complexity of symmetric predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Izvestiya: Mathematics, 67(1):145, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [11] Scott Aaronson and Andris Ambainis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Quantum search of spatial regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Theory of Computing, 1(4):47–79, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [12] Richard Cleve, Wim van Dam, Michael Nielsen, and Alain Tapp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Quantum entanglement and the commu- nication complexity of the inner product function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In Selected Papers from the First NASA International Conference on Quantum Computing and Quantum Communications, QCQC ’98, pages 61–74, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [13] Harry Buhrman and Ronald de Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Communication complexity lower bounds by polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In 16th Annual IEEE Conference on Computational Complexity, pages 120–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' IEEE, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [14] John F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Clauser, Michael A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Horne, Abner Shimony, and Richard A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Holt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Proposed experiment to test local hidden-variable theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Physical Review Letters, 23:880–884, Oct 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 9 [15] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' David Mermin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Simple unified form for the major no-hidden-variables theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Physical Review Letters, 65:3373–3376, Dec 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [16] Dmitry Gavinsky, Julia Kempe, Oded Regev, and Ronald de Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Bounded-error quantum state identification and exponential separations in communication complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In Proceedings of the thirty-eighth annual ACM symposium on Theory of Computing, pages 594–603, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [17] Dmitry Gavinsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' On the role of shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Quantum Information & Computation, 8(1):82–95, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [18] Yaoyun Shi and Yufan Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Quantum communication complexity of block-composed functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Quantum Information & Computation, 9(5):444–460, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [19] Troy Lee and Shengyu Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Composition theorems in communication complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In Automata, Languages and Programming, pages 475–489, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [20] Anil Ada, Arkadev Chattopadhyay, Omar Fawzi, and Phuong Nguyen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The nof multiparty communication complexity of composed functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In Automata, Languages, and Programming, pages 13–24, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [21] Harry Buhrman, Richard Cleve, and Avi Wigderson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Quantum vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' classical communication and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In 30th annual ACM symposium on Theory of computing, pages 63–68, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [22] Peter Høyer and Ronald de Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Improved quantum communication complexity bounds for disjointness and equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In Annual Symposium on Theoretical Aspects of Computer Science, pages 299–310, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [23] Sourav Chakraborty, Arkadev Chattopadhyay, Nikhil S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Mande, and Manaswi Paraashar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Quantum query- to-communication simulation needs a logarithmic overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In 35th Computational Complexity Conference, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [24] Sourav Chakraborty, Arkadev Chattopadhyay, Peter Høyer, Nikhil S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Mande, Manaswi Paraashar, and Ronald de Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Symmetry and quantum query-to-communication simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In 39th International Symposium on Theoretical Aspects of Computer Science, pages 20:1–20:23, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [25] Alexander A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Sherstov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The pattern matrix method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' SIAM Journal on Computing, 40(6):1969–2000, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [26] Ramamohan Paturi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' On the degree of polynomials that approximate symmetric boolean functions (preliminary version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In 24th annual ACM symposium on Theory of computing, pages 468–474, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [27] Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Quantum lower bounds by polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Journal of the ACM (JACM), 48(4):778–797, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [28] Wei Huang, Yaoyun Shi, Shengyu Zhang, and Yufan Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The communication complexity of the hamming distance problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Information Processing Letters, 99(4):149–153, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [29] Joshua Brody, Amit Chakrabarti, Ranganath Kondapally, David P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Woodruff, and Grigory Yaroslavtsev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Beyond set disjointness: The communication complexity of finding the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In ACM Symposium on Principles of Distributed Computing, PODC ’14, pages 106–113, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [30] Joshua Brody, Amit Chakrabarti, Ranganath Kondapally, David P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Woodruff, and Grigory Yaroslavtsev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Certifying equality with limited interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Algorithmica, 76(3):796–845, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [31] Dawei Huang, Seth Pettie, Yixiang Zhang, and Zhijun Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' The communication complexity of set intersection and multiple equality testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In 31st Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1715–1732, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' [32] Ilan Kremer and Noam Nisan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Quantum communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Technical report, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 10 A Modification for Lemma 2 Here we describe how the protocol given in [11, Section 7] is modified to the protocol in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In [11, Section 7], the authors proposed a protocol that finds i ∈ [n] such that xi ∧ yi = 1 where Alice is given x ∈ {0, 1}n and Bob is given y ∈ {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In the protocol, Alice and Bob perform the query OAND : |i, z⟩A|i⟩B �→ |i, z ⊕ (xi ∧ yi)⟩A|i⟩B for O(√n) times and other operations which require O(√n) communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Since the query operation is imple- mented using 2-qubits of communication, this protocol requires 2O(√n) + O(√n) = O(√n) communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Our modification for finding i such that G(Xi, Yi) = 1 is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' We just replace the query OAND to OG : |i, z⟩A|i⟩B �→ |i, z ⊕ G(Xi, Yi)⟩A|i⟩B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This protocol indeed finds the desired coordinate i, which is shown in the same manner as in [11, Section 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Let us analyze the communication cost of this protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Since QCCE(G) denotes the exact communication complexity of G, the operation OG is implemented using 2QCCE(G) qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' (First QCCE(G) communication is used to compute G and the second QCCE(G) is used to compute reversely and clear the unwanted registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=') Other operations are the same as in the original protocol and therefore use O(√n) communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Considering that the operation OG is performed for O(√n) times, we see that our modified protocol uses O(√n) + QCCE(G)O(√n) = O(QCCE(G)√n) qubits of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' B Modification for Proposition 3 In [29, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='1], the authors originally showed the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Suppose the inputs x, y ∈ {0, 1}n satisfy max{|x|, |y|} ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' There exists an O( √ k)-round constructive randomized classical protocol that outputs the set {i | xi = yi = 1} with success probability 1 − 1/poly(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' In the model of shared randomness the total expected communication is O(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' To modify this theorem for Proposition 3, we need to take care of the success probability and the expected communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' To take care of the success probability, we first take a sufficiently large constant k0 such that for any k ≥ k0, 1/poly(k) ≤ 1/200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' If k < k0 holds, the parties perform the protocol in Theorem 4 with the constant k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This requires O(k0) expected communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Otherwise (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=', when k > k0 holds), the parties perform the protocol in Theorem 4 with the constant k, which requires O(k) expected communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Since k0 is a constant, the protocol by this modification still requires O(k) expected communication with error ≤ 1/200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' To convert the expected communication to the worst-case communication, we use the Markov’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Suppose this protocol requires C · k expected communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' Then the probability of “the communication cost ≥ 200C · k” is less than or equal to 1/200 by the Markov’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' We create the desired protocol by Alice and Bob aborting communication when its cost gets 200C · k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' This modified protocol still have the success probability ≥ 99/100, since the first modification has the error 1/200 and the second modification affects the error at most 1/200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} +page_content=' 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf'} diff --git a/PNFOT4oBgHgl3EQf4TQo/content/2301.12949v1.pdf b/PNFOT4oBgHgl3EQf4TQo/content/2301.12949v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a4d20bf4051469a13bde7e137877ac06c9bf6f71 --- 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information in massive gravity +using Dirac brackets +Joydeep Chakravarty,1,2 Diksha Jain,3 Akhil Sivakumar2,4 +1McGill University +845 Sherbrooke Street West, Montreal H3A 0G4, Canada +2International Centre for Theoretical Sciences (ICTS-TIFR) +Shivakote, Hesaraghatta, Bangalore 560089, India. +3Tata Institute of Fundamental Research +Dr Homi Bhabha Road, Navy Nagar, Mumbai, 400005, India +4Asia Pacific Center for Theoretical Physics (APCTP) +San 31, Hyoja-dong, Nam-gu, Pohang 790-784, South Korea +E-mail: joydeep.chakravarty@mail.mcgill.ca, diksha.2012jain@gmail.com, +akhil.sivakumar@apctp.org +Abstract: The principle of holography of information states that in massless gravity, it is +possible to extract bulk information using asymptotic boundary operators. In our work, we +study this principle in a linearized setting about empty flat space and formulate it using Dirac +brackets between boundary Hamiltonian and bulk operators. We then address whether the +storage of bulk information in flat space linearized massive gravity resembles that of massless +gravity. For linearized massless gravity, using Dirac brackets, we recover the necessary criteria +for the holography of information. In contrast, we show that the Dirac bracket of the relevant +boundary observable with bulk operators vanishes for massive gravity. We use this important +distinction to outline the canonical Hilbert space. This leads to split states, and consequently, +one cannot use asymptotic boundary observables to extract bulk information in massive gravity. +We also argue the split property directly without an explicit reference to the Hilbert space. The +result reflects that we can construct local bulk operators in massive gravity, which are obscured +from boundary observables due to the lack of diffeomorphism invariance. Our analysis sheds +some light on evaporating black holes in the context of the islands proposal. +arXiv:2301.01075v1 [hep-th] 3 Jan 2023 + +Contents +1 +Introduction +1 +2 +Physical observables and Dirac brackets +3 +2.1 +Constraints and Dirac brackets +4 +3 +Linearized massless gravity: Dirac matrix +6 +4 +Linearized massive gravity: Dirac matrix +9 +5 +Boundary observables and Dirac brackets +11 +5.1 +Boundary observables and Dirac brackets for massless gravity +11 +5.2 +Boundary Observables and Dirac brackets for massive gravity +12 +6 +Vacua structure and split states +13 +6.1 +Split states in electrodynamics +13 +6.2 +Hilbert space and vacua structure in flat space gravity +15 +6.3 +Split states in massless gravity +17 +7 +Split states in massive gravity +18 +7.1 +Split states from the vacuum structure +18 +7.2 +Split states without the vacuum structure +19 +8 +Conclusion and discussion +21 +A Dirac brackets for electrodynamics +23 +B Constraints in linearized gravity with matter +26 +B.1 Massless Gravity with minimally coupled matter +26 +B.2 Minimally coupled matter to massive graviton +28 +C Massive gravity constraints by substitution +31 +1 +Introduction +The question of whether information about a bulk state can be extracted using boundary +operators in a theory of gravity is an important one [1–4]. In this light, our motivation for +this work is two-fold. Since a theory of gravity is a constrained system, it is only natural to +– 1 – + +ask whether such statements can be understood using the Dirac bracket formalism [5–7]. The +second goal is to use this formalism and apply it to Fierz Pauli massive gravity [8], which is +an interesting modification to gravity and has been a subject of recent interest (see [9–26] for +a sampling of works and references therein). Our chief motivation lies in the recent discussions +of massive gravity in the context of islands for evaporating black holes [3, 27–33]. +Whether information is holographically stored at the boundary is addressed by the following +question: given access to asymptotic boundary operators, can we precisely determine the bulk +state? This version of holography of information exists in massless gravity and is an essential +consequence of the Gauss constraint. The crucial ingredient involved here is the boundary +Hamiltonian, using which one can construct boundary operators that probe bulk physics [1– +4, 34–37]. +Related works discussing the localization of information in massless gravity are +[38–41]. +In massless gravity, the principle of holography of information implies that specifying a +bulk state |ψ⟩ outside a bounded region B uniquely fixes it inside B [2, 4]. As a result, it is +convenient to introduce split states, i.e., states which can be arbitrary inside B but are fixed +on the complement of B. Generally, all field theories apart from massless gravity obey a split +property that the set of such states is non-empty [42]. However, the holography of information +in massless gravity implies that the set of split states is empty. Keeping this in mind, in our +work, we investigate the following objectives: +1. To understand holography of information and split property using Dirac brackets by +verifying known cases of linearized massless gravity and electrodynamics. +2. To determine whether the property holds in massive gravity at a linearized level. +Brief description of results +In our work, we account for constraints using Dirac brackets and use them to demonstrate the +information stored at a linearized level for different constrained theories. Some related works on +the phase space structure and the computation of Dirac brackets in massive gravity are given +in [18, 43–46]. +In §2, we discuss physical observables in constrained theories and briefly review the Dirac +bracket formalism for considering the constraints. Based on our discussion of physical observ- +ables, we develop a schematic argument for why we may be able to create local bulk operators +in massive gravity. However, the presence of second-class constraints can render this picture +wrong, and we need to verify the same by computing the Dirac bracket between the boundary +Hamiltonian and an arbitrary bulk operator. We also argue that flat space massive gravity does +not have asymptotic symmetries such as BMS supertranslations. +We argue that coupling linearized gravity (both massless and Fierz-Pauli massive gravity) +to matter fields introduces inconsistencies in the structure of the constraints, including failure +to close. As a demonstration, see Appendix B.1 for the case of massless gravity and Appendix +B.2 for massive gravity, where in both cases, the constraints fail to close. Thus in principle, a +– 2 – + +complete calculation involves taking the full Einstein Hilbert action coupled to matter in the +massless case. Similarly, we should couple matter covariantly to the full non-linear action for +massive gravity [19–22]. +In our work, we argue that the issues regarding split states can be understood even using +a linearized analysis. We show that the Dirac matrix involving only the graviton phase space +is sufficient to understand split states, while the remaining Poisson brackets are defined over +the phase space of the complete gravity-matter theory. In other words, we demand that the +brackets between constraints are computed only over the gravity phase space, which allows +the constraints to close correctly. However, we do not put this restriction while computing +the rest of the brackets, where the matter insertions are addressed adequately. Following this +restriction, we also comment upon the vacua structure of massive gravity. +Intuitively this restriction is in line with our general expectation that the addition of matter +should not drastically change the nature of the gravity constraint structure. Analogously in +electrodynamics, the constraint analysis with or without including charged matter gives rise to +the same Dirac matrix shown in Appendix A. +In §3 and §4, we compute the Dirac matrix necessary to compute the brackets for massless +and massive gravity, respectively. Using this, we address the extraction of bulk information +using boundary operators for electrodynamics and massless/ massive gravity at leading order +in perturbation theory. In §5, we define relevant boundary observables for massless and massive +gravity and calculate the Dirac brackets. We also perform an alternate derivation of the Dirac +brackets in Appendix C. +For electrodynamics, using the Dirac matrix obtained in Appendix A, we find in §6.1 that +one cannot use boundary operators to determine the bulk state, hence obeying the expected +split property. For massless gravity, building upon the computation of the Dirac brackets in +§3 and §5, we obtain the necessary conditions for the lack of split states upon taking the Dirac +bracket of boundary Hamiltonian with a bulk operator insertion in §6.3. +For massive gravity, following §4 and §5, we argue in §5.2 that the computation of the +Dirac bracket between the boundary operator with bulk operators vanishes. Upon quantization, +lifting the Dirac bracket to the commutator between relevant operators acting on the Hilbert +space implies that the commutator is zero. Due to this, in §7, we argue that in contrast to +the principle of holography of information in massless gravity, we do not have an analogous +statement in massive gravity. We argue this in two different ways: with and without an explicit +reference to the Hilbert space of massive gravity. In §8, we discuss the potential limitations of +our work, the implications of our results for evaporating black holes, and list some interesting +directions. +2 +Physical observables and Dirac brackets +In gauge theory, there are different ways to address gauge redundancy. The general method +to fix the redundancy is by defining gauge invariant observables. A useful subclass is to work +– 3 – + +with gauge invariant observables, which we can construct by fixing a good gauge choice, hence +removing the redundancy (up to residual gauge, if any). +In massless gravity, there is a gauge redundancy, i.e., small diffeomorphisms, which die off +at the asymptotic boundary of the spacetime. In flat space linearized gravity, these are +δhµν = ∂µζν + ∂νζµ + O +�� +GN +� +, +(2.1) +where ζ parametrizes the diffeomorphisms at the linearized order. Hence the construction of +physical observables in gravity is accomplished by demanding invariance under small diffeomor- +phisms characterized by (2.1), either by gauge fixing or by defining observables that commute +with constraints. +More generally, there are no local diffeomorphism invariant operators in +massless gravity. +We can use gravitational dressing to construct observables invariant under small diffeomor- +phisms, where we dress bulk observables to the boundary1. In gauge theories, one can similarly +construct similar gauge invariant observables, either by gauge fixing or by defining manifestly +gauge invariant observables like Wilson loops2. +The Fierz Pauli interaction term in massive gravity explicitly breaks the diffeomorphism +invariance given in equation (2.1). Since small diffeomorphisms are no longer a symmetry of +massive gravity, we need not define physical observables by methods such as dressing them +using the boundary. +In flat space massive gravity, since diffeomorphisms are no longer a symmetry, the sub- +group of diffeomorphism group generating asymptotic symmetries such as supertranslations are +absent. +From the phase space perspective, the fact that there is no gauge symmetry of the form +(2.1) for massive gravity is because the constraints of massive gravity are second-class and +hence do not have any redundancy in the phase space. This feature contrasts the first-class +constraints of massless gravity, which necessitate gauge fixing. Thus it naively seems that there +can be local bulk observables in massive gravity, which can completely hide from the boundary. +Despite the intuition from the gauge-fixing picture, we still need to consider the other +second-class constraints for a consistent description. Due to these constraints, bulk observables +may not be completely independent of the observables at the boundary. In this light, our work +aims to understand whether these second-class constraints are sufficient for a boundary observer +to fix the bulk state completely. +2.1 +Constraints and Dirac brackets +We will follow [5] in our discussion. Here we will denote our set of constraints as {Φi}. Given a +Lagrangian L for a constrained system, we have a set of primary constraints {ΦP +i }, which are +1See [47] for a detailed construction of gauge invariant operators using dressing in massless gravity and gauge +theories. +2Note that Wilson loops are not good observables in gravity beyond leading order, since the loops possess +stress energy and hence backreact. However, they can undergo further gravitational dressing at subleading order +and become good observables up to that order. +– 4 – + +independent relations between the fields h and their canonical momenta Π. Let H0 denote the +Hamiltonian obtained by taking the Legendre transform of the Lagrangian L. We define the +Dirac Hamiltonian to be +H = H0 + viΦP +i +(2.2) +Recall that the Poisson bracket between two observables F(x) and G(y) is given by +{F(x), G(y)} = +� +dD−1z +�δF(x) +δh(z) +δG(y) +δΠ(z) − δG(y) +δh(z) +δF(x) +δΠ(z) +� +(2.3) +We first need to ensure whether the primary constraints are stable and use the stability to +determine the parameters vi from (2.2). We check the stability by taking the Poisson brackets +of primary constraints with the constrained Hamiltonian, i.e. {ΦP +i , H}, which either vanishes or +gives us secondary constraints. Next, we need to check the stability of the secondary constraints, +which may give us tertiary constraints. +The process should be repeated for consistency of +the constrained system until we have determined all possible constraints {Φi} and fixed the +parameters vi. +We can further classify the set of constraints {Φi} into two subsets: first-class and second- +class. Second-class constraints are defined as constraints that do not commute among them- +selves i.e. +{Φs +i, Φs +j} ̸= 0 +on the constrained surface, while first-class constraints are defined as constraints that commute +among themselves i.e. +{Φf +i , Φf +j } = 0 +where we denote the first class constraints by Φf, and the second class constraints by Φs. +The presence of first-class constraints in the system indicates the presence of gauge sym- +metry. Hence we need to fix a gauge corresponding to each of the first-class constraints. The +set of first-class constraints {Φf +i } and the gauge conditions {Gi} together form a system of +second-class constraints. Once we obtain a system of second-class constraints, we can define +the Dirac matrix as follows: +C (Φi, Φj) ≡ {Φi, Φj}. +(2.4) +This matrix is now invertible since any constraint Φi gives non-zero Poisson with at least +one other constraint3. We then invert this matrix (not always), thereby obtaining the inverse +C−1 (Φi, Φj) +C (Φi, Φk) C−1 (Φk, Φj) = δij +(2.5) +With C−1 +ij ≡ C−1 (Φi, Φj), the Dirac bracket between two observables F(x1) and G(x2) defined +on the phase space is given by +{F(x1), G(x2)}D.B. = {F(x1), G(x2)} − +� +y1 +� +y2 +{F(x1), Φi(y1)} C−1 +ij (y1, y2) {Φj(y2), G(x2)}. +(2.6) +3The first class constraints give non-zero Poisson brackets with the gauge constraints +– 5 – + +where we have used the notation +� +y1 +� +y2 ≡ +� +dd−1y1 +� +dd−1y2. Note that in (2.6), apart from +the first term (i.e., the standard Poisson bracket), we also have the second term, which is the +contribution due to the constraints. +3 +Linearized massless gravity: Dirac matrix +Before addressing massive gravity, we will warm up with the Dirac matrix calculation for +linearized massless gravity without matter, which will also help contrast results with the massive +gravity calculation. +Let us begin with a convenient form for the action of the massless graviton: +Lg = 1 +κ2 +� +−1 +2∂λhµν∂λhµν + ∂µhνλ∂νhµλ − ∂µhµν∂νh + 1 +2∂λh∂λh +� ++ boundary terms (3.1) +where the coefficient κ2 is given by κ2 = 32πGN, where GN is Newton’s constant. The boundary +terms in the Lagrangian (3.1) are chosen to simplify the momenta and the constraints, thereby +giving us Π00 = Π0i = 0.4 Using (3.1), we compute the canonical momenta corresponding to +hµν: +Π00 = 0, +Π0i = 0 +Πij = ∂L +∂ ˙hij += 1 +κ2 +� +˙hij − ˙hkkδij − 2∂(ihj)0 + 2∂kh0kδij +� +(3.2) +The first line gives us D primary constraints. Then the Hamiltonian for massless gravity is +given by taking the Legendre transform of (3.1): +H0 = κ2 +�Π2 +ij +2 − +Π2 +ii +2(D − 2) +� ++ 1 +κ2 +�1 +2∂khij∂khij − ∂ihjk∂jhik + ∂ihij∂jhk +k − 1 +2∂ihj +j∂ihk +k +� +− 2h0i∂jΠij − h00 +� +∇2hi +i − ∂i∂jhij +� +(3.3) +The constraints with Hamiltonian (3.3) are given by +Π00 = 0 +(3.4) +Π0i = 0 +(3.5) +χ0 = {Π00, Htot} = ∇2hi +i − ∂i∂jhij +(3.6) +χi = {Π0i, Htot} = 2∂jΠij, +(3.7) +Since we have two primary constraints, the Dirac Hamiltonian is given by +Htot = H0 + v0Π00 + viΠ0i +(3.8) +where v0 and vi are undetermined constants that will be fixed. We can check that this system +of constraints is first class since their Poisson brackets with themselves and the Hamiltonian +vanish. +4Since we are working in asymptotically flat space, we can ignore possible boundary contributions to the +pointwise constraints. +– 6 – + +Gauge choice +Given the above first-class constraints, we need to fix the redundancy in phase space. We do +that by implementing constraints arising from fixing the gauge (i.e. small diffeomorphisms) +and the undetermined constants v0 and vi in the Hamiltonian. +A good gauge choice is fixing them so that the gauge constraints are orthogonal to the set +of first-class constraints. Thus a natural guess for gauge conditions is the following5: +G0 : h00 = 0, +Gi : h0i = 0, +K0 : +Πk +k +D − 2 = 0, +and +Kj : ∂ihij = 0. +(3.9) +In the rest of this section, we will use this choice to implement the Dirac procedure. +Dirac brackets +Given the set of gauge conditions in (3.9), we need to ensure their stability under time evolution, +i.e., whether the above constraints give rise to new constraints after time evolution. +{G0, Htot} = v0 +{Gi, Htot} = vi +{K0, Htot} = −h00 ≈ 0 +{Kj, Htot} = ∂iΠij − ∂jΠk +k +D − 2 + 2∂j∂ih0i ≈ 0 +(3.10) +where ≈ denotes that the equation is valid on the constraint surface. From (3.10), we see that +a consistent choice of implementing the Dirac procedure is by setting v0 = vi = 0 since, in this +case, we do not get any new constraints. From the perspective of counting degrees of freedom, +we now have 4D second class constraints on an originally D(D + 1) dimensional phase space, +thereby reducing the phase space dimensionality to D(D−3). This reduction is consistent with +the fact that the graviton has D(D−3) +2 +degrees of freedom6. +In hindsight, we will find that the above choice of gauge conditions is designed such that +each gauge condition gives a non-zero commutator with exactly one of the first-class constraints, +thereby helping us obtain a simpler yet non-singular Dirac matrix. Specifically, the non-zero +elements of the constraint matrix are given by: +{Π00(x), G0(y)} = −δD−1(x − y) +{Π0i(x), Gj(y)} = −δi +jδD−1(x − y) +{χ0(x), K0(y)} = ∇2δD−1(x − y) +{χi(x), Kj(y)} = +� +δij∇2 + ∂i∂j +� +δD−1(x − y) +{K0(x), Ki(y)} = +1 +D − 2∂iδD−1(x − y) +(3.11) +5The numerical multiplicative factor in K0 is chosen for later convenience. +6The degrees of freedom in massless gravity in D-dimensions can be found out by counting the symmetric +traceless representations of the little group SO(D − 2), giving rise to +D(D−3) +2 +polarizations of the standard +graviton. Note here that D ≥ 3. +– 7 – + +For later convenience, we will rename the constraints as follows: +C0 : χ0, +Ci : χi, +CD : Π00 +CD+i : Π0i, +C2D : K0, +C2D+i : Ki, +C3D : G0, +C3D+i : Gi. +(3.12) +In this new notation, we label the constraint matrix as +Cab = {Ca, Cb} +where a and b run from 0 · · · 4D − 1. +Writing the matrix using the representation in the +momentum space, we obtain the following: +C(p) = +� +������������� +0 +0j +0 0j −p2 +0j +0 +0j +0i +0i +j +0i 0i +j +0i +−(pipj + p2δi +j) 0i +0i +j +0 +0j +0 0j +0 +0j +−1 +0j +0i +0i +j +0i 0i +j +0i +0i +j +0i −δi +j +p2 +0j +0 0j +0 +ipj +D−2 +0 +0j +0i (pipj + p2δi +j) 0i 0i +j +ipj +D−2 +0i +j +0i +0i +j +0 +0j +1 0j +0 +0j +0 +0j +0i +0i +j +0i δi +j +0i +0i +j +0i +0i +j +� +������������� +, +(3.13) +where we have used raised (lowered) indices on the matrix elements to abbreviate entries worth +a column (row) array. Since the matrix given in (3.13) is non-singular, we can use it to compute +the inverse matrix +C−1(p) = +1 +2p2 +� +������������� +0 +1 +D−2 +ipj +p2 +0 +0j +2 +0j +0 +0j +1 +D−2 +ipi +p2 +0i +j +0i +0i +j +0i 2δi +j − pipj +p2 +0i +0i +j +0 +0j +0 +0j +0 +0j +2p2 +0j +0i +0i +j +0i +0i +j +0i +0i +j +0i 2p2δi +j +−2 +0j +0 +0j +0 +0j +0 +0j +0i +−2δi +j + pipj +p2 +0i +0i +j +0j +0i +j +0i +0i +j +0 +0j +−2p2 +0j +0 +0j +0 +0j +0i +0i +j +0i +−2p2δi +j 0i +0i +j +0i +0i +j +� +������������� +. +(3.14) +Notice that the inverse of the constraint matrix is non-local. Such non-localities are essential +ingredients of a gauge invariant theory and encodes the structure of its Gauss law. We will +later find that this feature gives rise to the property that the energy of field excitations within +a spatial region is detectable from the boundary of the region. +This concludes our analysis of the Dirac matrix for linearised gravity without matter. +– 8 – + +4 +Linearized massive gravity: Dirac matrix +We will now move on to computing the Dirac matrix for massive gravity without matter. The +Fierz-Pauli action for massive graviton is given by: +Lg = 1 +κ2 +� +−1 +2∂λhµν∂λhµν + ∂µhνλ∂νhµλ − ∂µhµν∂νh + 1 +2∂λh∂λh − 1 +2m2(hµνhµν − h2) +� ++ boundary terms. +(4.1) +Again, as in the massless case, we have chosen boundary terms such that Π00 = Π0i = 0 and +κ2 = 32πGN. In addition to the massless gravity Lagrangian, we now have the Fierz Pauli +coupling term, with m denoting the mass of the graviton. +We can easily extend our analysis from the massless case to the massive case and similarly +determine the remaining canonical momenta and the Hamiltonian. Since the kinetic part of +the Lagrangian remains the same, the canonical momenta of the massive case are the same as +for the massless case and are given by (3.2). The massive gravity Hamiltonian is given by: +Hg = κ2 +�Π2 +ij +2 − +Π2 +ii +2(D − 2) +� ++ 1 +κ2 +�1 +2∂khij∂khij − ∂ihjk∂jhik + ∂ihij∂jhk +k − 1 +2∂ihj +j∂ihk +k +1 +2m2(hijhij − hi +ihj +j) − m2h2 +0i − h00 +� +∇2hi +i − ∂i∂jhij − m2hk +k +�� +− 2h0i∂jΠij +(4.2) +As before, we again have two primary constraints, i.e. Π00 = Π0i = 0. Using these primary +constraints, the Dirac Hamiltonian is given by +Htot = Hg + v0Π00 + viΠ0i. +(4.3) +Constraints and Dirac matrix +As for the massless case, we systematically determine the constraints and repeat the Dirac +procedure. Demanding stability of primary constraints under the action of the Hamiltonian, +we find the following secondary constraints: +C0 = {Π00, Htot} = (∇2 − m2)hj +j − ∂i∂jhij +Ci = {Π0i, Htot} = ∂jΠji + m2h0i +(4.4) +Next, we demand the stability of these secondary constraints under the Hamiltonian and thereby +obtain the following: +C−1 = {C0, Htot} ≈ m2 +� Πk +k +D − 2 − ∂ihi0 +� +C−2 = {C−1, Htot} ≈ m4h +(4.5) +where ≈ denotes that the corresponding equation is valid on the constraint surface. +The +Poisson brackets of Ci and C−1 with the Hamiltonian can be set to zero by fixing the Lagrange +– 9 – + +multipliers vi and v0, respectively. Thus, we have no tertiary constraints, and the system is +consistent. +The above procedure leads us to a system of 2(D + 1) second class constraints provided we +fix the Lagrange multipliers (v0 and vi) accompanying the primary constraints as follows +Htot = Hg + ∂ihi0Π00 + +� +∂jhji − ∂ih +� +Πi0. +(4.6) +Note that in (4.5), we have retained the m scaling of these constraints since this allows us to +keep track of the m → 0 limit of the Dirac matrix. In the limit m → 0 only the constraints +Ca≥0 are relevant. Therefore the massive theory has two additional constraints to the massless +theory, albeit second class. As a cross-check, the above analysis leads to a correct determination +of the degrees of freedom.7 +Next, we define the constraint matrix +Cab(x, y) ≡ {Ca(x), Cb(y)} +(4.7) +where a, b now spans −2, −1, . . . , 2D − 1. +We have defined {CD, CD+i} = {Π00, Π0i} and +together with the constraints (4.4) and (4.5) they generate the constraint matrix +C(x − y) = m2 +� +�������� +0 +d +d−1m4 +0 +−m2∂j −m2 0j +− +d +d−1m4 +0 +−∇2 + dm2 +d−1 +0j +0 +−∂j +0 +∇2 − dm2 +d−1 +0 +∂j +0 +0j +−m2∂i +0i +∂i +0i +j +0i +δi +j +m2 +0 +0 +0j +0 +0j +0i +−∂i +0i +−δi +j +0i +0i +j +� +�������� +δd(x − y) , +(4.8) +where derivatives are taken with respect to the coordinate x and d = D − 1 denotes the +dimension of the Cauchy slice.8 Here the raised and lowered indices denote rows and columns +respectively. Fourier transforming C(x − y), we get the momentum space constraint matrix +C(p) = m2 +� +�������� +0 +d +d−1m4 +0 +−m2ipj −m2 +0j +− +d +d−1m4 +0 +p2 + dm2 +d−1 +0j +0 +−ipj +0 +−p2 − dm2 +d−1 +0 +ipj +0 +0j +−m2ipi +0i +ipi +0i +j +0i +δi +j +m2 +0 +0 +0j +0 +0j +0i +−ipi +0i +−δi +j +0i +0i +j +� +�������� +, +(4.9) +7For theories with massive graviton, one needs to look at the symmetric traceless representation of the group +SO(D − 1), which gives us D2−D−2 +2 +polarizations. This is valid for D ≥ 2, and in particular massive gravity in +three dimensions has a propagating degree of freedom, whereas, in four dimensions, we have five polarizations. +8Note that the factors of m2 in C(x − y) shows that in the limit m → 0, the 2D constraints Ca≥0 are first +class, while the remaining two constraints in (4.5) identically vanish. +– 10 – + +where we have used the momentum space representation of the delta function, (2π)dδd(x−y) = +� +p eip.(x−y). The matrix C(p) can be easily inverted to obtain the Dirac constraint matrix +C−1(p) = 1 +m4 +d − 1 +d +� +�������� +0 +0 +0 +0j +d +d−1 +0j +0 +0 +−1 +0j +0 +−ipj +0 +1 +0 +ipj +−p2 + dm2 +d−1 +0j +0i +0i +ipi +0i +j +0i +−pipj − dm2 +d−1δi +j +− +d +d−1 +0 +p2 − dm2 +d−1 +0j +0 +ipjp2 +0i +−ipi +0i +pipj + dm2 +d−1δi +j +ipip2 +0i +j +� +�������� +. +(4.10) +The main takeaway from the above analysis is that, unlike in the case of massless gravity, +the Dirac matrix of a massive gravity theory has a local expression. +This observation has +important implications for the statement of holography of information. In particular, in §5.2 +we demonstrate that in contrast to the situation in massless gravity, massive gravity theories +can hide information about bulk operator insertions from boundary operators. +5 +Boundary observables and Dirac brackets +We will now utilize the Dirac matrices obtained for various theories, i.e. +electrodynamics, +massless gravity, and massive gravity, to calculate the Dirac brackets. +5.1 +Boundary observables and Dirac brackets for massless gravity +As in electrodynamics, the relevant boundary operator for massless gravity can be obtained +from the Gauss constraint. The constraints for linearized massless gravity with matter are +given in Appendix B.1, and from (B.13), the Gauss constraint χm +0 for massless gravity with +matter insertion is given by +∇2hk +k − ∂i∂jhij = −16πGNT00. +(5.1) +Given any bounded spatial region V , the Gauss constraint makes it possible to encode the +energy of matter fields supported on it +� +V T00 via an equivalent boundary operator given by: +H∂ = +1 +16πGN +� +V +dD−1x ∂i.(∂jhij − ∂ihk +k) = +1 +16πGN +� +∂V +dD−2x ni.(∂jhij − ∂ihk +k) , +(5.2) +where n denotes the unit normal to the boundary ∂V of V (we review the analogous construction +for electrodynamics in Appendix A). The Hamiltonian of the full theory can be obtained from +H∂ by taking the limit where V includes the entire Cauchy slice containing it. +Let us compute the Dirac bracket for the boundary operator H∂ with some bulk matter +insertion O(z). Using the gravity constraints (3.12), since H∂ only depends on hij, we see that +– 11 – + +H∂ has nonzero commutators only with the constraint C2D = K0 given in (3.9). The relevant +commutator is given by: +{H∂, C2D(y)} = +� +ddz +∂H∂ +∂hij(z) +∂C2D(y) +∂Πij(z) += − +1 +16πGN +� +V +ddx ∇2δd(x − y) = +1 +16πGN +� +V +ddx +� +ddp +(2π)d p2eip.(x−y) , +(5.3) +where we have set d = D − 1. The Dirac bracket of the boundary operator H∂ with a bulk +matter operator insertion O(z) is given by: +{H∂, O(z)}D.B. = − +� +ddy ddz′ {H∂, Ca(y)} C−1 +ab (y, z′) {Cb(z′), O(z)} += +1 +16πGN +� +V +ddx +� +ddp +(2π)d +ddq +(2π)d ddy ddz′ eip.(x−y)eiq.(y−z′)p2 +q2 {χm +0 (z′), O(z)} += +1 +16πGN +� +V +ddx {χm +0 (x), O(z)} = +� +V +ddx {T00(x), O(z)} , +(5.4) +where χm +0 is the Hamiltonian constraint in the presence of matter, given in (B.13). In the second +step of (5.4), we have used the fact that only the component C−1 +2D 0 of the inverse constraint +matrix contributes. Notice that in the limit where V approaches the full spatial slice containing +it, the right-hand side of (5.3) vanishes. However, the non-local factor arising from the inverse +constraint matrix provides a measured counter-effect which makes (5.4) valid even for the full +spatial slice. Thus the Dirac bracket of the boundary Hamiltonian with the observable O(z) is +equal to the Poisson bracket of the observable with the integrated Gauss constraint. This in +turn, as expected, is equivalent to the Poisson bracket of O(z) with the matter Hamiltonian. +5.2 +Boundary Observables and Dirac brackets for massive gravity +Like massless gravity, the relevant boundary Hamiltonian for massive gravity can be obtained +from the Gauss constraint. From (B.20) and (B.21), the Gauss constraint χ0 for massive gravity +with matter insertion is given by +∇2hk +k − ∂i∂jhij = m2hk +k − 16πGNT00. +(5.5) +As in massless gravity, we can integrate the LHS of the Gauss constraint within a spacelike +region V and obtain the boundary operator H∂, which is given by +H∂ = +1 +16πGN +� +V +dD−1x ∂i.(∂jhij − ∂ihk +k) = +1 +16πGN +� +∂V +dD−2x ni.(∂jhij − ∂ihk +k). +(5.6) +From the massive gravity constraints, we see that the boundary Hamiltonian fails to commute +only with the constraint C−1 (see Eqn (4.5)). The relevant commutator is given by: +{H∂, C−1(y)} = +� +ddz +∂H∂ +∂hij(z) +∂C−1(y) +∂Πij(z) += − +m2 +16πGN +� +V +ddx ∇2δd(x − y) = +m2 +16πGN +� +V +ddx +� +ddp +(2π)d p2eip.(x−y) , +(5.7) +– 12 – + +which is identical to (5.3) up to a factor of m2. The Dirac bracket of the boundary operator +with a bulk matter operator O(z) is given by: +{H∂, O(z)}D.B. = − +� +ddy ddz′ {H∂, Ca(y)} C−1 +ab (y, z′) {Cb(z′), O(z)} += d − 1 +m2d +1 +16πGN +� +V +ddx +� +ddp +(2π)d +ddq +(2π)dddy ddz′ eip.(x−y)eiq.(y−z′) +p2� +{C0(z′), O(z)} + iqi{CD+i(z′), O(z)} +� += −d − 1 +m2d +1 +16πGN +� +V +ddx ∇2� +{C0(x), O(z)} + ∂i{Π0i(x), O(z)} +� += −d − 1 +m2d +1 +16πGN +� +∂V +dd−1x ni∂i +� +{C0(x), O(z)} +� +. +(5.8) +In the second equation above, we have identified the only contributing terms to be from C−1 +−1 0 +and C−1 +−1 D+i. In the final step we have neglected the contribution from the C−1 +−1 D+i terms as +O(z) is assumed to be a pure matter operator with trivial Poisson brackets with Π0i. +The result (5.8) differs from that of (5.4) fundamentally due to fact that, unlike in the case +of massless gravity, the inverse constraint matrix contributing to (5.8) is local, thus allowing +us to reduce the Dirac bracket to a pure boundary term. Therefore, when O(z) is taken to +be an operator insertion strictly in the bulk, we find that its Dirac bracket with the boundary +operator H∂ vanishes. +In Appendix C, we treat the constraints of the free theory but substitute the equation of +motion of h0i back into the action. We argue that at the classical level, the substitution makes +sense. We use this to alternatively demonstrate that {H∂, O(z)}D.B. = 0. +6 +Vacua structure and split states +We will now utilize the Dirac brackets obtained for various theories, i.e. +electrodynamics, +massless gravity, and massive gravity, to investigate the existence of split states. In order to +set up the stage for further discussions of constraints using Dirac brackets and their relation +with split states, let us first begin with the case of electrodynamics. Readers familiar with split +states in electrodynamics can directly skip ahead to §6.2. +6.1 +Split states in electrodynamics +We minimally couple a charged scalar φ to the electrodynamic field. The Hamiltonian for this +system is given by +HJ = +� +dd−1x +� +−1 +2ΠiΠi + 1 +4FijF ij − ∂iΠiA0 + ΠφΠφ∗ − ieA0 (φΠφ − Πφ∗φ∗) + (Diφ)∗Diφ +� +(6.1) +– 13 – + +where Diφ ≡ ∂iφ + ieAiφ is the covariant derivative with respect to the gauge field, and Πi +denotes the momentum conjugate to the electrodynamic field, while Πφ denotes the momentum +conjugate to the scalar field and is given by +Πφ = ˙φ∗ − ieA0φ∗ +(6.2) +In terms of the gauge invariant fields, Πi = Ei. +Using (6.1), one can compute the Gauss +constraint, which is given by: +∂iΠi − J0 = 0 +where +J0 = −ie (φΠφ − Πφ∗φ∗) +is the matter current. As a consequence, the Gauss constraint implies that given a codimension +one spacelike slicing Σ, measuring the integral of the electric field over the boundary gives us +the total charge Q = +� +Σ J0, i.e. +� +Σ +∂iΠi = +� +∂Σ +niΠi = Q. +(6.3) +where ni is the outward pointing normal vector. +The boundary operator in (6.3) is unique to electrodynamics due to the Gauss constraint +and gives us the total charge. Now consider some matter insertion in bulk, denoted by the +action of an operator O(x). We want to determine whether the information content of the +insertion can be obtained using a relevant boundary observable, i.e., the boundary operator +defined in (6.3). +The computation of the Dirac matrix for electrodynamics, both with and without matter, +is performed in Appendix A. Using our analysis there, we are in a position to investigate the +Dirac bracket of +� +Σ ∂iΠi with O(x), which gives us +� � +Σ +∂iΠi(x), O(z) +� +D.B. = +� � +Σ +∂iΠi(x), O(z) +� +− +� +y1 +� +y2 +� +Σ +∂2δ(x − y1) 1 +∂2δ(y1 − y2) +� +∂iΠi(y2) − J0(y2), O(z) +� (6.4) +We will work with purely matter insertions O(z) in the following discussion. Then the first +Poisson bracket on the RHS of (6.4) is zero, while the second bracket, upon integration by +parts and using the Gauss law, takes the form +� � +Σ +∂iΠi(x), O(z) +� +D.B. = − +� +Σ +� +∂iΠi(x) − J0(x), O(z) +� += {Q, O(z)} +(6.5) +where we have used {∂iΠi(x), O(z)} = 0. Thus we obtain an order one contribution due to the +matter insertion. If we have a chargeless state, the Dirac bracket expectedly vanishes, whereas +we obtain a finite contribution for the charged state. +– 14 – + +Lifting the Dirac brackets to operators acting on the Hilbert space, equation (6.5) takes +the following form 9 +� +Σ +� +∂iΠi(x), O(z) +� += [Q, O(z)] +(6.6) +Equation (6.6) leads to the existence of split states in electrodynamics. To see this, note that +the order one contribution on RHS is insufficient to specify the state of the bulk on the spacelike +slicing. This is because the boundary operator +� +Σ ∂iΠi(x) can only measure the charge of the +state. +Consequently, there is an infinite degeneracy of states with a given electromagnetic charge, +which all evaluate to the same value on the RHS of (6.5). In this way, the Gauss constraint in +electrodynamics cannot specify the state in question. Hence the split property holds since one +cannot distinguish bulk states using the relevant boundary operator. +6.2 +Hilbert space and vacua structure in flat space gravity +In this subsection, we will revisit the canonical Hilbert space of asymptotically flat massless +gravity and use it to construct the massive gravity Hilbert space analogously. The Hilbert +space will be important in understanding the split property of flat space gravity in the later +subsections. +Massless gravity +We begin by studying the vacua and boundary operators for flat space massless gravity [48–50]. +The asymptotic symmetries are implemented by subgroups of the diffeomorphism group, such +as the BMS group, which generates supertranslations [51–56]. +Supertranslations (in D = 4) are generated using supertranslation charges Qlm constructed +by spherically smearing the Bondi mass aspect mB(u, Ω) at I+ +−: +Qlm = +1 +4πGN +� √γ d2Ω mB(u = −∞, Ω) Yl,m(Ω). +(6.7) +We can separate Qlm into hard and soft components. Thus a specification of vacuum involves +annihilation under positive modes of matter and gravity together with the eigenvalue under +soft mode: +Qlm |0, {s}⟩ = sl,m |0, {s}⟩ +(6.8) +Here in |0, {s}⟩, the first label denotes the energy, while the second label specifies the super- +translation sector, i.e. the eigenvalue under the soft mode. Supertranslation sectors serve as +different superselection sectors for the theory, and hence the flat space massless gravity Hilbert +space is given by +H = +� +{s} +H{s} +(6.9) +9The lifting from phase space observables to operators acting on the Hilbert space is subject to the assumption +that a suitable operator ordering prescription exists which removes any anomalous terms. This issue arises not +only in electrodynamics but also in our later analysis of gravity, and we discuss more about this in §8. +– 15 – + +The ADM Hamiltonian H∂ defined in (5.2) is simply Q00 charge defined in (6.7) [57, 58]. We +now introduce an abstract projector onto the vacua subspace of the massless gravity theory +[1, 35]: +P0 = lim +a→∞ exp (−aH∂) . +(6.10) +We can also define a more physical projector Pδ which projects onto energies below an IR cutoff +δ [37]: +Pδ = Θ (δ − H) . +(6.11) +In §6.3, we will revisit the significance of H∂ and the projectors by computing the Dirac bracket +between H∂ and a bulk matter insertion O(z). Specifically, the projectors allow us to reconstruct +bulk operators using the boundary. +Massive gravity +In order to construct the Hilbert space of linearized massive gravity, we note the following +points: +1. From our calculation in (5.8), the Dirac bracket of H∂ with a bulk matter insertion O(z) +is zero. Upon lifting operators to the phase space, we replace the Dirac bracket with +commutators, thereby giving us +[H∂, O(z)] = 0, +(6.12) +where H∂ is given by the standard expression: +H∂ = +1 +16πGN +� +dD−2x ni.(∂jhij − ∂ihk +k). +(6.13) +2. Since asymptotic symmetries are absent in massive gravity, in contrast to the massless +theory, the vacua subspace of the flat space massive theory is labelled by a single sector +rather than a direct sum over infinite supertranslation sectors as in (6.9). +Using these above points and from equations (6.12) and (6.13), we can use the boundary +operator H∂ to label the states in the Hilbert space as |E, M⟩ such that +H∂ |E, M⟩ = E |E, M⟩ . +(6.14) +Here the label E denotes the eigenvalue under H∂, and M is a quantum number that labels the +bulk matter and gravity insertions. In contrast to the massless case, the second label in (6.14) +does not denote the supertranslation sector but is closely related to the longitudinal mode of +graviton polarization in massive gravity. +Due to the absence of asymptotic symmetries, we expect that the S-matrix defined over +the state space {|E, M⟩} is infrared finite, i.e. as expected, there are no infrared divergences in +pure massive gravity. +Projectors onto the vacuum: Using H∂, we can again try to construct boundary pro- +jectors onto the vacua subspace using the boundary Hamiltonian H∂, as in (6.10) and (6.11). +– 16 – + +The construction works within our linearized analysis since, by definition, the lowest value that +H∂ can take is zero. The boundedness of energy is crucial to defining the projectors at the +linearized level. +However, from §5.2, the action of the projector P0 in (6.10) on a generic state does not +project onto just the empty state |0, 0⟩, but projects onto a subspace of states labelled by +|0, M⟩.10 +This is a significant departure from the flat space massless gravity case, and the +above vacua structure will be important in addressing the questions regarding split states in +massive gravity11. +However, defining projectors in this fashion makes sense only in linearized gravity. We +comment on going beyond our linearized analysis in §8. +6.3 +Split states in massless gravity +Equation (5.4) states that the boundary Hamiltonian H∂ knows about bulk insertions since +the Poisson bracket of the matter insertion with the integrated stress energy component T00 +is the same as the commutator of the H∂ with the matter insertion. This is in line with the +Hamiltonian constraint analysis in equations (4.57) and (4.58) of [2], where expanding the +constraint at the second order in perturbation theory, H∂ is related to the bulk energy. The +bulk energy has a contribution from both gravity and matter, with the matter contribution +being T00. In our case, on the RHS of (5.4), O(z) clicks only with T00 in the Poisson bracket. +Thus the commutator of H∂ with matter insertion gives us the same result as expected from +the order-by-order expansion of the Hamiltonian constraint. +Given this consistency check, the statements about holography of information and split +states follow as in [1], which we briefly summarize. Using a Reeh Schlieder type argument, one +can construct operators QB supported near the boundary, which can act on a supertranslation +vacuum |0, {s}⟩ to create any arbitrary state |B, {s}⟩ in the supertranslation sector {s} +QB |0, {s}⟩ = |B, {s}⟩ + O +�� +GN +� +. +(6.15) +Henceforth we will ignore O +�√GN +� +corrections. Thus using (6.15) any hard bulk operator12 +O(z) can be written as +O(z) = +� +mn +cmn |m, {s}⟩ ⟨n, {s}| = +� +mn +cmn Qm |0, {s}⟩ ⟨0, {s}| Q† +n. +(6.16) +From equation (5.4), we see that the matter insertion O(z) leaves an imprint on the ADM +energy since it does not commute with H∂. Hence H∂ can be labelled using the bulk matter- +energy, which is positive definite in well-behaved theories. Thus one can use H∂ to construct a +boundary projector (6.10) onto the vacuum of the theory on the lines of [1]. +10This is in striking contrast to the massless gravity case where graviton and bulk matter insertions labelled +by M completely fix the boundary ADM energy E. We do not use M as a separate quantum number in massless +gravity since one label is enough to specify the state. +11At this stage, the fastidious reader may correctly anticipate that one can exploit the above-mentioned +contrasting feature to construct split states in massive gravity. +12We only consider hard bulk operators here whose action does not change the superselection sector {s}. +– 17 – + +Using the Fock space representation of the projector (6.10), we can write the operator +representation in (6.16) as +O(z) = +� +mn +cmn QmP0 Q† +n. +(6.17) +which is completely expressed in boundary operators Qm and P0. Using the arguments of [3, 4], +various statements regarding the holography of information follow from the above representa- +tion. Since bulk operators can be written as combinations of boundary operators, there cannot +be split states since one can always utilize the boundary expression for the bulk operators to +probe bulk physics. Thus the decomposition of the massless gravity Hilbert space H into +H = HA ⊗ H ¯ +A +(6.18) +is not allowed13, and correspondingly, the notion of split states in massless gravity does not +exist [4]. +One can also use the more physical boundary projector Pδ defined in [37] to express O(z) in +terms of boundary operators. However, this is a slightly more difficult task since this involves +delicately tuning smearing functions on the complement of the bulk region we are interested in. +7 +Split states in massive gravity +Since the Dirac bracket of H∂ with bulk operator O(z) is zero in massive gravity from equation +(5.8), we can hide any bulk matter insertion O(z) from detection using the boundary operators. +In this section, we will demonstrate the existence of split states in massive gravity in two ways: +firstly by making an explicit reference to the Hilbert space outlined in §6.2, and secondly by +not making an explicit reference to the Hilbert space. +7.1 +Split states from the vacuum structure +Following our notation introduced in §6.2, from equation (5.8) we have +[H∂, O(z)] = 0. +(7.1) +which implies that the we have simultaneous eigenstates |E, M⟩. Thus there is no way to define +a vacuum projector using our relevant boundary operator H∂, which projects onto the empty +state, i.e. with no bulk matter. In other words, there are arbitrarily many states with bulk +matter insertions in the zero eigenvalue subspace of H∂. +This leads to split states since from eqn (7.1), H∂ or operators constructed using it can +never detect matter eigenvalue M in the bulk. In other words, one can shield bulk excitations +from any possible detection at the boundary. Hence there is no holography of bulk information +at the boundary, which could be detected using boundary operators. Let us now precisely define +what we mean by factorization of the Hilbert space. +13The precise sense in which we refer to the factorization of Hilbert space is explained in §7. +– 18 – + +Approximate factorization of Hilbert space +The notion of Hilbert space factorization in eqn (6.18) is approximate. Even if we ignore any +constraints, such a factorization is not allowed in a quantum field theory since the energy of +such product states lies outside effective field theory. To work with such states within effective +field theory, we should imagine some spatial separation between the regions A and ¯A larger +than the UV length scale of effective field theory. +Formally, introducing the spatial separation and momentarily ignoring the constraints, we +can approximately factorize the Hilbert space H upon spatially partitioning flat space into a +bounded region A and its complement ¯A as follows +H = HA ⊗ H ¯ +A. +(7.2) +Then the factorization in (7.2) implies that spatially partitioned split states of the following +form exist in the Hilbert space H: +|ψ⟩ = |ψA⟩ ⊗ |ψ ¯ +A⟩ . +(7.3) +In our present case, region A should be thought of as most of the bulk region while the com- +plement region ¯A is a small enough region sufficient to include the boundary. Then in massive +gravity, states in the Hilbert space spanned by |ψ⟩ = |E, M⟩ can be thought of as split states +living in the above approximately factorised Hilbert space. +This is roughly because we can modify the bulk state |ψA⟩ in region A thereby changing +the matter quantum number M, while preserving the boundary eigenvalue E in region ¯A at the +same time. Since there are no other relevant boundary operators that probe the bulk physics +in region A, changing |ψA⟩ has no effect on the state |ψ ¯ +A⟩ in region ¯A. Consequently states +of the form (7.3) are allowed in the Hilbert space of massive gravity taking into account the +constraints using Dirac brackets. +7.2 +Split states without the vacuum structure +More generally, we do not need to reconstruct the Hilbert space in order to understand the split +property. To see this, let us work with a non gravitational theory first, and outline the split +property. +Consider a density matrix ρ defined on the spatially partitioned system acting on H. We +will be working with the algebra of observables A and ¯ +A acting on the previously-defined regions +A and ¯A respectively. The algebras A and ¯ +A consist of products of bulk operators supported +on A and ¯A respectively. +We now look at operators OA ∈ A and O ¯ +A ∈ ¯ +A, where elements from the algebras mutually +commute, i.e., [OA, O ¯ +A] = 0. Let us consider a bulk observable of the form O = OAO ¯ +A. The +split property [59, 60] implies that the expectation value of operator O can be written as: +⟨O⟩ ≡ Tr (ρ O) = Tr (ρAOA) Tr (ρ ¯ +AO ¯ +A) +(7.4) +– 19 – + +where ρA and ρ ¯ +A are density matrices such that they are unconstrained in the traced out regions +¯A and A respectively. Effectively the density matrix ρ takes the following form +ρ = Tr ¯ +A (ρA) ⊗ TrA (ρ ¯ +A) , +(7.5) +which can be seen by substituting (7.5) into (7.4). +Support of boundary operator H∂ +In massless gravity, there are a couple of issues regarding the above construction. Firstly, from +eqn (5.4), we have +[H∂, OA] ̸= 0. +(7.6) +Hence the decomposition in (7.4) does not work, since there always exists an operator H∂ living +in the region ¯A which can be used to detect observables in A14. Correspondingly, the Hilbert +space does not factorize, and the density matrix cannot be written in the form (7.5) for massless +gravity. +The second issue arises since only small diffeomorphism invariant observables make sense. +For instance, diffeomorphism invariant dressed operators obey [OA, O ¯ +A] ̸= 0, and hence (7.4) +and (7.5) do not hold. +A more precise version involves defining the algebra of observables +asymptotically. We refer the reader to [1, 4] for a detailed treatment of this issue. +In massive gravity, we do not need to work with asymptotic observables since diffeomor- +phism invariance is absent, and hence we can work with the previously defined algebra of +observables. Since we do not need to dress the observables, we have [OA, O ¯ +A] = 0. Also for +an observable OA, the boundary operator H∂ simply commutes with the spatially partitioned +observables +[H∂, OA] = 0. +(7.7) +As a consequence, equations (7.4) and (7.5) follow, since we cannot use H∂ which is supported +in region ¯A, to detect the state supported in region A any more. Thus from (7.4) and (7.5), +specifying information in region ¯A does not fix the state in the region A. Consequently, the +existence of split states in massive gravity can be understood from lifting the Dirac brackets +onto operator commutators acting on the Hilbert space. +The appearance of split states is a significant departure from the case of massless gravity, +where any matter insertion has a specific signature in correlators involving boundary operators. +Massive gravity is not constrained enough to detect bulk insertions using boundary observables +and their correlators. From our analysis in this subsection, it is clear that massive gravity +resembles a non-gravitational QFT than massless gravity. +14From (5.4) since the commutator is non-zero, the ADM Hamiltonian H∂, though supported in the region +¯A, does not belong solely to the algebra ¯ +A. +– 20 – + +8 +Conclusion and discussion +In our work, we have computed Dirac brackets in different settings and used them to investigate +the issue of split states. More generally, we have addressed the following question regarding the +principle of holography of information: given access to boundary operators, can one use them +to identify a generic bulk state? In linearized massive gravity, using our analysis of the Dirac +brackets, it appears that such information is hidden from boundary operators, thereby allowing +for split states. +We find that the Dirac bracket between the relevant boundary operator from the Gauss +constraint and a generic bulk matter insertion is zero for massive gravity. This is consistent +with our intuitively expected picture resulting from the lack of small diffeomorphisms in massive +gravity. +Thus one can create local bulk operators which can never be detected using the +boundary operator H∂ since the commutator is simply zero. This is a significant departure +from massless gravity. We show that this leads to split states, and hence there is no holography +of information in linearized massive gravity. +Potential limitations of our analysis +Let us now discuss some potential limitations of our analysis: +1. Regarding the closure of constraints: As discussed in Appendix B, from the per- +spective of constraints, one cannot consistently couple matter to the linearized gravity +action since the constraint algebra does not close. The failure of linearised gravity-matter +constraints to close introduces further constraints on the phase space. Introducing these +additional constraints is inconsistent with the degree of freedom counting. In contrast, the +case of electrodynamics is much simpler, where the Dirac matrix, upon the inclusion of +charged matter, is the same as for the uncharged case (see Appendix A). The issues with +consistently coupling matter with linearized gravity are an important feature contrary +to our naive expectations. This issue is also demonstrated from the failure to impose +∂µT µν = 0 in [24].15 +Motivated by electrodynamics, we circumvent this issue by taking the Dirac matrix of +linearised gravity without matter to evaluate the Dirac brackets. This ensures that the +algebra closes, and we use this matrix to compute Dirac brackets between observables, +with subsequent Poisson brackets defined over the entire matter-gravity phase space. In +other words, we restrict ourselves to the gravity phase space whenever we take the Poisson +bracket between two different constraints but otherwise work in the entire matter-gravity +phase space. A more satisfactory procedure would be to consider the full non-linear action +coupled to matter [19–22] and study the Dirac brackets, which is an open question. +15Very loosely, we can also argue that there should not be any corrections to the linearised Dirac matrix upon +including matter since naively coupling matter order by order in perturbation theory is problematic, and as a +consequence, the constraints fail to close properly. +– 21 – + +2. Holography of information: In massless gravity, the principle of holography of infor- +mation is a non-perturbative statement. Perturbatively we can demonstrate holography of +information about low energy states. However, for heavy time-dependent states, as stud- +ied in [41], within perturbation theory, it may be possible to construct operators which +can commute with the Hamiltonian. We expect that there exist complicated observables +using which we can probe the bulk, which incorporate non-perturbative effects. +In our work, we restrict our analysis to linearized gravity over the empty flat background +and restrict our statements to low energy states about the vacuum. Though likely, we +are unable to concretely establish whether holography of information in massive gravity +is a non-perturbative statement since it is unclear whether the canonical vacuum satisfies +necessary properties such as boundedness and whether one can define projectors onto the +same. +3. Quantization ambiguities: Generally, lifting constraints from the phase space to op- +erator equations on the Hilbert space may introduce some corrections to the constraint +algebra. For instance, we may have ambiguities proportional to δ(0) for first-class con- +straints. If such ambiguities arise, we need to implement a suitable choice of normal +ordering that allows us to circumvent them. +In our analysis of massive gravity, some simple operator-ordering ambiguities, such as +ones resulting from the multiplication of canonical field with momenta, are absent since +the constraints are all linear. There are seemingly no such obstructions to quantization +at the level of our linearized analysis. +4. Higgs-type mechanism and localization of information: A straightforward implica- +tion regarding the localization of information is as follows: the localization of information +on the whole AdS5 boundary is different from the Karch Randall type setups [14], which +have a massive graviton. In such setups, the massive graviton arises from a higher dimen- +sion. However, the crucial feature is that the boundary of the complete theory is not the +same as the boundary of the dimensionally reduced theory, hence the difference in the +localization of information. +Another question that may arise is how a possible Higgs mechanism which gives the +graviton mass may change the localization of information. +The naive picture is that +breaking the diffeomorphism invariance by such a mechanism changes the localization of +information, and consequently, even at the non-perturbative level, one may not be able +to recover bulk information using just the boundary operators. Such issues still need to +be better understood. +Black hole evaporation +We now briefly discuss some other implications of our work regarding black hole evaporation. +Our analysis here indicates consistency with the arguments of [61] that massive gravity at a +linearized level allows for black hole evaporation using the islands formalism. This is because +– 22 – + +the Dirac bracket of operator insertions inside the disconnected entanglement wedge with the +boundary Hamiltonian is zero, indicating consistency with the subregion duality. +However, since the Dirac bracket in massless gravity between the boundary Hamiltonian +and the bulk Hamiltonian is not zero, operator insertions inside the entanglement wedge can +potentially be detected using the boundary Hamiltonian. Whether our formalism sheds some +light to circumvent this obstruction in massless gravity is an interesting question. +Other open questions +We conclude our work with some other open questions. For massless gravity, we observe that +the form of the constraint algebra of the linearised theory without matter and the complete +non-linear theory with matter look similar, provided we fix the gauge appropriately. It would +be interesting to investigate whether this observation also holds for the case of massive gravity. +Stated differently, the question is whether we expect the form of the constraint algebra of non- +linear massive gravity coupled to matter to be similar to the Fierz-Pauli case. Our analysis is +plausibly valid for the full non-linear theory coupled to matter in such a case. A related point +is whether our described vacua structure, which depends on our restrictive linearized analysis, +generalizes to the non-linear case. +Given the constraint algebra of the non-linear action, an interesting question is whether a +systematic procedure exists to truncate it to the constraint algebra from the quadratic action, +i.e. +to the linearized case. +As we argued, to include matter, we need to consider the full +non-linear Einstein action. However, is there any consistent truncation of the full non-linear +constraint algebra, which gives us the Dirac matrix of linearized gravity? +A slightly distant avenue is to understand whether there is any natural obstruction to +lifting massive gravity phase space observables to state-independent operators [62, 63] and +their subsequent dependence on late time effects [3, 64–66]. We hope to address some of these +issues in future work. +Acknowledgments +We thank Suvrat Raju for suggesting the problem and valuable suggestions regarding the work. +We thank Claudia de Rham and Alok Laddha for useful correspondence. We are also grateful +to Sayali Bhatkar, Simon Caron-Huot, Tuneer Chakraborty, Victor Godet and Priyadarshi Paul +for related useful discussions. We acknowledge support of the Department of Atomic Energy, +Government of India, under project number RTI4001. +A +Dirac brackets for electrodynamics +We will first look at Dirac brackets for electrodynamics without matter and then derive the +Dirac brackets after adding in the matter. +– 23 – + +Electrodynamics without charges +We will work with the Lagrangian given by +L = −1 +4FµνF µν, +(A.1) +where Fµν denotes the field strength. Using this Lagrangian, we arrive at the primary constraint +Π0 = 0, +(A.2) +where Π0 denotes the standard canonical momentum. Then the Hamiltonian H0 obtained from +the Legendre transformation of (A.1) is given by +H0 = +� +ddx +� +−1 +2ΠiΠi + 1 +4FijF ij − ∂iΠiA0 +� +. +(A.3) +To implement the Dirac bracket procedure, we will first add in the contribution from the primary +constraint, i.e. +H = H0 + v0Π0, +(A.4) +where our goal now is to fix v0. The condition for the stability of the primary constraint gives +us the Gauss constraint, which is a secondary constraint, +{Π0, H} = ∂iΠi. +(A.5) +We find that there are no further tertiary constraints because +{∂iΠi, H} = 0 +(A.6) +due to cancellations among terms resulting from integration by parts. Consequently, we have +v0 = 0 in (A.4), i.e. the constrained Hamiltonian is the same as obtained from Legendre trans- +formation of the electrodynamics Lagrangian. Thus we have a system of first class constraints +characterized by the Hamiltonian H, and constraints (A.2) and (A.5). +Gauge fixing +Since we have a first class system, we fix the gauge by putting in gauge conditions. A convenient +choice is to choose a gauge that is orthogonal to the first class constraints. In our case, this +amounts to +A0 = 0 +and +∂iAi = 0. +(A.7) +We rewrite the system of constraints given by (A.2), (A.5) and (A.7) in the following ordered +form +C0 = Π0(x) +C1 = ∂iΠi(x) +C2 = A0(x) +C3 = ∂iAi(x) +(A.8) +– 24 – + +Using the above constraints, we have the following non-zero Dirac brackets +{Π0(x), A0(y)} = −δd(x − y) +{∂iΠi(x), ∂jAj(y)} = ∇2δd(x − y). +(A.9) +Note that the plus sign in the expression for the second commutator in (A.9) arises due to the +shifting of derivatives while performing integration by parts. Then the constraint matrix with +the ordering in (A.8) is given by +M(x − y) = +� +� +� +� +� +0 +0 +−1 0 +0 +0 +0 ∇2 +1 +0 +0 +0 +0 −∇2 0 +0 +� +� +� +� +� δd(x − y) +(A.10) +The inverse matrix of M(x, y) from (A.10) is given by +M −1(x − y) = +� +� +� +� +� +0 +0 1 +0 +0 +0 0 − 1 +∇2 +−1 0 0 +0 +0 +1 +∇2 0 +0 +� +� +� +� +� δd(x − y) +(A.11) +Electrodynamics with charges +Recall from (6.1) that the Hamiltonian for a charged scalar coupled to electrodynamics is given +by +HJ = +� +ddx +� +−1 +2ΠiΠi + 1 +4FijF ij − ∂iΠiA0 + ΠφΠφ∗ − ieA0 (φΠφ − Πφ∗φ∗) + (Diφ)∗Diφ +� +(A.12) +Here we again have the primary constraint Π0 = 0. As previously done, we write the constrained +Hamiltonian as +H = HJ + v0Π0 +(A.13) +The stability of the primary constraint gives us the secondary Gauss constraint, which is given +by +∂iΠi − ie (φΠφ + φ∗Πφ∗) ≡ ∂iΠi − J0 = 0.16 +(A.14) +Recall that now the Poisson bracket involves taking derivatives with respect to the scalar field +and its conjugate momentum as well, since a complete specification of the phase space involves +accounting for the scalar field as well. We find that the secondary Gauss constraint is stable, +i.e. +{∂iΠi − J0, H} = 0 +(A.15) +due to cancellations among various terms, and use this to set v0 = 0, similar to the case of free +electrodynamics. +16As a comparison, for fermions, there is no electrodynamic contribution to the matter current. +– 25 – + +Gauge fixing and Dirac brackets +Since the primary constraint remains unchanged and the secondary Gauss constraint receives +an additive scalar contribution plus the contribution from A0, a good orthogonal choice is to +choose the same gauge conditions as previously chosen. This is due to the fact that the extra +charged contribution to the Gauss constraint commutes with the choice of gauge, which by +construction gives a non-zero commutator with the free part. As a consequence, we have the +constraints +C0 = Π0(x) +C1 = ∂iΠi(x) − J0 +C2 = A0(x) +C3 = ∂iAi(x) +(A.16) +which gives us the exact same Dirac matrix as in (A.10) and its inverse in (A.11). +B +Constraints in linearized gravity with matter +In this Appendix, we covariantly couple matter to linearized massless and massive gravity. We +show that the constraints do not close i.e., they become inconsistent with our expected counting +of the degrees of freedom. +B.1 +Massless Gravity with minimally coupled matter +We will now minimally couple a scalar field to massless gravity using the stress energy tensor. +Using this, we will compute the constraints of this theory and calculate the Dirac bracket in +this subsection. The action of minimally coupled matter to gravity is given by: +Sφ = −1 +2 +� +dDx √−g(gµν∂µφ∂νφ + m2φ2) +(B.1) +where g is the determinant of the space-time metric and indices µ, ν run from 0 to D − 1. +Expanding the metric about the flat background (i.e. gµν = ηµν+hµν and hence gµν = ηµν−hµν), +at leading order, we obtain: +Sφ = +� +dDx +� +1 + h +2 +� � +−ηµν +2 ∂µφ∂νφ − m2 +2 φ2 +� ++ hµν +2 ∂µφ∂νφ +(B.2) +where h = Tr(hµν). In terms of the energy-momentum tensor Tµν, above action can be re- +written as: +Sφ = +� +dDx +� +− +√−gb +2 +hµνT µν(φ, ˙φ) + √−gbLm(φ, ˙φ) +� +(B.3) +where Lm(φ, ˙φ) is the free-scalar Lagrangian, gb is the determinant of background metric (which +is ηµν in present case) and T µν is given by: +T µν = ∂µφ∂νφ − ηµν +2 (∂ρφ∂ρφ + m2φ2) +(B.4) +– 26 – + +The massless graviton Lagrangian Lg is given in (3.1). Hence the total action is given by: +S = Sφ + Sg +(B.5) +where Sg denotes the integral of Lg, with the GN dependence restored using an overall multi- +plicative factor κ2. +Sg = 1 +κ2 +� +dDx√−gLg +Momenta and Hamiltonian +Using the combined action in (B.5), we will now determine the canonical momenta and the +Hamiltonian for our scalar-gravity theory. From the gravity part we obtain the following ex- +pression for the momenta: +Π00 = 0, +Π0i = 0 +Πij = ∂L +∂ ˙hij += 1 +κ2 +� +˙hij − ˙hkkδij − 2∂(ihj)0 + 2∂kh0kδij +� +(B.6) +From (B.6), we find that we have two primary constraints Π00 and Π0i. The gravity Hamiltonian +from (3.3) is given by: +Hg = κ2 +�Π2 +ij +2 − +Π2 +ii +2(D − 2) +� ++ 1 +κ2 +�1 +2∂khij∂khij − ∂ihjk∂jhik + ∂ihij∂jhk +k − 1 +2∂ihj +j∂ihk +k +� +− h00 +κ2 +� +∂2 +khi +i − ∂i∂jhij +� +− 2h0i∂jΠij +(B.7) +Next, we determine the canonical momenta of scalar field theory from the scalar Lagrangian +given in (B.2), +πφ = ∂Lφ +∂ ˙φ += ˙φ +� +1 + h +2 +� ++ h00 ˙φ + h0i∂iφ = ˙φ +� +1 + h00 + hii +2 +� ++ h0i∂iφ. +(B.8) +We can invert (B.8) to determine ˙φ in terms of canonical variables, which will be useful to +obtain the scalar Hamiltonian +˙φ = πφ − h0i∂iφ +� +1 + h00+hii +2 +� +(B.9) +We can now obtain the Hamiltonian for the scalar field by Legendre transforming the scalar +Lagrangian, i.e. +Hφ = πφ ˙φ − Lφ. +Substituting equations (B.9) and (B.2) in the Legendre +transform, we obtain: +Hφ = E − h00 +2 E + h0iπφ∂iφ − hk +k +2 +�π2 +φ +2 − (∇φ)2 +2 +− m2φ2 +2 +� +− 1 +2hij∂iφ∂jφ + O(h2) (B.10) +where the energy E is given by +E = π2 +φ +2 + (∇φ)2 +2 ++ m2φ2 +2 +. +(B.11) +Consequently, from equations (B.7) and (B.10), and taking into account the primary con- +straints (B.6), the full Hamiltonian is given by: +Htot = Hg + Hφ + voΠ00 + viΠ0i +(B.12) +– 27 – + +Closure of constraints and the Dirac matrix +Let us now find the secondary constraints: +χm +0 = +� +Π00, +� +ddx Htot +� += χ0 + 1 +2E +χm +i = +� +Π0i, +� +ddx Htot +� += χi + πφ∂iφ +(B.13) +where χ0 and χi are the secondary constraints without matter. They are given by: +χ0 = 1 +κ2 +� +∂2 +i hk +k − ∂i∂jhij +� +χi = −2∂jΠij +(B.14) +Next, we compute the tertiary constraints: +ξm +0 = +� +χm +0 , +� +ddx Htot +� += +� +χ0, +� +ddx Htot +� ++ 1 +2 +� +E, +� +ddx Htot +� += ξ0 + 1 +2 +� +E, +� +ddx Hφ +� += −∂i∂jΠij + 1 +2∂i(πφ∂iφ) = −∂iχm +i ≈ 0 +(B.15) +Next, we compute the commutator of χm +i with the Hamiltonian. +ξm +i = +� +χm +i , +� +ddx Htot +� += +� +χi, +� +ddx Htot +� ++ +� +πφ∂iφ, +� +ddx Htot +� += +� +πφ∂iφ, +� +ddx Hφ +� += πφ∂iπφ + ∂iφ(∂2 +kφ − m2φ) +(B.16) +At the perturbative level, it seems that the constraint algebra does not close. This can be +seen from the fact that since ξm +i +is non-zero, its Poisson bracket with Hamiltonian we obtain +a non-zero answer. Further it also seems that the constraints (χm +0 and χm +i ) are no longer first +class as well since {χm +0 (x), χm +i (y)} = {T00(x), T0i(y)} ̸= 0. +However, this is a consequence of doing perturbation theory incorrectly. As an example, for +the commutator {χm +0 (x), χm +i (y)} without matter, the gravity part of the constraints commute. +However, if we keep the full non-linear correction in the gravity part of the Lagrangian, then +the gravity part of the constraints does receive corrections, which then makes the constraints +first class. +B.2 +Minimally coupled matter to massive graviton +We again minimally couple the scalar field to gravity but with the Fierz Pauli action (4.1). As +a consequence, the total action is given by: +S = Sφ + Sg +(B.17) +where Sg now denotes the Fierz Pauli massive gravity action. +– 28 – + +Momenta and Hamiltonian +In presence of matter, the full Hamiltonian is given by: +Htot = Hg + Hφ + voΠ00 + viΠ0i +(B.18) +where Hφ is given in (B.10). The addition of matter does not affect the primary constraints, +and consequently, they are still given by (3.2). The stability of primary constraints leads to +secondary constraints, which are given by +χm +0 = +� +Π00, +� +ddx Htot +� += χ0 + 1 +2E +χm +i = +� +Π0i, +� +ddx Htot +� += χi + πφ∂iφ +(B.19) +where χ0 and χi denote secondary constraints without matter and are given by +χ0 = 1 +κ2 +� +(∂2 +i − m2)hk +k − ∂i∂jhij +� +, +χi = −2 +� +∂jΠij + m2 +κ2 h0i +� +(B.20) +Next, we demand the stability of secondary constraints, which give rise to the tertiary con- +straints: +ξm +0 = +� +χm +0 , +� +ddx Htot +� += +� +χ0, +� +ddx Htot +� ++ 1 +2 +� +E, +� +ddx Htot +� += ξ0 + 1 +2 +� +E, +� +ddx Hφ +� += ξ0 + 1 +2∂i(πφ∂iφ) +(B.21) +where ξ0 is again the constraint without matter and it is given by: +ξ0 = −∂i∂jΠij + +m2 +D − 2Πk +k − 2m2 +κ2 ∂ihi0. +(B.22) +Hence the constraint ξm +0 , is given by: +ξm +0 ≡ ξ′ = −∂i∂jΠij + +m2 +D − 2Πk +k − 2m2 +κ2 ∂ihi0 + 1 +2∂i(πφ∂iφ) += 1 +2∂iχm +i + m2 +κ2 +� +κ2 +Πk +k +D − 2 − ∂ih0i +� +≈ m2 +� Πk +k +D − 2 − ∂ih0i +κ2 +� +(B.23) +Next, we compute the commutator of χm +i with the Hamiltonian (B.18). +ξm +i = +� +χm +i , +� +ddx Htot +� += +� +χi, +� +ddx Htot +� ++ +� +πφ∂iφ, +� +ddx Htot +� += 2m2 +κ2 (∂jhji − ∂ih − vi) + +� +πφ∂iφ, +� +ddx Hφ +� += 2m2 +κ2 (∂jhji − ∂ih − vi) + Bi +(B.24) +where Bi = +� +πφ∂iφ, +� +ddx Hφ +� +. We can solve for vi by demanding ξm +i +equals zero, which gives +us +vi = ∂jhji − ∂ih + κ2 +2m2Bi, +(B.25) +– 29 – + +and where upto the quadratic order in matter fields, Bi is given by: +Bi = πφ∂iπφ + ∂iφ(∂2 +kφ − m2φ) +(B.26) +Next, we compute the quartic constraints by computing the commutator of ξm +0 with Hamiltonian +(B.18). +˜ξm +0 = +� +ξm +0 , +� +ddx Htot +� += +m2 +D − 2χ0 + m4 +κ2 +D − 1 +D − 2h − 1 +2∂iBi +≈ ˜ξ0 − +m2 +2(D − 2)E − 1 +2∂iBi +(B.27) +where ˜ξ0 is given by +˜ξ0 = m4 +κ2 +D − 1 +D − 2h +Using the continuity equation, we have +∂iBi = ∂0∂iT0i = ∂2 +t T00, +(B.28) +Next, we find the Poisson bracket of the above constraints with total Hamiltonian: +� +˜ξm +0 , +� +ddx Htot +� += +� +˜ξ0, +� +ddx Htot +� +− 1 +2∂i +� +Bi, +� +ddx Htot +� ++ +m2 +2(D − 2) +� +E, +� +ddx Htot +� +(B.29) +Again we can solve for v0 by demanding the above equation to be zero. +Various non-zero +elements of the constraint matrix are given below: +{Π00(x), ˜ξm +0 (y)} = m4 +κ2 +D − 1 +D − 2 δ(x − y), +{Π0i(x), χm +j (y)} = 2m2 +κ2 δij δ(x − y) +{Π0i(x), ξ′(y)} = −m2 +κ2 ∂iδ(x − y), +{χm +0 (x), χm +i (y)} = 2m2 +κ2 ∂iδ(x − y) + 1 +2Qi +{χm +0 (x), ξ′ +0(y)} = m2 +κ2 +� +∂2 +i − D − 1 +D − 2m2 +� +δ(x − y) +{χm +i (x), �ξm +0 (y)} = 2m4 +κ2 +D − 1 +D − 2∂iδ(x − y) + 1 +2 +� +∂2 +t + +m2 +D − 2 +� +Qi +{ξ′ +0(x), �ξm +0 (y)} = −m6 +κ2 +�D − 1 +D − 2 +�2 +δ(x − y), +{χm +0 (x), χm +0 (y)} = 1 +4P +{χm +i (x), χm +j (y)} = Rij, +{�ξm +0 (x), �ξm +0 (y)} = −1 +4 +� +∂2 +t + +m2 +D − 2 +�2 +P +{χm +0 (x), �ξm +0 (y)} = −1 +4 +� +∂2 +t + +m2 +D − 2 +� +P +(B.30) +– 30 – + +where +P = {T00(x), T00(y)} = +� +Πφ(y)∂φ(x) +∂xi +− Πφ(x)∂φ(y) +∂yi +� ∂ +∂xiδ(x − y) +Qi = {T00(x), T0i(y)} = +� +∂iφ∂kφ ∂ +∂xk − π2 +φ +∂ +∂xi + m2φ∂iφ +� +δ(x − y) +Rij = {T0i(x), T0j(y)} = +� +Πφ(x)∂φ(y) +∂yj +∂ +∂xi − Πφ(y)∂φ(x) +∂xi +∂ +∂yj +� +δ(x − y) +(B.31) +By computing the inverse of the constraint matrix, we notice that we obtain the following type +of inverse derivative dependence in Dirac brackets: +1 +m2 − ∂i∂j(RijP + QiQj) +(B.32) +The above constraint algebra fails to close due to the presence of matter energy momentum +tensor on the RHS of the algebra. +This can be seen by computing the Poisson bracket of +RHS of any of the constraints above with the Hamiltonian. Since {P, Qi} (and other such +combinations) is non-zero, the above algebra is not stable under Hamiltonian evolution. +This extra term in the denominator of (B.32) is seemingly an artifact of perturbation +theory. Since the constraints do not close now, they pose an inconsistency in the counting of +the degrees of freedom. Consequently, we expect the extra term ∂i∂j(RijP + QiQj) to go away +as for the massless case upon the inclusion of higher order corrections [19–22]. +C +Massive gravity constraints by substitution +Let us look at a different way to compute Dirac brackets, where we substitute for some of the +constraints. This procedure was used in [18]. Notice that the metric component h0i appears +quadratically in the Fierz-Pauli lagrangian given in equation (4.1). Hence we can just solve +for h0i using its equation of motion and substitute it back in the Lagrangian. This is the key +difference from our previous treatment of constraints.The h0i EOM is given by: +h0i = − 1 +m2∂jΠji +(C.1) +This equation can also be obtained by setting the constraints Ci (given in (4.4)) to zero. Now +we can substitute h0i in the massive gravity lagrangian and then solve for constraints of the +corresponding system. The Hamiltonian of this system is given by: +Hg = κ2 +�Π2 +ij +2 − +Π2 +ii +2(D − 2) +� ++ 1 +κ2 +�1 +2∂khij∂khij − ∂ihjk∂jhik + ∂ihij∂jhk +k − 1 +2∂ihj +j∂ihk +k +1 +2m2(hijhij − h2 +kk) − m2h2 +0i − h00 +� +∂2 +khi +i − ∂i∂jhij − m2hk +k +�� ++ 1 +m2(∂jΠij)2 +(C.2) +– 31 – + +Since Π00 = 017, this system has one secondary constraint given by: +Φg +1 ≡ (∂i∂i − m2)hj +j − ∂i∂jhij +(C.3) +One can readily compute the tertiary constraint : +Φg +2 ≡ {Hg, Φg +1} = +m2 +D − 2Πii + ∂i∂jΠij. +(C.4) +The stability of the above constraint under time evolution gives us the following further con- +straint: +Φg +3 ≡ {Hg, Φg +2} ≈ −D − 1 +D − 2m2h +(C.5) +where h = −h00 + hk +k. These set of constraints form a closed algebra. The constraint matrix is +now a 4 ∗ 4 matrix whose various non-zero elements are given by +{Π00(p), Φg +3(q)}P.B. = −m2D − 1 +D − 2δD−1(p − q), +{Φg +2(p), Φg +1(q)}P.B. = −m4D − 1 +D − 2δD−1(p − q), +{Φg +1(p), Φg +3(q)}P.B. = 0, +{Φg +2(p), Φg +3(q)}P.B. = m4 +�D − 1 +D − 2 +� � +m2D − 1 +D − 2 − p2 +� +δD−1(p − q). +(C.6) +Hence the inverse of Dirac Matrix C−1(p) is given by: +C−1(p) = 1 +m4 +d − 1 +d +� +� +� +� +� +0 +dm4 +d−1 − p2m2 0 m2 +p2m2 − dm4 +d−1 +0 +−1 0 +0 +1 +0 +0 +−m2 +0 +0 +0 +� +� +� +� +� δd(p − q) +(C.7) +where d = D − 1 is the dimension of the Cauchy slice. As expected, the above matrix is just a +sub-matrix of (4.10) (up to a factor of m2 18). +We can now use this matrix to define the Dirac bracket. +Since the inverse constraint +matrix does not contain any derivatives in the denominator, using the analysis similar to the +one discussed in §5.2, one can readily see that the Poisson bracket of boundary operator H∂ +with any bulk insertion O(x) is zero. +Why substitution works classically? +Roughly our action is of the form +L = L0(xi, ˙xi) + m2h2 +0i + Xh0i +(C.8) +17Since h0i is no longer a degree of freedom of the system, the corresponding momenta Π0i does not exist. +18This factor is different because of the difference in the definition of tertiary constraint. +– 32 – + +such that X is a function of xi, ˙xi. In the case of standard gravity with m = 0, we have the +constraint X = 0, with h0i acting as a Lagrange multiplier. When m2 ̸= 0, we can rewrite the +action as +L = L0(xi, ˙xi) + m2 +� +h0i + X +2m2 +�2 +− X2 +4m2. +(C.9) +Note that action is decomposed into a separate part for the h0i field, and a part containing +L0(xi, ˙xi). Setting the positive definite second term in above equation to zero gives us the +equation of motion for h0i. One can now always redefine the h0i field independently of xi, ˙xi +H0i = h0i + X +2 +(C.10) +such that there are no terms coupling the fields xi and H0i. Thus we can independently minimize +H0i without interfering with the variations of L0(xi, ˙xi) − X2 +4m2. Hence this substitution of the +equation of motion is allowed. This is also confirmed by the counting of degrees of freedom. +Note: This substitution is correct at a classical level but may pose some difficulties in +quantum mechanics when we vary over the whole space of paths with weightage. +References +[1] A. Laddha, S. G. Prabhu, S. Raju, and P. Shrivastava, The Holographic Nature of Null Infinity, +SciPost Phys. 10 (2021) 041, [arXiv:2002.02448]. +[2] C. Chowdhury, V. Godet, O. Papadoulaki, and S. Raju, Holography from the Wheeler-DeWitt +equation, JHEP 03 (2022) 019, [arXiv:2107.14802]. +[3] S. Raju, Lessons from the information paradox, Phys. Rept. 943 (2022) 1–80, +[arXiv:2012.05770]. +[4] S. Raju, Failure of the split property in gravity and the information paradox, Class. Quant. Grav. +39 (2022), no. 6 064002, [arXiv:2110.05470]. +[5] P. Dirac, Lectures on quantum mechanics. Belfer Graduate School Monograph Series. No.2 +(Yeshiva University, New York,). +[6] M. Henneaux and C. Teitelboim, Quantization of gauge systems. 1992. +[7] N. Mukunda and G. Sudarshan, Structure of the Dirac bracket in classical mechanics, J. Math. +Phys. 9 (1967) 411. +[8] M. Fierz and W. Pauli, On relativistic wave equations for particles of arbitrary spin in an +electromagnetic field, Proc. Roy. Soc. Lond. A 173 (1939) 211–232. +[9] A. Higuchi, Forbidden Mass Range for Spin-2 Field Theory in De Sitter Space-time, Nucl. Phys. +B 282 (1987) 397–436. +[10] M. Porrati, No van Dam-Veltman-Zakharov discontinuity in AdS space, Phys. Lett. B 498 +(2001) 92–96, [hep-th/0011152]. +[11] I. I. Kogan, S. Mouslopoulos, and A. Papazoglou, A New bigravity model with exclusively +positive branes, Phys. Lett. B 501 (2001) 140–149, [hep-th/0011141]. +– 33 – + +[12] N. Arkani-Hamed, H. Georgi, and M. D. Schwartz, Effective field theory for massive gravitons +and gravity in theory space, Annals Phys. 305 (2003) 96–118, [hep-th/0210184]. +[13] P. Creminelli, A. Nicolis, M. Papucci, and E. Trincherini, Ghosts in massive gravity, JHEP 09 +(2005) 003, [hep-th/0505147]. +[14] A. Karch and L. Randall, Locally localized gravity, JHEP 05 (2001) 008, [hep-th/0011156]. +[15] M. Porrati, Higgs phenomenon for the graviton in ADS space, Mod. Phys. Lett. A 18 (2003) +1793–1802, [hep-th/0306253]. +[16] A. S. Goldhaber and M. M. Nieto, Photon and Graviton Mass Limits, Rev. Mod. Phys. 82 +(2010) 939–979, [arXiv:0809.1003]. +[17] O. Aharony, A. B. Clark, and A. Karch, The CFT/AdS correspondence, massive gravitons and a +connectivity index conjecture, Phys. Rev. D 74 (2006) 086006, [hep-th/0608089]. +[18] K. Hinterbichler, Theoretical Aspects of Massive Gravity, Rev. Mod. Phys. 84 (2012) 671–710, +[arXiv:1105.3735]. +[19] C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli Action, Phys. Rev. D 82 +(2010) 044020, [arXiv:1007.0443]. +[20] C. de Rham, G. Gabadadze, and A. J. Tolley, Resummation of Massive Gravity, Phys. Rev. Lett. +106 (2011) 231101, [arXiv:1011.1232]. +[21] S. F. Hassan and R. A. Rosen, On Non-Linear Actions for Massive Gravity, JHEP 07 (2011) +009, [arXiv:1103.6055]. +[22] S. F. Hassan and R. A. Rosen, Resolving the Ghost Problem in non-Linear Massive Gravity, +Phys. Rev. Lett. 108 (2012) 041101, [arXiv:1106.3344]. +[23] M. Blake, D. Tong, and D. Vegh, Holographic Lattices Give the Graviton an Effective Mass, +Phys. Rev. Lett. 112 (2014), no. 7 071602, [arXiv:1310.3832]. +[24] C. de Rham, Massive Gravity, Living Rev. Rel. 17 (2014) 7, [arXiv:1401.4173]. +[25] F. Capela and G. Nardini, Hairy Black Holes in Massive Gravity: Thermodynamics and Phase +Structure, Phys. Rev. D 86 (2012) 024030, [arXiv:1203.4222]. +[26] M. Ghodrati, Complexity growth in massive gravity theories, the effects of chirality, and more, +Phys. Rev. D 96 (2017), no. 10 106020, [arXiv:1708.07981]. +[27] G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, +arXiv:1905.08255. +[28] A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao, The Page curve of Hawking radiation +from semiclassical geometry, JHEP 03 (2020) 149, [arXiv:1908.10996]. +[29] A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, The entropy of bulk quantum fields and +the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063, [arXiv:1905.08762]. +[30] G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, Replica wormholes and the black hole +interior, arXiv:1911.11977. +[31] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, Replica Wormholes +– 34 – + +and the Entropy of Hawking Radiation, JHEP 05 (2020) 013, [arXiv:1911.12333]. +[32] H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas, and S. Shashi, +Information Transfer with a Gravitating Bath, SciPost Phys. 10 (2021), no. 5 103, +[arXiv:2012.04671]. +[33] H. Geng and A. Karch, Massive islands, JHEP 09 (2020) 121, [arXiv:2006.02438]. +[34] C. Chowdhury, O. Papadoulaki, and S. Raju, A physical protocol for observers near the +boundary to obtain bulk information in quantum gravity, SciPost Phys. 10 (2021), no. 5 106, +[arXiv:2008.01740]. +[35] S. Banerjee, J.-W. Bryan, K. Papadodimas, and S. Raju, A toy model of black hole +complementarity, JHEP 05 (2016) 004, [arXiv:1603.02812]. +[36] S. Raju, A Toy Model of the Information Paradox in Empty Space, SciPost Phys. 6 (2019), no. 6 +073, [arXiv:1809.10154]. +[37] T. Chakraborty, J. Chakravarty, and P. Paul, Monogamy paradox in empty flat space, Phys. Rev. +D 106 (2022), no. 8 086002, [arXiv:2107.06919]. +[38] R. de Mello Koch and G. Kemp, Holography of Information in AdS/CFT, arXiv:2210.11066. +[39] S. B. Giddings, On the questions of asymptotic recoverability of information and subsystems in +quantum gravity, JHEP 08 (2022) 227, [arXiv:2112.03207]. +[40] C. Chowdhury and O. Papadoulaki, Recovering information in an asymptotically flat spacetime +in quantum gravity, arXiv:2203.07449. +[41] E. Bahiru, A. Belin, K. Papadodimas, G. Sarosi, and N. Vardian, State-dressed local operators in +AdS/CFT, arXiv:2209.06845. +[42] D. Buchholz, PRODUCT STATES FOR LOCAL ALGEBRAS, Commun. Math. Phys. 36 +(1974) 287–304. +[43] J. Kluson, Note About Hamiltonian Structure of Non-Linear Massive Gravity, JHEP 01 (2012) +013, [arXiv:1109.3052]. +[44] R. Jalali and A. Shirzad, Hamiltonian structure of Fierz-Pauli gravitons, partially massless fields +and gauge symmetry, Nucl. Phys. B 976 (2022) 115711, [arXiv:2010.04503]. +[45] O. Rodríguez-Tzompantzi, Canonical description of exotic general massive gravity, Eur. Phys. J. +C 82 (2022), no. 5 482, [arXiv:2201.03151]. +[46] S. Deser, Unconstrained canonical action for, and positive energy of, massive spin 2, Can. J. +Phys. 93 (2015), no. 4 395–397, [arXiv:1407.7178]. +[47] W. Donnelly and S. B. Giddings, Diffeomorphism-invariant observables and their nonlocal +algebra, Phys. Rev. D 93 (2016), no. 2 024030, [arXiv:1507.07921]. [Erratum: Phys.Rev.D 94, +029903 (2016)]. +[48] A. Ashtekar, Asymptotic quantization of the gravitational field, Phys. Rev. Lett. 46 (Mar, 1981) +573–576. +[49] A. Ashtekar, Radiative Degrees of Freedom of the Gravitational Field in Exact General +– 35 – + +Relativity, J. Math. Phys. 22 (1981) 2885–2895. +[50] A. Ashtekar, M. Campiglia, and A. Laddha, Null infinity, the BMS group and infrared issues, +Gen. Rel. Grav. 50 (2018), no. 11 140–163, [arXiv:1808.07093]. +[51] H. Bondi, Gravitational Waves in General Relativity, Nature 186 (1960), no. 4724 535–535. +[52] H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, Gravitational waves in general +relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) +21–52. +[53] R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851–2864. +[54] R. K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat +space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103–126. +[55] A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, +arXiv:1703.05448. +[56] G. Compère and A. Fiorucci, Advanced Lectures on General Relativity, arXiv:1801.07064. +[57] R. L. Arnowitt, S. Deser, and C. W. Misner, The Dynamics of general relativity, Gen. Rel. Grav. +40 (2008) 1997–2027, [gr-qc/0405109]. +[58] T. Regge and C. Teitelboim, Role of Surface Integrals in the Hamiltonian Formulation of +General Relativity, Annals Phys. 88 (1974) 286. +[59] R. Haag, Local quantum physics: Fields, particles, algebras. 1992. +[60] C. J. Fewster, The split property for quantum field theories in flat and curved spacetimes, 2016. +[61] H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas, and S. Shashi, +Inconsistency of Islands in Theories with Long-Range Gravity, arXiv:2107.03390. +[62] K. Papadodimas and S. Raju, Local Operators in the Eternal Black Hole, Phys. Rev. Lett. 115 +(2015), no. 21 211601, [arXiv:1502.06692]. +[63] K. Papadodimas and S. Raju, Remarks on the necessity and implications of state-dependence in +the black hole interior, Phys. Rev. D 93 (2016), no. 8 084049, [arXiv:1503.08825]. +[64] J. Chakravarty, Overcounting of interior excitations: A resolution to the bags of gold paradox in +AdS, JHEP 02 (2021) 027, [arXiv:2010.03575]. +[65] J. Chakravarty, Small corrections to semiclassical gravity and their role in information +paradoxes, arXiv:2105.09924. +[66] J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, D. Stanford, +A. Streicher, and M. Tezuka, Black Holes and Random Matrices, JHEP 05 (2017) 118, +[arXiv:1611.04650]. [Erratum: JHEP 09, 002 (2018)]. +– 36 – + diff --git a/R9AzT4oBgHgl3EQfJPtD/content/tmp_files/load_file.txt b/R9AzT4oBgHgl3EQfJPtD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1205e5864ebd50986887f82c6b25d5df73e84a8b --- /dev/null +++ b/R9AzT4oBgHgl3EQfJPtD/content/tmp_files/load_file.txt @@ -0,0 +1,1226 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf,len=1225 +page_content='Holography of information in massive gravity using Dirac brackets Joydeep Chakravarty,1,2 Diksha Jain,3 Akhil Sivakumar2,4 1McGill University 845 Sherbrooke Street West, Montreal H3A 0G4, Canada 2International Centre for Theoretical Sciences (ICTS-TIFR) Shivakote, Hesaraghatta, Bangalore 560089, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 3Tata Institute of Fundamental Research Dr Homi Bhabha Road, Navy Nagar, Mumbai, 400005, India 4Asia Pacific Center for Theoretical Physics (APCTP) San 31, Hyoja-dong, Nam-gu, Pohang 790-784, South Korea E-mail: joydeep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='chakravarty@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='mcgill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='ca, diksha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2012jain@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='com, akhil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='sivakumar@apctp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='org Abstract: The principle of holography of information states that in massless gravity, it is possible to extract bulk information using asymptotic boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In our work, we study this principle in a linearized setting about empty flat space and formulate it using Dirac brackets between boundary Hamiltonian and bulk operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We then address whether the storage of bulk information in flat space linearized massive gravity resembles that of massless gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' For linearized massless gravity, using Dirac brackets, we recover the necessary criteria for the holography of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In contrast, we show that the Dirac bracket of the relevant boundary observable with bulk operators vanishes for massive gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We use this important distinction to outline the canonical Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This leads to split states, and consequently, one cannot use asymptotic boundary observables to extract bulk information in massive gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We also argue the split property directly without an explicit reference to the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The result reflects that we can construct local bulk operators in massive gravity, which are obscured from boundary observables due to the lack of diffeomorphism invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Our analysis sheds some light on evaporating black holes in the context of the islands proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='01075v1 [hep-th] 3 Jan 2023 Contents 1 Introduction 1 2 Physical observables and Dirac brackets 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1 Constraints and Dirac brackets 4 3 Linearized massless gravity: Dirac matrix 6 4 Linearized massive gravity: Dirac matrix 9 5 Boundary observables and Dirac brackets 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1 Boundary observables and Dirac brackets for massless gravity 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2 Boundary Observables and Dirac brackets for massive gravity 12 6 Vacua structure and split states 13 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1 Split states in electrodynamics 13 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2 Hilbert space and vacua structure in flat space gravity 15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3 Split states in massless gravity 17 7 Split states in massive gravity 18 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1 Split states from the vacuum structure 18 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2 Split states without the vacuum structure 19 8 Conclusion and discussion 21 A Dirac brackets for electrodynamics 23 B Constraints in linearized gravity with matter 26 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1 Massless Gravity with minimally coupled matter 26 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2 Minimally coupled matter to massive graviton 28 C Massive gravity constraints by substitution 31 1 Introduction The question of whether information about a bulk state can be extracted using boundary operators in a theory of gravity is an important one [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In this light, our motivation for this work is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Since a theory of gravity is a constrained system, it is only natural to – 1 – ask whether such statements can be understood using the Dirac bracket formalism [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The second goal is to use this formalism and apply it to Fierz Pauli massive gravity [8], which is an interesting modification to gravity and has been a subject of recent interest (see [9–26] for a sampling of works and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Our chief motivation lies in the recent discussions of massive gravity in the context of islands for evaporating black holes [3, 27–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Whether information is holographically stored at the boundary is addressed by the following question: given access to asymptotic boundary operators, can we precisely determine the bulk state?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This version of holography of information exists in massless gravity and is an essential consequence of the Gauss constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The crucial ingredient involved here is the boundary Hamiltonian, using which one can construct boundary operators that probe bulk physics [1– 4, 34–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Related works discussing the localization of information in massless gravity are [38–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In massless gravity, the principle of holography of information implies that specifying a bulk state |ψ⟩ outside a bounded region B uniquely fixes it inside B [2, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' As a result, it is convenient to introduce split states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=', states which can be arbitrary inside B but are fixed on the complement of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Generally, all field theories apart from massless gravity obey a split property that the set of such states is non-empty [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' However, the holography of information in massless gravity implies that the set of split states is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Keeping this in mind, in our work, we investigate the following objectives: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' To understand holography of information and split property using Dirac brackets by verifying known cases of linearized massless gravity and electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' To determine whether the property holds in massive gravity at a linearized level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Brief description of results In our work, we account for constraints using Dirac brackets and use them to demonstrate the information stored at a linearized level for different constrained theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Some related works on the phase space structure and the computation of Dirac brackets in massive gravity are given in [18, 43–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In §2, we discuss physical observables in constrained theories and briefly review the Dirac bracket formalism for considering the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Based on our discussion of physical observ- ables, we develop a schematic argument for why we may be able to create local bulk operators in massive gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' However, the presence of second-class constraints can render this picture wrong, and we need to verify the same by computing the Dirac bracket between the boundary Hamiltonian and an arbitrary bulk operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We also argue that flat space massive gravity does not have asymptotic symmetries such as BMS supertranslations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We argue that coupling linearized gravity (both massless and Fierz-Pauli massive gravity) to matter fields introduces inconsistencies in the structure of the constraints, including failure to close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' As a demonstration, see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1 for the case of massless gravity and Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2 for massive gravity, where in both cases, the constraints fail to close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Thus in principle, a – 2 – complete calculation involves taking the full Einstein Hilbert action coupled to matter in the massless case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Similarly, we should couple matter covariantly to the full non-linear action for massive gravity [19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In our work, we argue that the issues regarding split states can be understood even using a linearized analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We show that the Dirac matrix involving only the graviton phase space is sufficient to understand split states, while the remaining Poisson brackets are defined over the phase space of the complete gravity-matter theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In other words, we demand that the brackets between constraints are computed only over the gravity phase space, which allows the constraints to close correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' However, we do not put this restriction while computing the rest of the brackets, where the matter insertions are addressed adequately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Following this restriction, we also comment upon the vacua structure of massive gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Intuitively this restriction is in line with our general expectation that the addition of matter should not drastically change the nature of the gravity constraint structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Analogously in electrodynamics, the constraint analysis with or without including charged matter gives rise to the same Dirac matrix shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In §3 and §4, we compute the Dirac matrix necessary to compute the brackets for massless and massive gravity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Using this, we address the extraction of bulk information using boundary operators for electrodynamics and massless/ massive gravity at leading order in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In §5, we define relevant boundary observables for massless and massive gravity and calculate the Dirac brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We also perform an alternate derivation of the Dirac brackets in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' For electrodynamics, using the Dirac matrix obtained in Appendix A, we find in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1 that one cannot use boundary operators to determine the bulk state, hence obeying the expected split property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' For massless gravity, building upon the computation of the Dirac brackets in §3 and §5, we obtain the necessary conditions for the lack of split states upon taking the Dirac bracket of boundary Hamiltonian with a bulk operator insertion in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' For massive gravity, following §4 and §5, we argue in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2 that the computation of the Dirac bracket between the boundary operator with bulk operators vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Upon quantization, lifting the Dirac bracket to the commutator between relevant operators acting on the Hilbert space implies that the commutator is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Due to this, in §7, we argue that in contrast to the principle of holography of information in massless gravity, we do not have an analogous statement in massive gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We argue this in two different ways: with and without an explicit reference to the Hilbert space of massive gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In §8, we discuss the potential limitations of our work, the implications of our results for evaporating black holes, and list some interesting directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 2 Physical observables and Dirac brackets In gauge theory, there are different ways to address gauge redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The general method to fix the redundancy is by defining gauge invariant observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' A useful subclass is to work – 3 – with gauge invariant observables, which we can construct by fixing a good gauge choice, hence removing the redundancy (up to residual gauge, if any).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In massless gravity, there is a gauge redundancy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=', small diffeomorphisms, which die off at the asymptotic boundary of the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In flat space linearized gravity, these are δhµν = ∂µζν + ∂νζµ + O �� GN � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1) where ζ parametrizes the diffeomorphisms at the linearized order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Hence the construction of physical observables in gravity is accomplished by demanding invariance under small diffeomor- phisms characterized by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1), either by gauge fixing or by defining observables that commute with constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' More generally, there are no local diffeomorphism invariant operators in massless gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We can use gravitational dressing to construct observables invariant under small diffeomor- phisms, where we dress bulk observables to the boundary1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In gauge theories, one can similarly construct similar gauge invariant observables, either by gauge fixing or by defining manifestly gauge invariant observables like Wilson loops2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The Fierz Pauli interaction term in massive gravity explicitly breaks the diffeomorphism invariance given in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Since small diffeomorphisms are no longer a symmetry of massive gravity, we need not define physical observables by methods such as dressing them using the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In flat space massive gravity, since diffeomorphisms are no longer a symmetry, the sub- group of diffeomorphism group generating asymptotic symmetries such as supertranslations are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' From the phase space perspective, the fact that there is no gauge symmetry of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1) for massive gravity is because the constraints of massive gravity are second-class and hence do not have any redundancy in the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This feature contrasts the first-class constraints of massless gravity, which necessitate gauge fixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Thus it naively seems that there can be local bulk observables in massive gravity, which can completely hide from the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Despite the intuition from the gauge-fixing picture, we still need to consider the other second-class constraints for a consistent description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Due to these constraints, bulk observables may not be completely independent of the observables at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In this light, our work aims to understand whether these second-class constraints are sufficient for a boundary observer to fix the bulk state completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1 Constraints and Dirac brackets We will follow [5] in our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Here we will denote our set of constraints as {Φi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Given a Lagrangian L for a constrained system, we have a set of primary constraints {ΦP i }, which are 1See [47] for a detailed construction of gauge invariant operators using dressing in massless gravity and gauge theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 2Note that Wilson loops are not good observables in gravity beyond leading order, since the loops possess stress energy and hence backreact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' However, they can undergo further gravitational dressing at subleading order and become good observables up to that order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 4 – independent relations between the fields h and their canonical momenta Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Let H0 denote the Hamiltonian obtained by taking the Legendre transform of the Lagrangian L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We define the Dirac Hamiltonian to be H = H0 + viΦP i (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2) Recall that the Poisson bracket between two observables F(x) and G(y) is given by {F(x), G(y)} = � dD−1z �δF(x) δh(z) δG(y) δΠ(z) − δG(y) δh(z) δF(x) δΠ(z) � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3) We first need to ensure whether the primary constraints are stable and use the stability to determine the parameters vi from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We check the stability by taking the Poisson brackets of primary constraints with the constrained Hamiltonian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' {ΦP i , H}, which either vanishes or gives us secondary constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Next, we need to check the stability of the secondary constraints, which may give us tertiary constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The process should be repeated for consistency of the constrained system until we have determined all possible constraints {Φi} and fixed the parameters vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We can further classify the set of constraints {Φi} into two subsets: first-class and second- class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Second-class constraints are defined as constraints that do not commute among them- selves i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' {Φs i, Φs j} ̸= 0 on the constrained surface, while first-class constraints are defined as constraints that commute among themselves i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' {Φf i , Φf j } = 0 where we denote the first class constraints by Φf, and the second class constraints by Φs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The presence of first-class constraints in the system indicates the presence of gauge sym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Hence we need to fix a gauge corresponding to each of the first-class constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The set of first-class constraints {Φf i } and the gauge conditions {Gi} together form a system of second-class constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Once we obtain a system of second-class constraints, we can define the Dirac matrix as follows: C (Φi, Φj) ≡ {Φi, Φj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) This matrix is now invertible since any constraint Φi gives non-zero Poisson with at least one other constraint3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We then invert this matrix (not always), thereby obtaining the inverse C−1 (Φi, Φj) C (Φi, Φk) C−1 (Φk, Φj) = δij (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5) With C−1 ij ≡ C−1 (Φi, Φj), the Dirac bracket between two observables F(x1) and G(x2) defined on the phase space is given by {F(x1), G(x2)}D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' = {F(x1), G(x2)} − � y1 � y2 {F(x1), Φi(y1)} C−1 ij (y1, y2) {Φj(y2), G(x2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='6) 3The first class constraints give non-zero Poisson brackets with the gauge constraints – 5 – where we have used the notation � y1 � y2 ≡ � dd−1y1 � dd−1y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Note that in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='6), apart from the first term (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=', the standard Poisson bracket), we also have the second term, which is the contribution due to the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 3 Linearized massless gravity: Dirac matrix Before addressing massive gravity, we will warm up with the Dirac matrix calculation for linearized massless gravity without matter, which will also help contrast results with the massive gravity calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Let us begin with a convenient form for the action of the massless graviton: Lg = 1 κ2 � −1 2∂λhµν∂λhµν + ∂µhνλ∂νhµλ − ∂µhµν∂νh + 1 2∂λh∂λh � + boundary terms (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1) where the coefficient κ2 is given by κ2 = 32πGN, where GN is Newton’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The boundary terms in the Lagrangian (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1) are chosen to simplify the momenta and the constraints, thereby giving us Π00 = Π0i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4 Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1), we compute the canonical momenta corresponding to hµν: Π00 = 0, Π0i = 0 Πij = ∂L ∂ ˙hij = 1 κ2 � ˙hij − ˙hkkδij − 2∂(ihj)0 + 2∂kh0kδij � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2) The first line gives us D primary constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Then the Hamiltonian for massless gravity is given by taking the Legendre transform of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1): H0 = κ2 �Π2 ij 2 − Π2 ii 2(D − 2) � + 1 κ2 �1 2∂khij∂khij − ∂ihjk∂jhik + ∂ihij∂jhk k − 1 2∂ihj j∂ihk k � − 2h0i∂jΠij − h00 � ∇2hi i − ∂i∂jhij � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3) The constraints with Hamiltonian (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3) are given by Π00 = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) Π0i = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5) χ0 = {Π00, Htot} = ∇2hi i − ∂i∂jhij (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='6) χi = {Π0i, Htot} = 2∂jΠij, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='7) Since we have two primary constraints, the Dirac Hamiltonian is given by Htot = H0 + v0Π00 + viΠ0i (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='8) where v0 and vi are undetermined constants that will be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We can check that this system of constraints is first class since their Poisson brackets with themselves and the Hamiltonian vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 4Since we are working in asymptotically flat space, we can ignore possible boundary contributions to the pointwise constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 6 – Gauge choice Given the above first-class constraints, we need to fix the redundancy in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We do that by implementing constraints arising from fixing the gauge (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' small diffeomorphisms) and the undetermined constants v0 and vi in the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' A good gauge choice is fixing them so that the gauge constraints are orthogonal to the set of first-class constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Thus a natural guess for gauge conditions is the following5: G0 : h00 = 0, Gi : h0i = 0, K0 : Πk k D − 2 = 0, and Kj : ∂ihij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='9) In the rest of this section, we will use this choice to implement the Dirac procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Dirac brackets Given the set of gauge conditions in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='9), we need to ensure their stability under time evolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=', whether the above constraints give rise to new constraints after time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' {G0, Htot} = v0 {Gi, Htot} = vi {K0, Htot} = −h00 ≈ 0 {Kj, Htot} = ∂iΠij − ∂jΠk k D − 2 + 2∂j∂ih0i ≈ 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10) where ≈ denotes that the equation is valid on the constraint surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10), we see that a consistent choice of implementing the Dirac procedure is by setting v0 = vi = 0 since, in this case, we do not get any new constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' From the perspective of counting degrees of freedom, we now have 4D second class constraints on an originally D(D + 1) dimensional phase space, thereby reducing the phase space dimensionality to D(D−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This reduction is consistent with the fact that the graviton has D(D−3) 2 degrees of freedom6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In hindsight, we will find that the above choice of gauge conditions is designed such that each gauge condition gives a non-zero commutator with exactly one of the first-class constraints, thereby helping us obtain a simpler yet non-singular Dirac matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Specifically, the non-zero elements of the constraint matrix are given by: {Π00(x), G0(y)} = −δD−1(x − y) {Π0i(x), Gj(y)} = −δi jδD−1(x − y) {χ0(x), K0(y)} = ∇2δD−1(x − y) {χi(x), Kj(y)} = � δij∇2 + ∂i∂j � δD−1(x − y) {K0(x), Ki(y)} = 1 D − 2∂iδD−1(x − y) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='11) 5The numerical multiplicative factor in K0 is chosen for later convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 6The degrees of freedom in massless gravity in D-dimensions can be found out by counting the symmetric traceless representations of the little group SO(D − 2), giving rise to D(D−3) 2 polarizations of the standard graviton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Note here that D ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 7 – For later convenience, we will rename the constraints as follows: C0 : χ0, Ci : χi, CD : Π00 CD+i : Π0i, C2D : K0, C2D+i : Ki, C3D : G0, C3D+i : Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='12) In this new notation, we label the constraint matrix as Cab = {Ca, Cb} where a and b run from 0 · · · 4D − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Writing the matrix using the representation in the momentum space, we obtain the following: C(p) = � ������������� 0 0j 0 0j −p2 0j 0 0j 0i 0i j 0i 0i j 0i −(pipj + p2δi j) 0i 0i j 0 0j 0 0j 0 0j −1 0j 0i 0i j 0i 0i j 0i 0i j 0i −δi j p2 0j 0 0j 0 ipj D−2 0 0j 0i (pipj + p2δi j) 0i 0i j ipj D−2 0i j 0i 0i j 0 0j 1 0j 0 0j 0 0j 0i 0i j 0i δi j 0i 0i j 0i 0i j � ������������� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='13) where we have used raised (lowered) indices on the matrix elements to abbreviate entries worth a column (row) array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Since the matrix given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='13) is non-singular, we can use it to compute the inverse matrix C−1(p) = 1 2p2 � ������������� 0 1 D−2 ipj p2 0 0j 2 0j 0 0j 1 D−2 ipi p2 0i j 0i 0i j 0i 2δi j − pipj p2 0i 0i j 0 0j 0 0j 0 0j 2p2 0j 0i 0i j 0i 0i j 0i 0i j 0i 2p2δi j −2 0j 0 0j 0 0j 0 0j 0i −2δi j + pipj p2 0i 0i j 0j 0i j 0i 0i j 0 0j −2p2 0j 0 0j 0 0j 0i 0i j 0i −2p2δi j 0i 0i j 0i 0i j � ������������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='14) Notice that the inverse of the constraint matrix is non-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Such non-localities are essential ingredients of a gauge invariant theory and encodes the structure of its Gauss law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We will later find that this feature gives rise to the property that the energy of field excitations within a spatial region is detectable from the boundary of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This concludes our analysis of the Dirac matrix for linearised gravity without matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 8 – 4 Linearized massive gravity: Dirac matrix We will now move on to computing the Dirac matrix for massive gravity without matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The Fierz-Pauli action for massive graviton is given by: Lg = 1 κ2 � −1 2∂λhµν∂λhµν + ∂µhνλ∂νhµλ − ∂µhµν∂νh + 1 2∂λh∂λh − 1 2m2(hµνhµν − h2) � + boundary terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1) Again, as in the massless case, we have chosen boundary terms such that Π00 = Π0i = 0 and κ2 = 32πGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In addition to the massless gravity Lagrangian, we now have the Fierz Pauli coupling term, with m denoting the mass of the graviton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We can easily extend our analysis from the massless case to the massive case and similarly determine the remaining canonical momenta and the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Since the kinetic part of the Lagrangian remains the same, the canonical momenta of the massive case are the same as for the massless case and are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The massive gravity Hamiltonian is given by: Hg = κ2 �Π2 ij 2 − Π2 ii 2(D − 2) � + 1 κ2 �1 2∂khij∂khij − ∂ihjk∂jhik + ∂ihij∂jhk k − 1 2∂ihj j∂ihk k 1 2m2(hijhij − hi ihj j) − m2h2 0i − h00 � ∇2hi i − ∂i∂jhij − m2hk k �� − 2h0i∂jΠij (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2) As before, we again have two primary constraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Π00 = Π0i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Using these primary constraints, the Dirac Hamiltonian is given by Htot = Hg + v0Π00 + viΠ0i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3) Constraints and Dirac matrix As for the massless case, we systematically determine the constraints and repeat the Dirac procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Demanding stability of primary constraints under the action of the Hamiltonian, we find the following secondary constraints: C0 = {Π00, Htot} = (∇2 − m2)hj j − ∂i∂jhij Ci = {Π0i, Htot} = ∂jΠji + m2h0i (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) Next, we demand the stability of these secondary constraints under the Hamiltonian and thereby obtain the following: C−1 = {C0, Htot} ≈ m2 � Πk k D − 2 − ∂ihi0 � C−2 = {C−1, Htot} ≈ m4h (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5) where ≈ denotes that the corresponding equation is valid on the constraint surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The Poisson brackets of Ci and C−1 with the Hamiltonian can be set to zero by fixing the Lagrange – 9 – multipliers vi and v0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Thus, we have no tertiary constraints, and the system is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The above procedure leads us to a system of 2(D + 1) second class constraints provided we fix the Lagrange multipliers (v0 and vi) accompanying the primary constraints as follows Htot = Hg + ∂ihi0Π00 + � ∂jhji − ∂ih � Πi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='6) Note that in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5), we have retained the m scaling of these constraints since this allows us to keep track of the m → 0 limit of the Dirac matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In the limit m → 0 only the constraints Ca≥0 are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Therefore the massive theory has two additional constraints to the massless theory, albeit second class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' As a cross-check, the above analysis leads to a correct determination of the degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='7 Next, we define the constraint matrix Cab(x, y) ≡ {Ca(x), Cb(y)} (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='7) where a, b now spans −2, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' , 2D − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We have defined {CD, CD+i} = {Π00, Π0i} and together with the constraints (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5) they generate the constraint matrix C(x − y) = m2 � �������� 0 d d−1m4 0 −m2∂j −m2 0j − d d−1m4 0 −∇2 + dm2 d−1 0j 0 −∂j 0 ∇2 − dm2 d−1 0 ∂j 0 0j −m2∂i 0i ∂i 0i j 0i δi j m2 0 0 0j 0 0j 0i −∂i 0i −δi j 0i 0i j � �������� δd(x − y) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='8) where derivatives are taken with respect to the coordinate x and d = D − 1 denotes the dimension of the Cauchy slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='8 Here the raised and lowered indices denote rows and columns respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Fourier transforming C(x − y), we get the momentum space constraint matrix C(p) = m2 � �������� 0 d d−1m4 0 −m2ipj −m2 0j − d d−1m4 0 p2 + dm2 d−1 0j 0 −ipj 0 −p2 − dm2 d−1 0 ipj 0 0j −m2ipi 0i ipi 0i j 0i δi j m2 0 0 0j 0 0j 0i −ipi 0i −δi j 0i 0i j � �������� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='9) 7For theories with massive graviton, one needs to look at the symmetric traceless representation of the group SO(D − 1), which gives us D2−D−2 2 polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This is valid for D ≥ 2, and in particular massive gravity in three dimensions has a propagating degree of freedom, whereas, in four dimensions, we have five polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 8Note that the factors of m2 in C(x − y) shows that in the limit m → 0, the 2D constraints Ca≥0 are first class, while the remaining two constraints in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5) identically vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 10 – where we have used the momentum space representation of the delta function, (2π)dδd(x−y) = � p eip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='(x−y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The matrix C(p) can be easily inverted to obtain the Dirac constraint matrix C−1(p) = 1 m4 d − 1 d � �������� 0 0 0 0j d d−1 0j 0 0 −1 0j 0 −ipj 0 1 0 ipj −p2 + dm2 d−1 0j 0i 0i ipi 0i j 0i −pipj − dm2 d−1δi j − d d−1 0 p2 − dm2 d−1 0j 0 ipjp2 0i −ipi 0i pipj + dm2 d−1δi j ipip2 0i j � �������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10) The main takeaway from the above analysis is that, unlike in the case of massless gravity, the Dirac matrix of a massive gravity theory has a local expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This observation has important implications for the statement of holography of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In particular, in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2 we demonstrate that in contrast to the situation in massless gravity, massive gravity theories can hide information about bulk operator insertions from boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 5 Boundary observables and Dirac brackets We will now utilize the Dirac matrices obtained for various theories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' electrodynamics, massless gravity, and massive gravity, to calculate the Dirac brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1 Boundary observables and Dirac brackets for massless gravity As in electrodynamics, the relevant boundary operator for massless gravity can be obtained from the Gauss constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The constraints for linearized massless gravity with matter are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1, and from (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='13), the Gauss constraint χm 0 for massless gravity with matter insertion is given by ∇2hk k − ∂i∂jhij = −16πGNT00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1) Given any bounded spatial region V , the Gauss constraint makes it possible to encode the energy of matter fields supported on it � V T00 via an equivalent boundary operator given by: H∂ = 1 16πGN � V dD−1x ∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (∂jhij − ∂ihk k) = 1 16πGN � ∂V dD−2x ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (∂jhij − ∂ihk k) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2) where n denotes the unit normal to the boundary ∂V of V (we review the analogous construction for electrodynamics in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The Hamiltonian of the full theory can be obtained from H∂ by taking the limit where V includes the entire Cauchy slice containing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Let us compute the Dirac bracket for the boundary operator H∂ with some bulk matter insertion O(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Using the gravity constraints (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='12), since H∂ only depends on hij, we see that – 11 – H∂ has nonzero commutators only with the constraint C2D = K0 given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The relevant commutator is given by: {H∂, C2D(y)} = � ddz ∂H∂ ∂hij(z) ∂C2D(y) ∂Πij(z) = − 1 16πGN � V ddx ∇2δd(x − y) = 1 16πGN � V ddx � ddp (2π)d p2eip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (x−y) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3) where we have set d = D − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The Dirac bracket of the boundary operator H∂ with a bulk matter operator insertion O(z) is given by: {H∂, O(z)}D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' = − � ddy ddz′ {H∂, Ca(y)} C−1 ab (y, z′) {Cb(z′), O(z)} = 1 16πGN � V ddx � ddp (2π)d ddq (2π)d ddy ddz′ eip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='(x−y)eiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (y−z′)p2 q2 {χm 0 (z′), O(z)} = 1 16πGN � V ddx {χm 0 (x), O(z)} = � V ddx {T00(x), O(z)} , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) where χm 0 is the Hamiltonian constraint in the presence of matter, given in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In the second step of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4), we have used the fact that only the component C−1 2D 0 of the inverse constraint matrix contributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Notice that in the limit where V approaches the full spatial slice containing it, the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' However, the non-local factor arising from the inverse constraint matrix provides a measured counter-effect which makes (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) valid even for the full spatial slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Thus the Dirac bracket of the boundary Hamiltonian with the observable O(z) is equal to the Poisson bracket of the observable with the integrated Gauss constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This in turn, as expected, is equivalent to the Poisson bracket of O(z) with the matter Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2 Boundary Observables and Dirac brackets for massive gravity Like massless gravity, the relevant boundary Hamiltonian for massive gravity can be obtained from the Gauss constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' From (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='20) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='21), the Gauss constraint χ0 for massive gravity with matter insertion is given by ∇2hk k − ∂i∂jhij = m2hk k − 16πGNT00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5) As in massless gravity, we can integrate the LHS of the Gauss constraint within a spacelike region V and obtain the boundary operator H∂, which is given by H∂ = 1 16πGN � V dD−1x ∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (∂jhij − ∂ihk k) = 1 16πGN � ∂V dD−2x ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (∂jhij − ∂ihk k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='6) From the massive gravity constraints, we see that the boundary Hamiltonian fails to commute only with the constraint C−1 (see Eqn (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The relevant commutator is given by: {H∂, C−1(y)} = � ddz ∂H∂ ∂hij(z) ∂C−1(y) ∂Πij(z) = − m2 16πGN � V ddx ∇2δd(x − y) = m2 16πGN � V ddx � ddp (2π)d p2eip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (x−y) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='7) – 12 – which is identical to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3) up to a factor of m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The Dirac bracket of the boundary operator with a bulk matter operator O(z) is given by: {H∂, O(z)}D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' = − � ddy ddz′ {H∂, Ca(y)} C−1 ab (y, z′) {Cb(z′), O(z)} = d − 1 m2d 1 16πGN � V ddx � ddp (2π)d ddq (2π)dddy ddz′ eip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='(x−y)eiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (y−z′) p2� {C0(z′), O(z)} + iqi{CD+i(z′), O(z)} � = −d − 1 m2d 1 16πGN � V ddx ∇2� {C0(x), O(z)} + ∂i{Π0i(x), O(z)} � = −d − 1 m2d 1 16πGN � ∂V dd−1x ni∂i � {C0(x), O(z)} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='8) In the second equation above, we have identified the only contributing terms to be from C−1 −1 0 and C−1 −1 D+i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In the final step we have neglected the contribution from the C−1 −1 D+i terms as O(z) is assumed to be a pure matter operator with trivial Poisson brackets with Π0i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The result (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='8) differs from that of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) fundamentally due to fact that, unlike in the case of massless gravity, the inverse constraint matrix contributing to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='8) is local, thus allowing us to reduce the Dirac bracket to a pure boundary term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Therefore, when O(z) is taken to be an operator insertion strictly in the bulk, we find that its Dirac bracket with the boundary operator H∂ vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In Appendix C, we treat the constraints of the free theory but substitute the equation of motion of h0i back into the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We argue that at the classical level, the substitution makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We use this to alternatively demonstrate that {H∂, O(z)}D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 6 Vacua structure and split states We will now utilize the Dirac brackets obtained for various theories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' electrodynamics, massless gravity, and massive gravity, to investigate the existence of split states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In order to set up the stage for further discussions of constraints using Dirac brackets and their relation with split states, let us first begin with the case of electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Readers familiar with split states in electrodynamics can directly skip ahead to §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1 Split states in electrodynamics We minimally couple a charged scalar φ to the electrodynamic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The Hamiltonian for this system is given by HJ = � dd−1x � −1 2ΠiΠi + 1 4FijF ij − ∂iΠiA0 + ΠφΠφ∗ − ieA0 (φΠφ − Πφ∗φ∗) + (Diφ)∗Diφ � (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1) – 13 – where Diφ ≡ ∂iφ + ieAiφ is the covariant derivative with respect to the gauge field, and Πi denotes the momentum conjugate to the electrodynamic field, while Πφ denotes the momentum conjugate to the scalar field and is given by Πφ = ˙φ∗ − ieA0φ∗ (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2) In terms of the gauge invariant fields, Πi = Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1), one can compute the Gauss constraint, which is given by: ∂iΠi − J0 = 0 where J0 = −ie (φΠφ − Πφ∗φ∗) is the matter current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' As a consequence, the Gauss constraint implies that given a codimension one spacelike slicing Σ, measuring the integral of the electric field over the boundary gives us the total charge Q = � Σ J0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' � Σ ∂iΠi = � ∂Σ niΠi = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3) where ni is the outward pointing normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The boundary operator in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3) is unique to electrodynamics due to the Gauss constraint and gives us the total charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Now consider some matter insertion in bulk, denoted by the action of an operator O(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We want to determine whether the information content of the insertion can be obtained using a relevant boundary observable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=', the boundary operator defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The computation of the Dirac matrix for electrodynamics, both with and without matter, is performed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Using our analysis there, we are in a position to investigate the Dirac bracket of � Σ ∂iΠi with O(x), which gives us � � Σ ∂iΠi(x), O(z) � D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' = � � Σ ∂iΠi(x), O(z) � − � y1 � y2 � Σ ∂2δ(x − y1) 1 ∂2δ(y1 − y2) � ∂iΠi(y2) − J0(y2), O(z) � (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) We will work with purely matter insertions O(z) in the following discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Then the first Poisson bracket on the RHS of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) is zero, while the second bracket, upon integration by parts and using the Gauss law, takes the form � � Σ ∂iΠi(x), O(z) � D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' = − � Σ � ∂iΠi(x) − J0(x), O(z) � = {Q, O(z)} (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5) where we have used {∂iΠi(x), O(z)} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Thus we obtain an order one contribution due to the matter insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' If we have a chargeless state, the Dirac bracket expectedly vanishes, whereas we obtain a finite contribution for the charged state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 14 – Lifting the Dirac brackets to operators acting on the Hilbert space, equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5) takes the following form 9 � Σ � ∂iΠi(x), O(z) � = [Q, O(z)] (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='6) Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='6) leads to the existence of split states in electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' To see this, note that the order one contribution on RHS is insufficient to specify the state of the bulk on the spacelike slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This is because the boundary operator � Σ ∂iΠi(x) can only measure the charge of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Consequently, there is an infinite degeneracy of states with a given electromagnetic charge, which all evaluate to the same value on the RHS of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In this way, the Gauss constraint in electrodynamics cannot specify the state in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Hence the split property holds since one cannot distinguish bulk states using the relevant boundary operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2 Hilbert space and vacua structure in flat space gravity In this subsection, we will revisit the canonical Hilbert space of asymptotically flat massless gravity and use it to construct the massive gravity Hilbert space analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The Hilbert space will be important in understanding the split property of flat space gravity in the later subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Massless gravity We begin by studying the vacua and boundary operators for flat space massless gravity [48–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The asymptotic symmetries are implemented by subgroups of the diffeomorphism group, such as the BMS group, which generates supertranslations [51–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Supertranslations (in D = 4) are generated using supertranslation charges Qlm constructed by spherically smearing the Bondi mass aspect mB(u, Ω) at I+ −: Qlm = 1 4πGN � √γ d2Ω mB(u = −∞, Ω) Yl,m(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='7) We can separate Qlm into hard and soft components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Thus a specification of vacuum involves annihilation under positive modes of matter and gravity together with the eigenvalue under soft mode: Qlm |0, {s}⟩ = sl,m |0, {s}⟩ (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='8) Here in |0, {s}⟩, the first label denotes the energy, while the second label specifies the super- translation sector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' the eigenvalue under the soft mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Supertranslation sectors serve as different superselection sectors for the theory, and hence the flat space massless gravity Hilbert space is given by H = � {s} H{s} (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='9) 9The lifting from phase space observables to operators acting on the Hilbert space is subject to the assumption that a suitable operator ordering prescription exists which removes any anomalous terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This issue arises not only in electrodynamics but also in our later analysis of gravity, and we discuss more about this in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 15 – The ADM Hamiltonian H∂ defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2) is simply Q00 charge defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='7) [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We now introduce an abstract projector onto the vacua subspace of the massless gravity theory [1, 35]: P0 = lim a→∞ exp (−aH∂) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10) We can also define a more physical projector Pδ which projects onto energies below an IR cutoff δ [37]: Pδ = Θ (δ − H) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='11) In §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3, we will revisit the significance of H∂ and the projectors by computing the Dirac bracket between H∂ and a bulk matter insertion O(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Specifically, the projectors allow us to reconstruct bulk operators using the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Massive gravity In order to construct the Hilbert space of linearized massive gravity, we note the following points: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' From our calculation in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='8), the Dirac bracket of H∂ with a bulk matter insertion O(z) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Upon lifting operators to the phase space, we replace the Dirac bracket with commutators, thereby giving us [H∂, O(z)] = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='12) where H∂ is given by the standard expression: H∂ = 1 16πGN � dD−2x ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (∂jhij − ∂ihk k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='13) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Since asymptotic symmetries are absent in massive gravity, in contrast to the massless theory, the vacua subspace of the flat space massive theory is labelled by a single sector rather than a direct sum over infinite supertranslation sectors as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Using these above points and from equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='12) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='13), we can use the boundary operator H∂ to label the states in the Hilbert space as |E, M⟩ such that H∂ |E, M⟩ = E |E, M⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='14) Here the label E denotes the eigenvalue under H∂, and M is a quantum number that labels the bulk matter and gravity insertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In contrast to the massless case, the second label in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='14) does not denote the supertranslation sector but is closely related to the longitudinal mode of graviton polarization in massive gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Due to the absence of asymptotic symmetries, we expect that the S-matrix defined over the state space {|E, M⟩} is infrared finite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' as expected, there are no infrared divergences in pure massive gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Projectors onto the vacuum: Using H∂, we can again try to construct boundary pro- jectors onto the vacua subspace using the boundary Hamiltonian H∂, as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 16 – The construction works within our linearized analysis since, by definition, the lowest value that H∂ can take is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The boundedness of energy is crucial to defining the projectors at the linearized level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' However, from §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2, the action of the projector P0 in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10) on a generic state does not project onto just the empty state |0, 0⟩, but projects onto a subspace of states labelled by |0, M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10 This is a significant departure from the flat space massless gravity case, and the above vacua structure will be important in addressing the questions regarding split states in massive gravity11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' However, defining projectors in this fashion makes sense only in linearized gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We comment on going beyond our linearized analysis in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3 Split states in massless gravity Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) states that the boundary Hamiltonian H∂ knows about bulk insertions since the Poisson bracket of the matter insertion with the integrated stress energy component T00 is the same as the commutator of the H∂ with the matter insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This is in line with the Hamiltonian constraint analysis in equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='57) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='58) of [2], where expanding the constraint at the second order in perturbation theory, H∂ is related to the bulk energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The bulk energy has a contribution from both gravity and matter, with the matter contribution being T00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In our case, on the RHS of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4), O(z) clicks only with T00 in the Poisson bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Thus the commutator of H∂ with matter insertion gives us the same result as expected from the order-by-order expansion of the Hamiltonian constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Given this consistency check, the statements about holography of information and split states follow as in [1], which we briefly summarize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Using a Reeh Schlieder type argument, one can construct operators QB supported near the boundary, which can act on a supertranslation vacuum |0, {s}⟩ to create any arbitrary state |B, {s}⟩ in the supertranslation sector {s} QB |0, {s}⟩ = |B, {s}⟩ + O �� GN � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='15) Henceforth we will ignore O �√GN � corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Thus using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='15) any hard bulk operator12 O(z) can be written as O(z) = � mn cmn |m, {s}⟩ ⟨n, {s}| = � mn cmn Qm |0, {s}⟩ ⟨0, {s}| Q† n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='16) From equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4), we see that the matter insertion O(z) leaves an imprint on the ADM energy since it does not commute with H∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Hence H∂ can be labelled using the bulk matter- energy, which is positive definite in well-behaved theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Thus one can use H∂ to construct a boundary projector (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10) onto the vacuum of the theory on the lines of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 10This is in striking contrast to the massless gravity case where graviton and bulk matter insertions labelled by M completely fix the boundary ADM energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We do not use M as a separate quantum number in massless gravity since one label is enough to specify the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 11At this stage, the fastidious reader may correctly anticipate that one can exploit the above-mentioned contrasting feature to construct split states in massive gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 12We only consider hard bulk operators here whose action does not change the superselection sector {s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 17 – Using the Fock space representation of the projector (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10), we can write the operator representation in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='16) as O(z) = � mn cmn QmP0 Q† n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='17) which is completely expressed in boundary operators Qm and P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Using the arguments of [3, 4], various statements regarding the holography of information follow from the above representa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Since bulk operators can be written as combinations of boundary operators, there cannot be split states since one can always utilize the boundary expression for the bulk operators to probe bulk physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Thus the decomposition of the massless gravity Hilbert space H into H = HA ⊗ H ¯ A (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='18) is not allowed13, and correspondingly, the notion of split states in massless gravity does not exist [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' One can also use the more physical boundary projector Pδ defined in [37] to express O(z) in terms of boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' However, this is a slightly more difficult task since this involves delicately tuning smearing functions on the complement of the bulk region we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 7 Split states in massive gravity Since the Dirac bracket of H∂ with bulk operator O(z) is zero in massive gravity from equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='8), we can hide any bulk matter insertion O(z) from detection using the boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In this section, we will demonstrate the existence of split states in massive gravity in two ways: firstly by making an explicit reference to the Hilbert space outlined in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2, and secondly by not making an explicit reference to the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1 Split states from the vacuum structure Following our notation introduced in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2, from equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='8) we have [H∂, O(z)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1) which implies that the we have simultaneous eigenstates |E, M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Thus there is no way to define a vacuum projector using our relevant boundary operator H∂, which projects onto the empty state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' with no bulk matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In other words, there are arbitrarily many states with bulk matter insertions in the zero eigenvalue subspace of H∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This leads to split states since from eqn (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1), H∂ or operators constructed using it can never detect matter eigenvalue M in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In other words, one can shield bulk excitations from any possible detection at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Hence there is no holography of bulk information at the boundary, which could be detected using boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Let us now precisely define what we mean by factorization of the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 13The precise sense in which we refer to the factorization of Hilbert space is explained in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 18 – Approximate factorization of Hilbert space The notion of Hilbert space factorization in eqn (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='18) is approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Even if we ignore any constraints, such a factorization is not allowed in a quantum field theory since the energy of such product states lies outside effective field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' To work with such states within effective field theory, we should imagine some spatial separation between the regions A and ¯A larger than the UV length scale of effective field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Formally, introducing the spatial separation and momentarily ignoring the constraints, we can approximately factorize the Hilbert space H upon spatially partitioning flat space into a bounded region A and its complement ¯A as follows H = HA ⊗ H ¯ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2) Then the factorization in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2) implies that spatially partitioned split states of the following form exist in the Hilbert space H: |ψ⟩ = |ψA⟩ ⊗ |ψ ¯ A⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3) In our present case, region A should be thought of as most of the bulk region while the com- plement region ¯A is a small enough region sufficient to include the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Then in massive gravity, states in the Hilbert space spanned by |ψ⟩ = |E, M⟩ can be thought of as split states living in the above approximately factorised Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This is roughly because we can modify the bulk state |ψA⟩ in region A thereby changing the matter quantum number M, while preserving the boundary eigenvalue E in region ¯A at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Since there are no other relevant boundary operators that probe the bulk physics in region A, changing |ψA⟩ has no effect on the state |ψ ¯ A⟩ in region ¯A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Consequently states of the form (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3) are allowed in the Hilbert space of massive gravity taking into account the constraints using Dirac brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2 Split states without the vacuum structure More generally, we do not need to reconstruct the Hilbert space in order to understand the split property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' To see this, let us work with a non gravitational theory first, and outline the split property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Consider a density matrix ρ defined on the spatially partitioned system acting on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We will be working with the algebra of observables A and ¯ A acting on the previously-defined regions A and ¯A respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The algebras A and ¯ A consist of products of bulk operators supported on A and ¯A respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We now look at operators OA ∈ A and O ¯ A ∈ ¯ A, where elements from the algebras mutually commute, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=', [OA, O ¯ A] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Let us consider a bulk observable of the form O = OAO ¯ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The split property [59, 60] implies that the expectation value of operator O can be written as: ⟨O⟩ ≡ Tr (ρ O) = Tr (ρAOA) Tr (ρ ¯ AO ¯ A) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) – 19 – where ρA and ρ ¯ A are density matrices such that they are unconstrained in the traced out regions ¯A and A respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Effectively the density matrix ρ takes the following form ρ = Tr ¯ A (ρA) ⊗ TrA (ρ ¯ A) , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5) which can be seen by substituting (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5) into (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Support of boundary operator H∂ In massless gravity, there are a couple of issues regarding the above construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Firstly, from eqn (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4), we have [H∂, OA] ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='6) Hence the decomposition in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) does not work, since there always exists an operator H∂ living in the region ¯A which can be used to detect observables in A14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Correspondingly, the Hilbert space does not factorize, and the density matrix cannot be written in the form (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5) for massless gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The second issue arises since only small diffeomorphism invariant observables make sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' For instance, diffeomorphism invariant dressed operators obey [OA, O ¯ A] ̸= 0, and hence (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5) do not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' A more precise version involves defining the algebra of observables asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We refer the reader to [1, 4] for a detailed treatment of this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In massive gravity, we do not need to work with asymptotic observables since diffeomor- phism invariance is absent, and hence we can work with the previously defined algebra of observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Since we do not need to dress the observables, we have [OA, O ¯ A] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Also for an observable OA, the boundary operator H∂ simply commutes with the spatially partitioned observables [H∂, OA] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='7) As a consequence, equations (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5) follow, since we cannot use H∂ which is supported in region ¯A, to detect the state supported in region A any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Thus from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5), specifying information in region ¯A does not fix the state in the region A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Consequently, the existence of split states in massive gravity can be understood from lifting the Dirac brackets onto operator commutators acting on the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The appearance of split states is a significant departure from the case of massless gravity, where any matter insertion has a specific signature in correlators involving boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Massive gravity is not constrained enough to detect bulk insertions using boundary observables and their correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' From our analysis in this subsection, it is clear that massive gravity resembles a non-gravitational QFT than massless gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 14From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) since the commutator is non-zero, the ADM Hamiltonian H∂, though supported in the region ¯A, does not belong solely to the algebra ¯ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 20 – 8 Conclusion and discussion In our work, we have computed Dirac brackets in different settings and used them to investigate the issue of split states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' More generally, we have addressed the following question regarding the principle of holography of information: given access to boundary operators, can one use them to identify a generic bulk state?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In linearized massive gravity, using our analysis of the Dirac brackets, it appears that such information is hidden from boundary operators, thereby allowing for split states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We find that the Dirac bracket between the relevant boundary operator from the Gauss constraint and a generic bulk matter insertion is zero for massive gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This is consistent with our intuitively expected picture resulting from the lack of small diffeomorphisms in massive gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Thus one can create local bulk operators which can never be detected using the boundary operator H∂ since the commutator is simply zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This is a significant departure from massless gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We show that this leads to split states, and hence there is no holography of information in linearized massive gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Potential limitations of our analysis Let us now discuss some potential limitations of our analysis: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Regarding the closure of constraints: As discussed in Appendix B, from the per- spective of constraints, one cannot consistently couple matter to the linearized gravity action since the constraint algebra does not close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The failure of linearised gravity-matter constraints to close introduces further constraints on the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Introducing these additional constraints is inconsistent with the degree of freedom counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In contrast, the case of electrodynamics is much simpler, where the Dirac matrix, upon the inclusion of charged matter, is the same as for the uncharged case (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The issues with consistently coupling matter with linearized gravity are an important feature contrary to our naive expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This issue is also demonstrated from the failure to impose ∂µT µν = 0 in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='15 Motivated by electrodynamics, we circumvent this issue by taking the Dirac matrix of linearised gravity without matter to evaluate the Dirac brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This ensures that the algebra closes, and we use this matrix to compute Dirac brackets between observables, with subsequent Poisson brackets defined over the entire matter-gravity phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In other words, we restrict ourselves to the gravity phase space whenever we take the Poisson bracket between two different constraints but otherwise work in the entire matter-gravity phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' A more satisfactory procedure would be to consider the full non-linear action coupled to matter [19–22] and study the Dirac brackets, which is an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 15Very loosely, we can also argue that there should not be any corrections to the linearised Dirac matrix upon including matter since naively coupling matter order by order in perturbation theory is problematic, and as a consequence, the constraints fail to close properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 21 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Holography of information: In massless gravity, the principle of holography of infor- mation is a non-perturbative statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Perturbatively we can demonstrate holography of information about low energy states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' However, for heavy time-dependent states, as stud- ied in [41], within perturbation theory, it may be possible to construct operators which can commute with the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We expect that there exist complicated observables using which we can probe the bulk, which incorporate non-perturbative effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In our work, we restrict our analysis to linearized gravity over the empty flat background and restrict our statements to low energy states about the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Though likely, we are unable to concretely establish whether holography of information in massive gravity is a non-perturbative statement since it is unclear whether the canonical vacuum satisfies necessary properties such as boundedness and whether one can define projectors onto the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Quantization ambiguities: Generally, lifting constraints from the phase space to op- erator equations on the Hilbert space may introduce some corrections to the constraint algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' For instance, we may have ambiguities proportional to δ(0) for first-class con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' If such ambiguities arise, we need to implement a suitable choice of normal ordering that allows us to circumvent them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In our analysis of massive gravity, some simple operator-ordering ambiguities, such as ones resulting from the multiplication of canonical field with momenta, are absent since the constraints are all linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' There are seemingly no such obstructions to quantization at the level of our linearized analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Higgs-type mechanism and localization of information: A straightforward implica- tion regarding the localization of information is as follows: the localization of information on the whole AdS5 boundary is different from the Karch Randall type setups [14], which have a massive graviton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In such setups, the massive graviton arises from a higher dimen- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' However, the crucial feature is that the boundary of the complete theory is not the same as the boundary of the dimensionally reduced theory, hence the difference in the localization of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Another question that may arise is how a possible Higgs mechanism which gives the graviton mass may change the localization of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The naive picture is that breaking the diffeomorphism invariance by such a mechanism changes the localization of information, and consequently, even at the non-perturbative level, one may not be able to recover bulk information using just the boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Such issues still need to be better understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Black hole evaporation We now briefly discuss some other implications of our work regarding black hole evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Our analysis here indicates consistency with the arguments of [61] that massive gravity at a linearized level allows for black hole evaporation using the islands formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This is because – 22 – the Dirac bracket of operator insertions inside the disconnected entanglement wedge with the boundary Hamiltonian is zero, indicating consistency with the subregion duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' However, since the Dirac bracket in massless gravity between the boundary Hamiltonian and the bulk Hamiltonian is not zero, operator insertions inside the entanglement wedge can potentially be detected using the boundary Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Whether our formalism sheds some light to circumvent this obstruction in massless gravity is an interesting question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Other open questions We conclude our work with some other open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' For massless gravity, we observe that the form of the constraint algebra of the linearised theory without matter and the complete non-linear theory with matter look similar, provided we fix the gauge appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' It would be interesting to investigate whether this observation also holds for the case of massive gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Stated differently, the question is whether we expect the form of the constraint algebra of non- linear massive gravity coupled to matter to be similar to the Fierz-Pauli case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Our analysis is plausibly valid for the full non-linear theory coupled to matter in such a case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' A related point is whether our described vacua structure, which depends on our restrictive linearized analysis, generalizes to the non-linear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Given the constraint algebra of the non-linear action, an interesting question is whether a systematic procedure exists to truncate it to the constraint algebra from the quadratic action, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' to the linearized case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' As we argued, to include matter, we need to consider the full non-linear Einstein action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' However, is there any consistent truncation of the full non-linear constraint algebra, which gives us the Dirac matrix of linearized gravity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' A slightly distant avenue is to understand whether there is any natural obstruction to lifting massive gravity phase space observables to state-independent operators [62, 63] and their subsequent dependence on late time effects [3, 64–66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We hope to address some of these issues in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Acknowledgments We thank Suvrat Raju for suggesting the problem and valuable suggestions regarding the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We thank Claudia de Rham and Alok Laddha for useful correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We are also grateful to Sayali Bhatkar, Simon Caron-Huot, Tuneer Chakraborty, Victor Godet and Priyadarshi Paul for related useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We acknowledge support of the Department of Atomic Energy, Government of India, under project number RTI4001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' A Dirac brackets for electrodynamics We will first look at Dirac brackets for electrodynamics without matter and then derive the Dirac brackets after adding in the matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 23 – Electrodynamics without charges We will work with the Lagrangian given by L = −1 4FµνF µν, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1) where Fµν denotes the field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Using this Lagrangian, we arrive at the primary constraint Π0 = 0, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2) where Π0 denotes the standard canonical momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Then the Hamiltonian H0 obtained from the Legendre transformation of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1) is given by H0 = � ddx � −1 2ΠiΠi + 1 4FijF ij − ∂iΠiA0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3) To implement the Dirac bracket procedure, we will first add in the contribution from the primary constraint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' H = H0 + v0Π0, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) where our goal now is to fix v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The condition for the stability of the primary constraint gives us the Gauss constraint, which is a secondary constraint, {Π0, H} = ∂iΠi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5) We find that there are no further tertiary constraints because {∂iΠi, H} = 0 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='6) due to cancellations among terms resulting from integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Consequently, we have v0 = 0 in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' the constrained Hamiltonian is the same as obtained from Legendre trans- formation of the electrodynamics Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Thus we have a system of first class constraints characterized by the Hamiltonian H, and constraints (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Gauge fixing Since we have a first class system, we fix the gauge by putting in gauge conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' A convenient choice is to choose a gauge that is orthogonal to the first class constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In our case, this amounts to A0 = 0 and ∂iAi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='7) We rewrite the system of constraints given by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='7) in the following ordered form C0 = Π0(x) C1 = ∂iΠi(x) C2 = A0(x) C3 = ∂iAi(x) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='8) – 24 – Using the above constraints, we have the following non-zero Dirac brackets {Π0(x), A0(y)} = −δd(x − y) {∂iΠi(x), ∂jAj(y)} = ∇2δd(x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='9) Note that the plus sign in the expression for the second commutator in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='9) arises due to the shifting of derivatives while performing integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Then the constraint matrix with the ordering in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='8) is given by M(x − y) = � � � � � 0 0 −1 0 0 0 0 ∇2 1 0 0 0 0 −∇2 0 0 � � � � � δd(x − y) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10) The inverse matrix of M(x, y) from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10) is given by M −1(x − y) = � � � � � 0 0 1 0 0 0 0 − 1 ∇2 −1 0 0 0 0 1 ∇2 0 0 � � � � � δd(x − y) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='11) Electrodynamics with charges Recall from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1) that the Hamiltonian for a charged scalar coupled to electrodynamics is given by HJ = � ddx � −1 2ΠiΠi + 1 4FijF ij − ∂iΠiA0 + ΠφΠφ∗ − ieA0 (φΠφ − Πφ∗φ∗) + (Diφ)∗Diφ � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='12) Here we again have the primary constraint Π0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' As previously done, we write the constrained Hamiltonian as H = HJ + v0Π0 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='13) The stability of the primary constraint gives us the secondary Gauss constraint, which is given by ∂iΠi − ie (φΠφ + φ∗Πφ∗) ≡ ∂iΠi − J0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='16 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='14) Recall that now the Poisson bracket involves taking derivatives with respect to the scalar field and its conjugate momentum as well, since a complete specification of the phase space involves accounting for the scalar field as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We find that the secondary Gauss constraint is stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' {∂iΠi − J0, H} = 0 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='15) due to cancellations among various terms, and use this to set v0 = 0, similar to the case of free electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 16As a comparison, for fermions, there is no electrodynamic contribution to the matter current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 25 – Gauge fixing and Dirac brackets Since the primary constraint remains unchanged and the secondary Gauss constraint receives an additive scalar contribution plus the contribution from A0, a good orthogonal choice is to choose the same gauge conditions as previously chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This is due to the fact that the extra charged contribution to the Gauss constraint commutes with the choice of gauge, which by construction gives a non-zero commutator with the free part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' As a consequence, we have the constraints C0 = Π0(x) C1 = ∂iΠi(x) − J0 C2 = A0(x) C3 = ∂iAi(x) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='16) which gives us the exact same Dirac matrix as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10) and its inverse in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' B Constraints in linearized gravity with matter In this Appendix, we covariantly couple matter to linearized massless and massive gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We show that the constraints do not close i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=', they become inconsistent with our expected counting of the degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1 Massless Gravity with minimally coupled matter We will now minimally couple a scalar field to massless gravity using the stress energy tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Using this, we will compute the constraints of this theory and calculate the Dirac bracket in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The action of minimally coupled matter to gravity is given by: Sφ = −1 2 � dDx √−g(gµν∂µφ∂νφ + m2φ2) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1) where g is the determinant of the space-time metric and indices µ, ν run from 0 to D − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Expanding the metric about the flat background (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' gµν = ηµν+hµν and hence gµν = ηµν−hµν), at leading order, we obtain: Sφ = � dDx � 1 + h 2 � � −ηµν 2 ∂µφ∂νφ − m2 2 φ2 � + hµν 2 ∂µφ∂νφ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2) where h = Tr(hµν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In terms of the energy-momentum tensor Tµν, above action can be re- written as: Sφ = � dDx � − √−gb 2 hµνT µν(φ, ˙φ) + √−gbLm(φ, ˙φ) � (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3) where Lm(φ, ˙φ) is the free-scalar Lagrangian, gb is the determinant of background metric (which is ηµν in present case) and T µν is given by: T µν = ∂µφ∂νφ − ηµν 2 (∂ρφ∂ρφ + m2φ2) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) – 26 – The massless graviton Lagrangian Lg is given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Hence the total action is given by: S = Sφ + Sg (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5) where Sg denotes the integral of Lg, with the GN dependence restored using an overall multi- plicative factor κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Sg = 1 κ2 � dDx√−gLg Momenta and Hamiltonian Using the combined action in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5), we will now determine the canonical momenta and the Hamiltonian for our scalar-gravity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' From the gravity part we obtain the following ex- pression for the momenta: Π00 = 0, Π0i = 0 Πij = ∂L ∂ ˙hij = 1 κ2 � ˙hij − ˙hkkδij − 2∂(ihj)0 + 2∂kh0kδij � (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='6) From (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='6), we find that we have two primary constraints Π00 and Π0i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The gravity Hamiltonian from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3) is given by: Hg = κ2 �Π2 ij 2 − Π2 ii 2(D − 2) � + 1 κ2 �1 2∂khij∂khij − ∂ihjk∂jhik + ∂ihij∂jhk k − 1 2∂ihj j∂ihk k � − h00 κ2 � ∂2 khi i − ∂i∂jhij � − 2h0i∂jΠij (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='7) Next, we determine the canonical momenta of scalar field theory from the scalar Lagrangian given in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2), πφ = ∂Lφ ∂ ˙φ = ˙φ � 1 + h 2 � + h00 ˙φ + h0i∂iφ = ˙φ � 1 + h00 + hii 2 � + h0i∂iφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='8) We can invert (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='8) to determine ˙φ in terms of canonical variables, which will be useful to obtain the scalar Hamiltonian ˙φ = πφ − h0i∂iφ � 1 + h00+hii 2 � (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='9) We can now obtain the Hamiltonian for the scalar field by Legendre transforming the scalar Lagrangian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Hφ = πφ ˙φ − Lφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Substituting equations (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='9) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2) in the Legendre transform, we obtain: Hφ = E − h00 2 E + h0iπφ∂iφ − hk k 2 �π2 φ 2 − (∇φ)2 2 − m2φ2 2 � − 1 2hij∂iφ∂jφ + O(h2) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10) where the energy E is given by E = π2 φ 2 + (∇φ)2 2 + m2φ2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='11) Consequently, from equations (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='7) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10), and taking into account the primary con- straints (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='6), the full Hamiltonian is given by: Htot = Hg + Hφ + voΠ00 + viΠ0i (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='12) – 27 – Closure of constraints and the Dirac matrix Let us now find the secondary constraints: χm 0 = � Π00, � ddx Htot � = χ0 + 1 2E χm i = � Π0i, � ddx Htot � = χi + πφ∂iφ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='13) where χ0 and χi are the secondary constraints without matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' They are given by: χ0 = 1 κ2 � ∂2 i hk k − ∂i∂jhij � χi = −2∂jΠij (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='14) Next, we compute the tertiary constraints: ξm 0 = � χm 0 , � ddx Htot � = � χ0, � ddx Htot � + 1 2 � E, � ddx Htot � = ξ0 + 1 2 � E, � ddx Hφ � = −∂i∂jΠij + 1 2∂i(πφ∂iφ) = −∂iχm i ≈ 0 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='15) Next, we compute the commutator of χm i with the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' ξm i = � χm i , � ddx Htot � = � χi, � ddx Htot � + � πφ∂iφ, � ddx Htot � = � πφ∂iφ, � ddx Hφ � = πφ∂iπφ + ∂iφ(∂2 kφ − m2φ) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='16) At the perturbative level, it seems that the constraint algebra does not close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This can be seen from the fact that since ξm i is non-zero, its Poisson bracket with Hamiltonian we obtain a non-zero answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Further it also seems that the constraints (χm 0 and χm i ) are no longer first class as well since {χm 0 (x), χm i (y)} = {T00(x), T0i(y)} ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' However, this is a consequence of doing perturbation theory incorrectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' As an example, for the commutator {χm 0 (x), χm i (y)} without matter, the gravity part of the constraints commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' However, if we keep the full non-linear correction in the gravity part of the Lagrangian, then the gravity part of the constraints does receive corrections, which then makes the constraints first class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2 Minimally coupled matter to massive graviton We again minimally couple the scalar field to gravity but with the Fierz Pauli action (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' As a consequence, the total action is given by: S = Sφ + Sg (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='17) where Sg now denotes the Fierz Pauli massive gravity action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 28 – Momenta and Hamiltonian In presence of matter, the full Hamiltonian is given by: Htot = Hg + Hφ + voΠ00 + viΠ0i (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='18) where Hφ is given in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The addition of matter does not affect the primary constraints, and consequently, they are still given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The stability of primary constraints leads to secondary constraints, which are given by χm 0 = � Π00, � ddx Htot � = χ0 + 1 2E χm i = � Π0i, � ddx Htot � = χi + πφ∂iφ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='19) where χ0 and χi denote secondary constraints without matter and are given by χ0 = 1 κ2 � (∂2 i − m2)hk k − ∂i∂jhij � , χi = −2 � ∂jΠij + m2 κ2 h0i � (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='20) Next, we demand the stability of secondary constraints, which give rise to the tertiary con- straints: ξm 0 = � χm 0 , � ddx Htot � = � χ0, � ddx Htot � + 1 2 � E, � ddx Htot � = ξ0 + 1 2 � E, � ddx Hφ � = ξ0 + 1 2∂i(πφ∂iφ) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='21) where ξ0 is again the constraint without matter and it is given by: ξ0 = −∂i∂jΠij + m2 D − 2Πk k − 2m2 κ2 ∂ihi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='22) Hence the constraint ξm 0 , is given by: ξm 0 ≡ ξ′ = −∂i∂jΠij + m2 D − 2Πk k − 2m2 κ2 ∂ihi0 + 1 2∂i(πφ∂iφ) = 1 2∂iχm i + m2 κ2 � κ2 Πk k D − 2 − ∂ih0i � ≈ m2 � Πk k D − 2 − ∂ih0i κ2 � (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='23) Next, we compute the commutator of χm i with the Hamiltonian (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' ξm i = � χm i , � ddx Htot � = � χi, � ddx Htot � + � πφ∂iφ, � ddx Htot � = 2m2 κ2 (∂jhji − ∂ih − vi) + � πφ∂iφ, � ddx Hφ � = 2m2 κ2 (∂jhji − ∂ih − vi) + Bi (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='24) where Bi = � πφ∂iφ, � ddx Hφ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We can solve for vi by demanding ξm i equals zero, which gives us vi = ∂jhji − ∂ih + κ2 2m2Bi, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='25) – 29 – and where upto the quadratic order in matter fields, Bi is given by: Bi = πφ∂iπφ + ∂iφ(∂2 kφ − m2φ) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='26) Next, we compute the quartic constraints by computing the commutator of ξm 0 with Hamiltonian (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' ˜ξm 0 = � ξm 0 , � ddx Htot � = m2 D − 2χ0 + m4 κ2 D − 1 D − 2h − 1 2∂iBi ≈ ˜ξ0 − m2 2(D − 2)E − 1 2∂iBi (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='27) where ˜ξ0 is given by ˜ξ0 = m4 κ2 D − 1 D − 2h Using the continuity equation, we have ∂iBi = ∂0∂iT0i = ∂2 t T00, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='28) Next, we find the Poisson bracket of the above constraints with total Hamiltonian: � ˜ξm 0 , � ddx Htot � = � ˜ξ0, � ddx Htot � − 1 2∂i � Bi, � ddx Htot � + m2 2(D − 2) � E, � ddx Htot � (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='29) Again we can solve for v0 by demanding the above equation to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Various non-zero elements of the constraint matrix are given below: {Π00(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' ˜ξm 0 (y)} = m4 κ2 D − 1 D − 2 δ(x − y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' {Π0i(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' χm j (y)} = 2m2 κ2 δij δ(x − y) {Π0i(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' ξ′(y)} = −m2 κ2 ∂iδ(x − y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' {χm 0 (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' χm i (y)} = 2m2 κ2 ∂iδ(x − y) + 1 2Qi {χm 0 (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' ξ′ 0(y)} = m2 κ2 � ∂2 i − D − 1 D − 2m2 � δ(x − y) {χm i (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' �ξm 0 (y)} = 2m4 κ2 D − 1 D − 2∂iδ(x − y) + 1 2 � ∂2 t + m2 D − 2 � Qi {ξ′ 0(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' �ξm 0 (y)} = −m6 κ2 �D − 1 D − 2 �2 δ(x − y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' {χm 0 (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' χm 0 (y)} = 1 4P {χm i (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' χm j (y)} = Rij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' {�ξm 0 (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' �ξm 0 (y)} = −1 4 � ∂2 t + m2 D − 2 �2 P {χm 0 (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' �ξm 0 (y)} = −1 4 � ∂2 t + m2 D − 2 � P (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='30) – 30 – where P = {T00(x), T00(y)} = � Πφ(y)∂φ(x) ∂xi − Πφ(x)∂φ(y) ∂yi � ∂ ∂xiδ(x − y) Qi = {T00(x), T0i(y)} = � ∂iφ∂kφ ∂ ∂xk − π2 φ ∂ ∂xi + m2φ∂iφ � δ(x − y) Rij = {T0i(x), T0j(y)} = � Πφ(x)∂φ(y) ∂yj ∂ ∂xi − Πφ(y)∂φ(x) ∂xi ∂ ∂yj � δ(x − y) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='31) By computing the inverse of the constraint matrix, we notice that we obtain the following type of inverse derivative dependence in Dirac brackets: 1 m2 − ∂i∂j(RijP + QiQj) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='32) The above constraint algebra fails to close due to the presence of matter energy momentum tensor on the RHS of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This can be seen by computing the Poisson bracket of RHS of any of the constraints above with the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Since {P, Qi} (and other such combinations) is non-zero, the above algebra is not stable under Hamiltonian evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This extra term in the denominator of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='32) is seemingly an artifact of perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Since the constraints do not close now, they pose an inconsistency in the counting of the degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Consequently, we expect the extra term ∂i∂j(RijP + QiQj) to go away as for the massless case upon the inclusion of higher order corrections [19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' C Massive gravity constraints by substitution Let us look at a different way to compute Dirac brackets, where we substitute for some of the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This procedure was used in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Notice that the metric component h0i appears quadratically in the Fierz-Pauli lagrangian given in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Hence we can just solve for h0i using its equation of motion and substitute it back in the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This is the key difference from our previous treatment of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='The h0i EOM is given by: h0i = − 1 m2∂jΠji (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1) This equation can also be obtained by setting the constraints Ci (given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4)) to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Now we can substitute h0i in the massive gravity lagrangian and then solve for constraints of the corresponding system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The Hamiltonian of this system is given by: Hg = κ2 �Π2 ij 2 − Π2 ii 2(D − 2) � + 1 κ2 �1 2∂khij∂khij − ∂ihjk∂jhik + ∂ihij∂jhk k − 1 2∂ihj j∂ihk k 1 2m2(hijhij − h2 kk) − m2h2 0i − h00 � ∂2 khi i − ∂i∂jhij − m2hk k �� + 1 m2(∂jΠij)2 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2) – 31 – Since Π00 = 017, this system has one secondary constraint given by: Φg 1 ≡ (∂i∂i − m2)hj j − ∂i∂jhij (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3) One can readily compute the tertiary constraint : Φg 2 ≡ {Hg, Φg 1} = m2 D − 2Πii + ∂i∂jΠij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4) The stability of the above constraint under time evolution gives us the following further con- straint: Φg 3 ≡ {Hg, Φg 2} ≈ −D − 1 D − 2m2h (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='5) where h = −h00 + hk k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' These set of constraints form a closed algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' The constraint matrix is now a 4 ∗ 4 matrix whose various non-zero elements are given by {Π00(p), Φg 3(q)}P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' = −m2D − 1 D − 2δD−1(p − q), {Φg 2(p), Φg 1(q)}P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' = −m4D − 1 D − 2δD−1(p − q), {Φg 1(p), Φg 3(q)}P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' = 0, {Φg 2(p), Φg 3(q)}P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' = m4 �D − 1 D − 2 � � m2D − 1 D − 2 − p2 � δD−1(p − q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='6) Hence the inverse of Dirac Matrix C−1(p) is given by: C−1(p) = 1 m4 d − 1 d � � � � � 0 dm4 d−1 − p2m2 0 m2 p2m2 − dm4 d−1 0 −1 0 0 1 0 0 −m2 0 0 0 � � � � � δd(p − q) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='7) where d = D − 1 is the dimension of the Cauchy slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' As expected, the above matrix is just a sub-matrix of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10) (up to a factor of m2 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' We can now use this matrix to define the Dirac bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Since the inverse constraint matrix does not contain any derivatives in the denominator, using the analysis similar to the one discussed in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2, one can readily see that the Poisson bracket of boundary operator H∂ with any bulk insertion O(x) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Why substitution works classically?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Roughly our action is of the form L = L0(xi, ˙xi) + m2h2 0i + Xh0i (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='8) 17Since h0i is no longer a degree of freedom of the system, the corresponding momenta Π0i does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 18This factor is different because of the difference in the definition of tertiary constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 32 – such that X is a function of xi, ˙xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' In the case of standard gravity with m = 0, we have the constraint X = 0, with h0i acting as a Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' When m2 ̸= 0, we can rewrite the action as L = L0(xi, ˙xi) + m2 � h0i + X 2m2 �2 − X2 4m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='9) Note that action is decomposed into a separate part for the h0i field, and a part containing L0(xi, ˙xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Setting the positive definite second term in above equation to zero gives us the equation of motion for h0i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' One can now always redefine the h0i field independently of xi, ˙xi H0i = h0i + X 2 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10) such that there are no terms coupling the fields xi and H0i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Thus we can independently minimize H0i without interfering with the variations of L0(xi, ˙xi) − X2 4m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Hence this substitution of the equation of motion is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' This is also confirmed by the counting of degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Note: This substitution is correct at a classical level but may pose some difficulties in quantum mechanics when we vary over the whole space of paths with weightage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Laddha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Prabhu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Raju, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Shrivastava, The Holographic Nature of Null Infinity, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 10 (2021) 041, [arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='02448].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Chowdhury, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Godet, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Papadoulaki, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Raju, Holography from the Wheeler-DeWitt equation, JHEP 03 (2022) 019, [arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='14802].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Raju, Lessons from the information paradox, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 943 (2022) 1–80, [arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='05770].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Raju, Failure of the split property in gravity and the information paradox, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 39 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 6 064002, [arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='05470].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Dirac, Lectures on quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Belfer Graduate School Monograph Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='2 (Yeshiva University, New York,).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Henneaux and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Teitelboim, Quantization of gauge systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [7] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Mukunda and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Sudarshan, Structure of the Dirac bracket in classical mechanics, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 9 (1967) 411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Fierz and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' A 173 (1939) 211–232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Higuchi, Forbidden Mass Range for Spin-2 Field Theory in De Sitter Space-time, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' B 282 (1987) 397–436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Porrati, No van Dam-Veltman-Zakharov discontinuity in AdS space, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' B 498 (2001) 92–96, [hep-th/0011152].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [11] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Kogan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Mouslopoulos, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Papazoglou, A New bigravity model with exclusively positive branes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' B 501 (2001) 140–149, [hep-th/0011141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 33 – [12] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Arkani-Hamed, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Georgi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Schwartz, Effective field theory for massive gravitons and gravity in theory space, Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 305 (2003) 96–118, [hep-th/0210184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [13] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Creminelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Nicolis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Papucci, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Trincherini, Ghosts in massive gravity, JHEP 09 (2005) 003, [hep-th/0505147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Karch and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Randall, Locally localized gravity, JHEP 05 (2001) 008, [hep-th/0011156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Porrati, Higgs phenomenon for the graviton in ADS space, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' A 18 (2003) 1793–1802, [hep-th/0306253].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Goldhaber and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Nieto, Photon and Graviton Mass Limits, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 82 (2010) 939–979, [arXiv:0809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1003].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [17] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Aharony, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Clark, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Karch, The CFT/AdS correspondence, massive gravitons and a connectivity index conjecture, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' D 74 (2006) 086006, [hep-th/0608089].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [18] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Hinterbichler, Theoretical Aspects of Massive Gravity, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 84 (2012) 671–710, [arXiv:1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3735].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [19] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' de Rham and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Gabadadze, Generalization of the Fierz-Pauli Action, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' D 82 (2010) 044020, [arXiv:1007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='0443].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [20] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' de Rham, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Gabadadze, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Tolley, Resummation of Massive Gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 106 (2011) 231101, [arXiv:1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='1232].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [21] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Hassan and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rosen, On Non-Linear Actions for Massive Gravity, JHEP 07 (2011) 009, [arXiv:1103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='6055].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Hassan and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rosen, Resolving the Ghost Problem in non-Linear Massive Gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 108 (2012) 041101, [arXiv:1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3344].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Blake, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Tong, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Vegh, Holographic Lattices Give the Graviton an Effective Mass, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 112 (2014), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 7 071602, [arXiv:1310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3832].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [24] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' de Rham, Massive Gravity, Living Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 17 (2014) 7, [arXiv:1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4173].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [25] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Capela and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Nardini, Hairy Black Holes in Massive Gravity: Thermodynamics and Phase Structure, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' D 86 (2012) 024030, [arXiv:1203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='4222].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Ghodrati, Complexity growth in massive gravity theories, the effects of chirality, and more, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' D 96 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 10 106020, [arXiv:1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='07981].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [27] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Penington, Entanglement Wedge Reconstruction and the Information Paradox, arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='08255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Almheiri, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Mahajan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Maldacena, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Zhao, The Page curve of Hawking radiation from semiclassical geometry, JHEP 03 (2020) 149, [arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10996].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [29] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Almheiri, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Engelhardt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Marolf, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063, [arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='08762].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [30] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Penington, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Shenker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Stanford, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Yang, Replica wormholes and the black hole interior, arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='11977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [31] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Almheiri, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Hartman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Maldacena, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Shaghoulian, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Tajdini, Replica Wormholes – 34 – and the Entropy of Hawking Radiation, JHEP 05 (2020) 013, [arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='12333].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [32] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Geng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Karch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Perez-Pardavila, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Raju, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Randall, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Riojas, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Shashi, Information Transfer with a Gravitating Bath, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 10 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 5 103, [arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='04671].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [33] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Geng and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Karch, Massive islands, JHEP 09 (2020) 121, [arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='02438].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [34] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Chowdhury, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Papadoulaki, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Raju, A physical protocol for observers near the boundary to obtain bulk information in quantum gravity, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 10 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 5 106, [arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='01740].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [35] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Banerjee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Bryan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Papadodimas, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Raju, A toy model of black hole complementarity, JHEP 05 (2016) 004, [arXiv:1603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='02812].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [36] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Raju, A Toy Model of the Information Paradox in Empty Space, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 6 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 6 073, [arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='10154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [37] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Chakraborty, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Chakravarty, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Paul, Monogamy paradox in empty flat space, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' D 106 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 8 086002, [arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='06919].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [38] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' de Mello Koch and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Kemp, Holography of Information in AdS/CFT, arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='11066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [39] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Giddings, On the questions of asymptotic recoverability of information and subsystems in quantum gravity, JHEP 08 (2022) 227, [arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='03207].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [40] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Chowdhury and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Papadoulaki, Recovering information in an asymptotically flat spacetime in quantum gravity, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='07449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [41] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Bahiru, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Belin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Papadodimas, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Sarosi, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Vardian, State-dressed local operators in AdS/CFT, arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='06845.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [42] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Buchholz, PRODUCT STATES FOR LOCAL ALGEBRAS, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 36 (1974) 287–304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [43] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Kluson, Note About Hamiltonian Structure of Non-Linear Massive Gravity, JHEP 01 (2012) 013, [arXiv:1109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='3052].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [44] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Jalali and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Shirzad, Hamiltonian structure of Fierz-Pauli gravitons, partially massless fields and gauge symmetry, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' B 976 (2022) 115711, [arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='04503].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [45] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rodríguez-Tzompantzi, Canonical description of exotic general massive gravity, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' C 82 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 5 482, [arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='03151].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [46] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Deser, Unconstrained canonical action for, and positive energy of, massive spin 2, Can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 93 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 4 395–397, [arXiv:1407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='7178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [47] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Donnelly and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Giddings, Diffeomorphism-invariant observables and their nonlocal algebra, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' D 93 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 2 024030, [arXiv:1507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='07921].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='D 94, 029903 (2016)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [48] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Ashtekar, Asymptotic quantization of the gravitational field, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 46 (Mar, 1981) 573–576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [49] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Ashtekar, Radiative Degrees of Freedom of the Gravitational Field in Exact General – 35 – Relativity, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 22 (1981) 2885–2895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [50] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Ashtekar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Campiglia, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Laddha, Null infinity, the BMS group and infrared issues, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 50 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 11 140–163, [arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='07093].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [51] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Bondi, Gravitational Waves in General Relativity, Nature 186 (1960), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 4724 535–535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [52] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Bondi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' van der Burg, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Metzner, Gravitational waves in general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Waves from axisymmetric isolated systems, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' A 269 (1962) 21–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [53] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Sachs, Asymptotic symmetries in gravitational theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 128 (1962) 2851–2864.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [54] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Sachs, Gravitational waves in general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Waves in asymptotically flat space-times, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' A 270 (1962) 103–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [55] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='05448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [56] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Compère and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Fiorucci, Advanced Lectures on General Relativity, arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='07064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [57] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Arnowitt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Deser, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Misner, The Dynamics of general relativity, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 40 (2008) 1997–2027, [gr-qc/0405109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [58] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Regge and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Teitelboim, Role of Surface Integrals in the Hamiltonian Formulation of General Relativity, Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 88 (1974) 286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [59] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Haag, Local quantum physics: Fields, particles, algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [60] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Fewster, The split property for quantum field theories in flat and curved spacetimes, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [61] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Geng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Karch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Perez-Pardavila, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Raju, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Randall, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Riojas, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Shashi, Inconsistency of Islands in Theories with Long-Range Gravity, arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='03390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [62] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Papadodimas and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Raju, Local Operators in the Eternal Black Hole, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 115 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 21 211601, [arXiv:1502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='06692].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [63] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Papadodimas and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Raju, Remarks on the necessity and implications of state-dependence in the black hole interior, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' D 93 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' 8 084049, [arXiv:1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='08825].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [64] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Chakravarty, Overcounting of interior excitations: A resolution to the bags of gold paradox in AdS, JHEP 02 (2021) 027, [arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='03575].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [65] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Chakravarty, Small corrections to semiclassical gravity and their role in information paradoxes, arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='09924.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [66] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Cotler, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Gur-Ari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Hanada, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Polchinski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Saad, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Shenker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Stanford, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Streicher, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' Tezuka, Black Holes and Random Matrices, JHEP 05 (2017) 118, [arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content='04650].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' [Erratum: JHEP 09, 002 (2018)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} +page_content=' – 36 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AzT4oBgHgl3EQfJPtD/content/2301.01075v1.pdf'} diff --git a/R9E4T4oBgHgl3EQflg05/content/tmp_files/2301.05160v1.pdf.txt b/R9E4T4oBgHgl3EQflg05/content/tmp_files/2301.05160v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9df873a15780afef7f1f3063fdac419152654f6e --- /dev/null +++ b/R9E4T4oBgHgl3EQflg05/content/tmp_files/2301.05160v1.pdf.txt @@ -0,0 +1,1253 @@ +Venus, Phosphine and the Possibility of Life +David L Clementsa +aDepartment of Physics, Imperial College, Prince Consort Road, London, SW7 2AZ, UK +ARTICLE HISTORY +Compiled January 13, 2023 +ABSTRACT +The search for life elsewhere in the universe is one of the central aims of science in +the 21st century. While most of this work is aimed at planets orbiting other stars, +the search for life in our own Solar System is an important part of this endeavour. +Venus is often thought to have too harsh an environment for life, but it may have +been a more hospitable place in the distant past. If life evolved there in the past +then the cloud decks of Venus are the only remaining niche where life as we know it +might survive today. The discovery of the molecule phosphine, PH3, in these clouds +has reinvigorated research looking into the possibility of life in the clouds. In this +review we examine the background to studies of the possibility of life on Venus, +discuss the discovery of phosphine, review conflicting and confirming observations +and analyses, and then look forward to future observations and space missions that +will hopefully provide definitive answers as to the origin of phosphine on Venus and +to the question of whether life might exist there. +KEYWORDS +Venus; astrobiology; search for life; +1. Introduction +The search for life elsewhere in the universe is one of the major driving forces for +astronomy and astrophysics in the 21st century [1] with extremely large telescopes +on the ground and in space being designed and built to this end. The main goal of +these facilities is to study the atmospheres of rocky, terrestrial planets in orbit around +other stars - so-called exo-planets - and follows the explosive development of exoplanet +studies since their first discovery in 1995 [2]. We now know of over 5000 exoplanets, and +more are announced all the time (for the latest information on exoplanet discoveries +see www.exoplanet.eu). However, while convincing evidence for life on exoplanets +may arrive in the next 20 years, there are many questions that such observations will +not be able to answer, including the biochemical processes exoplanet life uses, how +and when it originated, and how it evolved into whatever form it has today - a form +that will also be unknowable. +If we want answers to these further questions the only places they may be answered +is in our own Solar System. While still very distant, planets such as Mars, the Jovian +moon Europa, and Saturn’s moons Enceladus and Titan, which might harbour signs +of life, are accessible to direct in situ studies that can answer these questions. And +any answers inevitably lead back to the question of our own origins on Earth, as well +as the more general issue of the prevalence of life in the universe. +CONTACT David L. Clements Email: d.clements@imperial.ac.uk +arXiv:2301.05160v1 [astro-ph.EP] 12 Jan 2023 + +The search for signs of life on Mars is well underway, with orbiting spacecraft look- +ing at its atmosphere (eg. the ExoMars Trace Gas Orbiter (TGO) [3]) and an ever- +increasing number of rovers scouring its surface [4]. Some of these are already preparing +samples of rock to be returned to Earth for laboratory analysis. Further out in the +Solar System, the first stages of the exploration of Jupiter’s moons Ganymede and +Europa, in part for signs of habitable environments, are already under development +with the European Space Agency’s (ESA’s) JUpiter ICy moons Explorer (JUICE) [5], +aimed primarily at Ganymede, due for launch in 2023, and NASA’s Europa Express, +aimed squarely at Europa, due for launch in 2024. Plans are at an earlier stage for the +exploration of the moons of Saturn that might harbour life, including Titan with its +thick atmosphere and Enceladus with its plumes of water vapour spewing into space, +but it is clear that these too will be visited sometime in the next few decades. +Until very recently this list of targets - Mars, Europa, Ganymede, Titan and Ence- +ladus - would have been considered the most likely places to find signs of life in the +Solar System. It was thus rather surprising to find Venus added to this list in late +2020 with the discovery of an unusual gas, phosphine, chemical formula PH3, in its +atmosphere [6]. Venus, as we will see below, has a surface that is completely hostile +to life and so had been largely discounted from these considerations. But, as we will +also see, there have long been thoughts that there might be niches in the atmosphere +of this planet that might be more favourable to the existence of life than its deeply +unpleasant surface [7,8]. In this article we discuss how life is sought using atmospheric +observations, why phosphine is a potential signature of biological activity, how it was +detected on Venus, and what its discovery might mean for our understanding of the +history of Venus and of life in the Solar System. We also look at the prospects for +future studies of Venus in search of further possible signs of life. +2. What is Life? +The formal definition of life adopted by NASA for searches for life elsewhere is that +‘Life is a self-sustaining chemical system capable of Darwinian evolution’ [9]. This +definition clearly applies to most things that we would consider alive on Earth, but it +does leave some things out. Viruses, for example, cannot reproduce on their own, but +instead require a host cell, whose reproductive machinery they take over. Alternative +definitions are available (eg. [10]), but the NASA definition seems a useful starting +point to begin any discussion of where life might exist in the Solar System or elsewhere. +Given this definition we can start to examine what the requirements might be for life +to exist. +From a physicist’s perspective the key thing that life needs to be able to support +itself is a source of energy. For most of the life we are familiar with on the Earth +that source of energy is the Sun - plants photosynthesise using sunlight, while animals +and other organisms consume plants in various ways, including eating things that +have eaten plants. However, sunlight is not the only source of energy used by life on +Earth. In the depths of Earth’s oceans, where sunlight never shines, there are thriving +communities of organisms surrounding and fed by hydrothermal vents [11]. These arise +where ocean water can enter the Earth’s crust and be heated by magma. The heated +water dissolves minerals from the rocks and circulates back into the ocean through +the vents. The primary producers of energy in these vents are chemosynthetic bacteria +that use a variety of processes to derive energy from the chemicals emerging from the +vents. These then support a diverse community of other organisms around the vents, +2 + +including giant tube worms that can be up to 3 metres long. Hydrothermal vents in the +young Earth are a possible site for the first emergence of life on our planet [12]. Similar +hydrothermal vent structures are thought to exist beneath the ice-covered surfaces of +moons like Europa and Enceladus [13]. However, life without light on Earth is not +limited to hydrothermal vents. It has even been suggested that most ecosystems on +Earth exist in the dark, deriving their energy from chemical processes separate from, +and independent of, photosynthesis [14]. This clearly has implications for the search +for life elsewhere. +What requirements for life are there beyond a source of energy? Revisiting NASA’s +definition of life we see that it is defined as a chemical system. The chemical basis for +all the life we know on Earth is the element carbon. This element is exceptional in the +Periodic Table as the lightest element in Group IV. It thus has a half-filled electron +shell giving it a valence of 4 so it can donate or receive up to 4 electrons, allowing +it to bind with itself to form chains, and with numerous other elements. Some of the +commonest are hydrogen, oxygen and nitrogen, contributing to the wide variety of +complex organic compounds found in living organisms. The next heaviest Group IV +element is silicon. It has similar high valence and has been proposed as an alternative +building block for life [15], but there are a number of issues which mean that carbon +is a better choice. +Given a chemical basis for life, there must be a way for the necessary chemical +reactions to take place so that the processes of life can operate. The best way to +achieve this is for the reacting chemicals to be dissolved by some solvent so that they +can easily combine and interact. On Earth the solvent behind all biology is water. +While other solvents have been suggested, especially in environments too hot or too +cold for liquid water to be available [16], water remains the only solvent which we +know is associated with life. The bulk of our searches for life elsewhere have thus been +focussed on places where liquid water might, or is known to, exist. +Our consideration of the nature of life thus leads us to the conclusion that we should +be looking for life on an astronomical body where there is a ready supply of energy, +where complex carbon-based chemistry can take place, and where liquid water can +exist. In our own Solar System this leads us to focus on the planet Mars and the icy +moons of gas giants, such as Europa and Ganymede orbiting Jupiter, and Titan and +Enceladus orbiting Saturn. Venus does not appear on this list because, as we will see +below, its current surface conditions are far too hostile for life as we know it, but, as +we will also see, this may not always have been the case, and there may be loopholes +that could allow any life that might have emerged on the surface of this planet to +persist today in the dense clouds above its surface. +3. Looking for Life +Life is easy to find on Earth - we are surrounded by it - but looking for life on astro- +nomical bodies, even our closest neighbours, is a rather more difficult task, especially +since we do not expect to find something as obviously alive as a tree or a cow. In the +absence of complex macroscopic life we have to look for signs of activity from microbial +life. What might these be? Using Earth as an example, the clearest sign that life exists +is the presence of oxygen in our atmosphere. This is generated by photosynthesising +organisms. Today we would associate this with plants, but in the ancient history of +our planet, the first oxygenating organisms were single-celled microbes known as blue- +green algae. These were responsible for the Great Oxygenation Event which took place +3 + +about 2.5 billion years ago [17]. Before this event oxygen was not a major constituent +of Earth’s atmosphere. After it, the Earth had an atmosphere much closer to what +we see today, with abundant free oxygen available across the planet. In principle, the +presence of oxygen in the atmosphere of Earthlike exoplanets will be detectable by +future instruments through the absorption of ozone, O3, at mid-infrared wavelengths +[18]. +Molecules that potentially indicate the presence of life, such as oxygen and ozone, +are known as biosignatures (see [19] and references therein). They include oxygen, +ozone, methane (CH4), N2O, C2H6 CH3Cl, CH3SH and more. A wider variety of +biosignatures than just oxygen and ozone are needed to cope with potentially dif- +ferent biospheres than the one we currently inhabit. In fact, for much of the history +of life on Earth, there would not have been sufficient oxygen or ozone for life to be +detectable since the Earth was dominated by anaerobic life (ie. life that does not re- +quire or produce oxygen). Searches for other small molecules that might be additional +biosignatures are also underway [20]. The common factor among all of these biosigna- +ture molecules is that they should not exist in the abundances seen unless biological +processes are maintaining their abundance ie. their abundance is out of equilibrium +with their environment. However, biological processes are not the only way of main- +taining an out-of-equlibrium abundance of some of these biosignatures. For example, +the splitting of water into hydrogen and oxygen by stellar ultraviolet radiation, and +the subsequent escape of hydrogen from the planet’s atmosphere, can produce a sig- +nificant partial pressure of oxygen on an abiotic (ie. lifeless) planet given certain other +conditions [21]. Care must therefore be taken in interpreting any unusual abundance +of a single molecule as an unambiguous biosignature without a good understanding of +the broader environment in which it is found. +Phosphine, PH3, is one of the small molecules suggested as a potential biosigna- +ture by the team investigating novel biosignature molecules [22]. On Earth it is ex- +clusively associated with anaerobic ecosystems, or with human industrial chemistry. +Sources have been found associated with anaerobic environments in ponds, marshes +and sludges, and specifically with piles of penguin guano and in the faeces of European +badgers. While it is clearly associated with anaerobic biology, the specific biochemical +pathway for phosphine production in anaerobic systems remains unclear. Its use as a +biosignature was originally envisioned in the context of a significant concentration of +the gas within the atmosphere of a terrestrial exoplanet with a large anaerobic bio- +sphere. While phosphine has been detected in the vast reducing (ie. hydrogen rich) +atmosphere of the gas giant Jupiter [23], where it is produced by normal chemical +processes in the very dense, hot inner regions then brought to the surface by convec- +tion, its presence in the oxidised atmosphere of a terrestrial planet would be difficult +to explain by equilibrium chemistry, making it a good candidate biosignature. Obser- +vational facilities able to find phosphine in the atmosphere of a distant exoplanet are +some way in the future, but, as we see below, a surprise was waiting for us much closer +to home. +4. Venus - an unlikely candidate for astrobiology +4.1. Venus Today +Venus has often been described as Earth’s evil twin since the two planets have very +similar sizes and masses, with Venus having a radius about 95%, and a mass about +4 + +Figure 1. +Left: Venus as seen in the optical by the Messenger mission. In the optical the planet appears +almost featureless because of the highly reflective clouds that cover the entire planet. Credit: NASA/Johns +Hopkins University Applied Physics Laboratory/Carnegie Institution of Washington Right: Venus as seen in +the ultraviolet by the Japanese space mission AKATSUKI. This image is produced by observations in two +ultraviolet bands, at 365 and 283 nm. The colours are the result of unexplained ultraviolet absorption by small +particles in the cloud layer. Credit: JAXA/ISAS/DARTS/Kevin M. Gill +82% of the Earth’s. It is also a little under 30% closer to the Sun than the Earth. +Viewed from a telescope, Venus appears as a featureless pale disk because it has a +thick atmosphere containing a permanent cloud layer, opaque to optical observations, +that reflects over 70% of the sunlight that falls on it (see Figure 1). Venus’ orbit closer +to the Sun, combined with the effects of these reflective clouds, might naively suggest a +surface temperature not too different from the Earth. Because of this, in the early 20th +century it was thought that the clouds might be water vapour and that the surface of +Venus could be habitable, inspiring visions of steamy tropical jungles filled with alien +life. Once more detailed observations became available, and space probes started to +visit in the 1960s, it became apparent that Venus was very different from these initial +speculations. +Rather than having an Earthlike atmosphere, early 1960s space probes like NASA’s +Mariner 2 and the Soviet Union’s Venera 4 found that Venus has an extremely dense +atmosphere dominated by carbon dioxide, CO2, with a small amount of nitrogen (3.5%) +and traces of other gases such as sulphur dioxide (SO2). The surface atmospheric +pressure on Venus is about 93 times higher than sea level pressure on the Earth. This +huge blanket of CO2 absorbs thermal radiation from the surface of the planet which +would otherwise be radiated away into space. The Sun’s radiation is thus trapped +beneath the clouds, leading to a runaway greenhouse effect and surface temperatures +of about 735 K (462 C), making it hotter than the maximum surface temperature of +Mercury [24]1. To make things even more unpleasant, the thick clouds are largely made +up of sulphuric acid as a result of atmospheric SO2 dissolving into droplets of water +that condense at altitudes of about 55 km above the surface. All of this combines to +make the surface of Venus today an utterly inhospitable place for anything we might +consider a biological system. The true nature of the surface of Venus in fact led leading +1It is worth noting that the discovery of the runaway greenhouse effect on Venus prompted early discussions +about the dangers of CO2 emissions from fossil fuel use on Earth and their possible impact on climate. +5 + +Figure 2. +A 360 degree panorama of the surface of Venus captured by the Venera 9 probe during its 53 +minutes of operation on the harsh surface of the planet. This is the first image ever obtained from the surface +of Venus. Part of the probe can be seen at the bottom of the image. +astronomer Carl Sagan to describe the planet as hell. +Spacecraft continued to visit Venus after the early missions, including a long series +of Venera probes launched by the Soviet Union. Several of these, as well as the Pioneer +Venus mission from NASA, sent probes into the atmosphere of Venus to better under- +stand its chemistry and constituents. The Venera and Vega projects also attempted +to land probes on the hostile surface of Venus. After a number of failures they were +eventually successful and conducted a number of studies. None of the landers lasted +very long, with the longest lived surviving only about 2 hours. Nevertheless, some of +these landers managed to send back images of the Venusian surface, one of which can +be seen in Figure 2. +While the surface of Venus is undoubtedly hostile to life at the present time, there +is a potential niche for biological processes in the clouds that obscure the planet since +they are cool enough for liquid water to be present, and at an atmospheric pressure +that matches that of the Earth at sea level [25,26] (see Figure 3). Speculation about +the possibility of life in the clouds of Venus dates back to the very earliest days of our +understanding of the true conditions on the planet [27] and continues to the present +day (eg. [28,29]). +4.2. Venus in the past +While the surface of Venus is undoubtedly hostile today, this might not have been the +case in the distant past. If the surface of Venus was warm and wet - in the sense that +liquid water could flow on the surface - then it is possible that life might have emerged +there. When conditions later become increasingly hostile to life on the surface it might +have evolved to seek sanctuary in clouds, the last habitable ecological niche on the +planet. But is there any evidence to suggest that Venus was ever less hostile than it is +today? +Sadly, we do not have direct access to information about the conditions on Venus +billions of years ago, but there are hints that suggest that Venus may once have had +much more water on its surface, and in its atmosphere, than it does today. Principal +among this evidence is the deuterium to hydrogen ratio (D/H ratio). Venus currently +has a D/H ratio that is 150±30 times that of the Earth [30] 2. This suggests that +Venus has lost a substantial fraction of its water through the escape of hydrogen, +which is favoured over the escape of deuterium since the latter has a higher mass. +Hydrogen is driven off Venus through interactions with the solar wind which can +directly interact with the upper layers of its atmosphere since, unlike the Earth, Venus +lacks a protective magnetic field. However, it is possible that Venus may have retained +2Earth’s D/H ratio is ∼ 1.6 × 10−4 +6 + +3 +Figure 3: Left: Image of Venus from the AKATSUKI UV imager, using two channels, 3650˚A in or- +ange, which tracks SO2, and 2830˚A in blue, which tracks the unknown absorber. +Note the highly +complex structure of the unknown absorber. Right: A schematic diagram of the vertical structure of +the atmosphere of Venus (from Seager et al., 2021). +of Venus so significant. PH3 has previously been detected in the atmospheres of three other planets +- Jupiter, Saturn and Earth (see Weisstein & Serabyn (1996) and references therein). In gas giants, +phosphine is produced in the deepest atmospheric layers where the temperature and pressure is high +enough for the formation to be thermodynamically favoured. The PH3 seen in Jupiter and Saturn is +thought to account for nearly the entire phosphorous content of these atmospheres (Visscher et al. +2006). +Some of this material is then dredged up to the upper atmosphere by convection currents, +where it can be detected (Noll & Marley 1997). This is not the situation on Earth, where phosphine +production is highly disfavoured. In fact, life is the sole source of phosphine on Earth, whether as a +result of industrial chemistry processes or through as-yet poorly understood biological processes in a +number of di↵erent anaerobic environments (Sousa-Silva et al., 2020). Phosphine production on rocky +terrestrial planets is in fact so disfavoured that it has been suggested as a biosignature gas for exoplanets +(Sousa-Silva et al., 2020 and references therein). +However, the discovery of phosphine in the atmosphere of Venus cannot, on its own be taken as +evidence for life. Before that can happen we must eliminate all other potential sources of PH3. Extensive +analysis of non-biological routes to the formation of phosphine on Venus was undertaken in Bains et +al (2020) which analysed potential thermodynamic, photochemical, surface, and sub-surface routes to +its production, as well as lightning, volcanism and injection from impactors. They concluded that all +potential sources of phosphine fell short of the observed levels by many orders of magnitude. +This leaves us in the uncomfortable position of having no way to explain the presence of phosphine +without either invoking unknown chemical processes or the presence of life, though the life hypothesis +itself has its own problems since it would require life to exist in a profoundly hostile environment . As a +first step in addressing this problem, we here propose a long term programme of monitoring observations +of phosphine and several other species: HDO, SO, HCO+ and SO2 (see Fig. 5 and Tech Case). We +already know that the abundance of several of these species in the atmosphere of Venus varies with time, +but we do not know the degree to which phosphine varies with time, or if any variation is correlated with +other species or with other factors such as orbital phase. Correlations, or anticorrelations, between other +species and the abundance of phosphine could provide clues to the chemical, or biochemical, processes +that produce phosphine. +These observations will also provide useful legacy information on the variability of other molecular +species. The same dataset will also provide higher S/N observations of the line wings of phosphine, +thanks to the improved stability of the ’¯U’¯u receiver over RxA and our multiple observations. This will +Figure 3. +The atmospheric structure of Venus, showing how temperature and pressure vary with height. The +cloud layer at around 55km altitude has temperature and atmospheric pressure levels that are comparable to +those on the surface of the Earth. [29] +a magnetic field, and thus the possibility of liquid water oceans, for several billion +years after its formation [31]. +Computer simulations have been used to asses whether a warm wet Venus in the first +billion years of the Solar System would have a stable climate that could be conducive +to the emergence of life [32]. While the conclusions of this work are still not agreed +- some suggest that even a wet Venus would never have been able to condense its +water into oceans, leading to a so-called ‘steam Earth’ scenario [33] - it is intriguing +to consider the possibility that Venus may in fact have been the first habitable planet +in the Solar System. A warm wet Venus (see Figure 4) might have remained habitable +until as recently as about 700 million years ago, allowing substantial time for life +to evolve and propagate across its surface given that life seems to have emerged on +Earth somewhere between 3.7 and 4.3 billion years ago [12]. If this is correct, it may +only be in the most recent 15% of the age of the Solar System that massive volcanic +activity on Venus established a runaway greenhouse effect on the planet, leading to +water stripping and the hot, dry, hostile surface that we see today. +5. Atmospheric mysteries +As well as uncertainty about the role of water in Venus’ past, there are also some +significant mysteries about the planet’s atmosphere today even before we get to the +detection of phosphine. Long-standing issues include ([34] and references therein): +• The variation of the abundance of water vapour and SO2 with altitude in and +above the cloud layers [35]. Water, H2O, persists throughout the atmosphere but +SO2 levels drop from parts per million abundances below the clouds to parts per +billion above. This is not what is expected given that both gases are thought +7 + +Altitude [km] +Temperature[°C] +Pressure[bar +100 +10-4 +-100 +10-3 +80 +Upperhaze +10-2 +-45 +Upper.cloud +10-1 +60 +.Temperatezone +Middle&Lower.cloud +60 +Y7 +40 +Lower-haze +220 +10 +20 +Surface haze +45 +0Figure 4. +An artist conception of what a warm, wet Venus might have looked like during earlier stages in +the evolution of the Solar System. Credit: NASA +to be released by volcanism at the surface and to be well mixed throughout the +atmosphere until both are destroyed by solar UV at altitudes of 70 km or higher. +The apparent depletion of SO2 in the clouds is currently not understood. +• Oxygen, O2, is present in the clouds of Venus where it was detected by gas +chromatographs on board Pioneer Venus Probe [36] and the Venera 13 and 14 +descent modules [37]. There is currently no known process by which oxygen can +be formed in the cloud layers so its origin is something of a mystery. +• Observations of the clouds of Venus in the ultraviolet reveal complex spatial +and temporal changes in absorption and reflection [8]. This is in contrast to +observations in the optical and near-infrared where the clouds of Venus appear +nearly featureless on the dayside (see eg. Figure 1). Significant cloud contrasts +are only seen in reflected sunlight at wavelengths shorter than about 400 nm, +and at near-infrared wavelengths (1.7 to 2.4 µm) in emission on the nightside. +These variations in ultraviolet absorption were first observed in photographic +observations in 1928 [38], but, despite all the ground-based observations, space- +craft observations from orbit, and descent probes sampling the atmosphere, the +chemical and physical origin of the absorber responsible remains unknown. +• The clouds of Venus contain a variety of constituents. Based on size analysis from +the Pioneer Venus Probe particle size spectrometer [39] they can be divided up +into three particle sizes. These correspond to aerosols, of size ∼0.4 µm, droplets +of size ∼2 µm, and larger particles of size ∼7 µm. The larger particles are present +only in the middle and lower cloud layers, at altitudes from 47.5 to 56.5 km above +the surface. The nature of these largest particles, which may have a substantial +solid component, and be non-spherical in shape, is currently unclear. +8 + +• Recent reanalysis of the chemistry of Venus’ atmosphere based on measurements +by mass spectrometers on descent probes suggest the possibility of chemical +disequilibrium in the middle cloud layers [40]. This result is based on the presence +of several species in the mass spectrometer data, including hydrogen sulphide, +nitrous, nitric & hydrochloric acids, carbon monoxide, ethane, and hydrogen +cyanide as well as phosphine (this reanalysis was conducted after the detection +of phosphine by ground based observations, of which more later) and possibly +ammonia. This chemical mix indicates that reducing chemistry is taking place +in the clouds. If so, then the processes behind this activity would be out of +equilibrium with the oxidising chemistry of Venus. In the context of the search +for life elsewhere, chemical disequilibrium is a potential biosignature, making +these results very interesting. More recently still, it seems that ammonia, NH3, +may have been independently detected in Venus’ atmosphere by ground based +observations (Greaves et al., private communication), confirming the tentative +results from the Pioneer Venus Probe mass spectrometer reanalysis, and adding +an extra piece of evidence in favour of chemical disequilibrium in the clouds of +Venus. +These problems with our understanding of the atmosphere of Venus, and specifically +its clouds, make the case for renewed interest in the planet as a possible astrobiology +target. Even if biological processes do not provide the explanation for these poorly +understood phenomena, it is clear that there are chemical and physical processes +underway in the clouds of Venus that we do not currently understand. Further obser- +vations to explore the chemistry of Venus and its atmosphere are thus needed. It is +in this context that a team of astronomers proposed to look for phosphine, PH3, on +Venus3. +6. The Search for Phosphine +Phosphine, as we have seen above, has been proposed as a potential biosignature +gas which might be present in significant quantities on inhabited planets orbiting +other stars [22]. Current observational facilities, however, are not yet able to make +phosphine observations of terrestrial exoplanets. While we know that phosphine is +present in the Earth’s atmosphere in small amounts, thanks to industrial processes +and anaerobic organisms, there were no limits on the amount of phosphine present +on other Solar System terrestrial planets. Mercury has essentially no atmosphere so +is an inappropriate target. Mars has a very thin atmosphere so is unlikely to have +much phosphine even if there is biological activity underway there, and any phosphine +present would be rapidly destroyed by solar UV radiation. This leaves Venus as the +only reasonable terrestrial planet in the Solar System where some test observations in +search of phosphine might be conducted. +Therefore in 2016 a team of astronomers led by Prof Jane Greaves put together a +proposal to the James Clerk Maxwell Telescope (JCMT) to conduct test observations +of Venus’ atmosphere to look for absorption from the J=1-0 rotational transition of +phosphine, which would produce an absorption line at a wavelength of 1.123 mm (∼ +267 GHz). There are other transitions of phosphine at other wavelengths, notably in the +far-IR and in the mid-IR, but this particular transition has some advantages, Firstly, +3The author of the current paper was part of this team and is part of the ongoing work to study phosphine +on Venus. +9 + +observations can be conducted from the ground. Observations of the next highest +rotational transition, J=2-1, would require observations from the stratosphere (see +below). Mid-IR observations of other transitions can be conducted from the ground, +of which more later, but the Greaves team did not have easy access to the necessary +mid-IR facilities. +The initial idea for the observations was to acquire a few hours of data to better +understand the observational issues with looking for a weak absorption line against +a very bright continuum source, Venus, with the eventual intent to propose a longer +series of observations to set a stringent upper limit, since phosphine was not expected +to be found. That is not, however, how things turned out. +6.1. JCMT Observations +The JCMT is a 15 m diameter mm/submm telescope on the mountain Mauna Kea on +the Big Island of Hawaii, at an altitude of about 4000 m. Mauna Kea is an ideal site for +mm/submm observations since it is both high and dry, and so avoids much of the water +vapour in the atmosphere of the Earth that would otherwise absorb and contaminate +observations at these wavelengths. It is equipped with an array of instruments that +operate both as continuum imagers and as high resolution spectrometers. For the first +set of JCMT phosphine observations [6] an instrument called Receiver A3 (RxA3) was +used. RxA3 was one of the early instruments used on the JCMT and was delivered to +the telescope in 1998. It was retired not long after the phosphine detection observations +reported in [6]. +Like many mm/submm spectroscopy receivers, RxA3 uses a heterodyne approach, +whereby the incoming astronomical signal, in this case at frequencies of around 267 +GHz, is multiplied by a pure sine wave signal at a nearby frequency, the so-called +local oscillator frequency, using a device called a mixer. This results in the production +of a signal at a frequency that corresponds to the difference in frequencies between +the received signal and the local oscillator frequency, and allows signals at frequencies +outside the frequency range of interest to be removed. This lower frequency signal, at +what is called the intermediate frequency, or IF, is then measured and dealt with by +later stages of processing. The technology used in most mm/submm receivers relies +on SIS mixers (superconductor-insulator-superconductor) to mix the astronomical and +local oscillator frequencies. For more information on how these operate, and on much +else in radio astronomy, see [41]. +The IF signal from RxA3 is then processed by the Auto-Correlation Spectral Imag- +ing System (ACSIS - this system is used by all spectral receivers at the JCMT, includ- +ing both RxA3 and its replacement ’¯U’¯u). This digitises the input IF signal, calculates +the autocorrelation of the signal with itself - essentially multiplying the signal by a +time delayed version of itself - and then calculates the Fourier transform of the auto- +correlated signal. According to the Wiener-Khinchin theorem [41], the autocorrelation +of a signal is the Fourier transform of the signal’s power spectral density ie. the amount +power received as a function of frequency. Fourier transforming the autocorrelation of +the IF signal thus gives us what we want - the spectrum of the source in the frequency +range of interest. +Venus was observed by the JCMT using RxA3 in search of phosphine on five morn- +ings in June 2017. These dates were chosen so that Venus appeared large enough to +fill the telescope beam, minimising any effects due to errors in pointing the telescope. +Venus is a strong continuum emitter at millimetre wavelengths, so phosphine would +10 + +Figure 5. +The James Clerk Maxwell Telescope, a 15 m diameter mm/submm telescope on Mauna Kea in +Hawaii, currently operated by the East Asian Observatory. The 15 m primary mirror is protected from wind +during observations by a large gortex screen which is why it cannot be seen directly even when the telescope +is taking observations, as in this picture. Credit: William Montgomerie/EAO/JCMT. +be detected as a weak absorption line against this strong continuum. This strong +continuum, however, leads to a number of problems with the quality of the data. A +number of effects, including reflections from the floor or roof of the telescope dome, or +in the receiver cabin itself, entering the beam, lead to strong, time varying baselines +in the output spectra. These have to be detected and removed. For the initial JCMT +detection of phosphine [6] these effects were removed by the usual method of fitting +polynomial functions to the data, excluding the region of the spectrum where phos- +phine might lie. Once this process was applied to each of the 140 spectra that made +up the observations, and despite the original assumption that only an upper limit +would be found, an absorption line ascribed to phosphine was detected, corresponding +to an abundance of about 20 to 25 parts per billion (ppb). The JCMT spectrum of +phosphine can be seen on the right hand side in Figure 6. +6.2. ALMA Observations +Following the rather surprising detection of phosphine at the JCMT some further +observations in search of independent confirmation of this discovery were needed. To +this end, observing time was granted on the Atacama Large Millimetre/Submillimetre +Array (ALMA) in March 2019. Despite operating at similar mm/submm wavelengths, +ALMA is a rather different telescope to the JCMT because it is an interferometer. It +is made up of 66 separate antennae, mostly 12m in diameter, the signals of which are +combined together to produce the final results. 43 of the 12 m antennae were used for +the Venus phosphine observations. +11 + + +Figure 6. +The Phosphine 1.123mm J=1-0 line as detected by ALMA (left) [42] and JCMT with RxA3 +(right) [6]. The black lines indicate the level of SO2 absorption derived from simultaneous (ALMA) and near- +simultaneous (JCMT) observations. As can be seen the PH3 detections are clear and the SO2 contamination is +minimal. These spectra are continuum subtracted, so zero on the y-axis represents the continuum level. We use +the standard astrophysics approach for presenting high resolution spectra in this Figure, where the spectrum +is centred on the line of interest at zero velocity and frequencies are indicated by the doppler velocity in km/s +needed to shift from this central value. +12 + +-d compound coordinate system +.0002 +1.0001 +ine:continuum +0.0001 +-0.0002 +-50-40-30-20 -10 +0 +10 +20 +30 +40 +50 +Venus-framevelocity(km/s).0002 +0.0001 +line:continuum +-0.0001 +-0.0002 +-0.0003 +-50 -40 -30 -20 -10 +0 +10 +20 +30 +40 +50 +Venus-frame velocity (km/s)Figure 7. Some of the 64 antennae that make up the ALMA telescope. Credit: ESO/C. Malin. +While the signals received by each ALMA antenna are dealt with in a manner +similar to RxA3 on the JCMT, using a heterodyne SIS mixer and local oscillator in +the receiver, these signals are then cross-correlated pairwise with those from each of the +other antennae in the array (where each pair of antennae forms a ‘baseline’) to produce +an interferometric map of the target. Interferometry allows angular resolutions to be +achieved that correspond to a telescope whose diameter equals the longest baseline +separating individual antennae. +The cross-correlation of signals detected by each pair of antennae produces a series +of ‘visibilities’ which are a measure of the two-dimensional Fourier transform of the sky +distribution of brightness. The visibilities at each observed frequency are then Fourier +transformed to produce a series of images at successive frequencies, i.e. a spectral +cube. However, since only a finite number of antennae pairs are available, even for +an array with as many antennae as ALMA, the Fourier plane is like a telescope with +lots of holes. Various methods are used in a process called ‘cleaning’ to derive the +actual image from the limited sampling in the Fourier plane. For more information on +interferometry see [41] or the ALMA Primer4. +Processing interferometric data involves different challenges to those encountered at +the JCMT. For example, the angular size of Venus was so great that even the shortest +ALMA baselines could not provide good images on the scale of the whole disc and the +imperfect sampling led to strong ripples so the data from the affected short baselines, +all less than 33 m in length, were removed. There were also strong spectral ripples on +some parts of the planet, such as the poles, which had to be excluded from further +analysis otherwise they would add noise to the spectra, reducing the sensitivity of the +final results. Further analysis of the ALMA data processing also found some errors in +4https://almascience.eso.org/documents-and-tools/cycle9/alma-science-primer +13 + +the standard reduction script used, see Section 6.3.1, which improved on the initial +detection. The end result of the ALMA observations once all these various effects are +taken into account is shown in Figure 6 - a good detection of phosphine absorption +at a level of ∼20 ppb that matches what was seen by the JCMT but with somewhat +higher signal-to-noise. +6.3. The Detection of Phosphine from the Ground +The detection of phosphine in the atmosphere of Venus was, to say the least, a surprise. +The observations from JCMT and ALMA thus prompted a considerable amount of +debate and further observations using other facilities. In this section we look at these +various discussions, their conclusions, and counter-arguments to the suggestion that +phosphine has not been detected or that whatever has been detected was not phos- +phine. +6.3.1. Reanalysis of the ALMA Data +One of the first responses to the Greaves et al. detection paper, [6], was a reanalysis +of the ALMA data by a separate group [43]. This analysis did not reproduce the phos- +phine detection of [6], and instead found an upper limit to the phosphine abundance of +about 1 ppb. They identified some processes used in the standard ALMA calibration +scripts which were not adequate for a very bright, time-varying, beam-filling target or +indeed for the correspondingly bright calibrator sources used. This led to reprocessing +of the raw data by the ALMA observatory and European Southern Observatory (ESO) +staff (independently of any of the research groups), who provided new scripts taking +these and additional problems into account. The reprocessing simplified the basic re- +moval of instrumental bandpass ripples using the moon Callisto as a calibrator, and +also avoided the chance of spectral averaging producing sharp edges which could mimic +an absorption line. The new scripts also accounted for the non-linear instrumental re- +sponse to the high intensity of Venus (the brightest source in the sky after the Sun at +these wavelengths) and its large angular size, although, since this exceeds the extent +of accurate models of the response of individual ALMA dishes, this is thought to be a +source of residual error. +Greaves et al. responded to this reanalysis [42,44] by employing the improved ob- +servatory scripts and updating their own processing, using three different independent +methods to obtain final images and spectra. The first step after observatory calibration +is to remove the shortest baselines as explained above, and then to make a simple, +linear spectral fit to the visibility data to remove the contribution of Venus. Next, +residual spectral ripples can be corrected either in the visibility data or after Fourier +transforming to make an image cube, and before or after cleaning. Spectra were ex- +tracted over different portions of the planet; small residual errors meant that only +those spectra extracted from regions symmetric about the planet centre were consid- +ered reliable. A range of parameters allowed the continued recovery of a phosphine +signal using all the updated methods, optimised at 7.7σ significance by excluding the +planetary poles [42]. They attributed the non-recovery of the phosphine signal by [43] +as a result of including baselines shorter than 33 m in most of their analyses, as well as +including parts of the image of the planet that had significant spectral artefacts that +raise the noise in the final combined spectrum. They concluded that the phosphine +detection in the ALMA data remained robust. +14 + +6.3.2. Was it a real line? +A common feature of both the original ALMA and JCMT data analyses in [6] was the +use of fairly high order polynomials to allow the removal of varying baselines. In doing +this, it is necessary to mask out the region of the spectrum around a suspected line +otherwise the polynomial fitting method might fit and remove a real line, mistaking it +for a small scale baseline ripple. Several authors suggested that this process can instead +lead to the creation of fake lines, and that this was in fact the origin of the claimed +phosphine detection [43,45,46]. There are two counter-arguments to this suggestion +that the detection is essentially a statistical false positive. +The argument that the claimed phosphine detection is a false positive is that when +you take the ripple-contaminated spectrum, block out a portion of it where there might +be a line, and use a sixth or higher order polynomial to fit the baseline, then some +noise spikes or contaminating ripples in the blocked out section may end up looking +like a line. This is in fact correct, and blind searches for line candidates at random +locations in the spectrum would indeed suffer from this effect, significantly reducing +confidence that any detections are real. However, the detection of phosphine in [6] +did not solely rely on measuring the depth of an absorption line at a random position. +Instead, it also relied on the wavelength of the line seen coinciding with that of the line +being searched for, phosphine. This significantly reduces the chance of a noise spike +or residual masquerading as a phosphine detection. Analysis in [47] shows that adding +the additional constraint that a fake line must be at a specific frequency reduces the +chance of a false positive for line detection to < 1.5%. +Furthermore, if the line was in fact a false positive then there would be no reason +for any such noise-generated feature to lie at exactly the same frequency in both +the ALMA and JCMT data. As pointed out in [6], the only feature at matching +wavelengths in both the ALMA and JCMT data lies at the expected frequency of +phosphine. This further bolsters our confidence that the detected phosphine line is +real, and not a statistical artefact resulting from the data processing approach. +6.3.3. Is it really phosphine? +The foregoing analysis suggests that the line discovered is in fact real and not a +statistical false positive. However, can we be sure that it is in fact phosphine and not +some other molecular species that happens to have an absorption feature at a similar +frequency? Sulphur dioxide, SO2, a known constituent of Venus’ atmosphere, has a +transition due to the (J = 309,21 − 318,24) transition at 266.943329 GHz, a frequency +shift from the PH3 J = 1-0 line at 266.944513 GHz that corresponds to a velocity +difference of just 1.3 km/s. The possibility that the claimed phosphine line is actually +a misidentification of this SO2 line was first suggested by [43] and has been further +explored by others [48,49]. While they have concluded that SO2 contamination or +misidentification is a possibility, a number of problems with this interpretation have +been pointed out by [42]. Firstly, while the line centres of PH3 J=1-0 and SO2 J = +309,21 − 318,24 are close, they are still 1.3 km/s apart, leading to a ∼ 3 σ discrepancy +between the measured line centre and that expected for the SO2 line. Furthermore, +simultaneous (in the case of ALMA) and near-simultaneous (in the case of JCMT) +observations of a different and stronger SO2 line [42] provide predictions of the relative +strength of the SO2 transition that might contaminate the phosphine line. They find +that the level of contamination of the phosphine line by SO2 is ∼ 10% for the JCMT +data and < 2% for the ALMA data. This level of contamination by SO2 is shown as +a black line in Figure 6. On this basis it seems likely that the detected line is indeed +15 + +phosphine, and that any contamination by the neighbouring SO2 line is insignificant. +6.3.4. Other Observations +The phosphine J=1-0 line at 1.123 mm is not the only line of this molecule. However, +many of the other transitions are at wavelengths that are more difficult to observe from +the ground. Nevertheless, observations have been attempted of other lines in search of +independent confirmation of the presence of phosphine. +The first of these used archival data from the TEXES (Texas Echelon Cross Echelle +Spectrograph) instrument, a 5 to 25µm high resolution mid-infrared spectrometer, on +the NASA Infrared Telescope Facility (IRTF) on Mauna Kea in Hawaii [50]. These +observations were part of a long term project to monitor SO2 and H2O in the cloud tops +of Venus, and involved observations at a range of frequencies. One of these datasets, +obtained in March 2015, fortuitously included a range of wavelengths where there +are some relatively strong phosphine transitions, at a wavelength around 10.471 µm +(corresponding to a frequency of 28.65 THz). No phosphine absorption is detected, +indicting an upper limit of about 5 ppb, which is substantially lower than the claimed +millimetre wave phosphine detection. +Further infrared data, this time from the Venus Express spacecraft, were analysed, +looking for absorption from phosphine lines at wavelengths around 4.125 µm above the +cloud layers [51]. This data was taken at various times from June 2006 to December +2014, and measured absorption against the light of the Sun as it rises or sets, rather +than against the emission of Venus itself. This means that only a small part of the +atmosphere is studied rather than the entire planetary disk as is the case, for example, +for the JCMT or TEXES observations. These Venus Express observations also failed +to find any phosphine absorption, setting limits on its abundance of 0.2 to 20 ppb +depending on the specific observations and the assumed altitude of the absorption, +ranging from 60 to 95 km. +A third approach to confirm the detection of phosphine is to search for absorption +lines in the far-infrared, at frequencies around 534 and 1067 GHz [52]. While the +Earth’s atmosphere is completely opaque at these frequencies at sea level and even on +tall mountains like Mauna Kea, the SOFIA observatory (Stratospheric Observatory +For Infrared Astronomy) - essentially a 747 Jumbo Jet with a hole cut in the fuselage +with a 2.5m telescope pointing out (see Figure 8) - can perform these observations +since it flies at an altitude of about 13 km, above much of the water vapour that +absorbs far-IR radiation in the Earth’s atmosphere5. +SOFIA observations of Venus in search of phosphine were carried out in November +2021 [52] using the GREAT (German REceiver At Terraherz frequencies) instrument, a +receiver similar to the JCMT receivers but operating at much higher frequencies. Data +reduction and analysis by the original authors failed to find any sign of phosphine, +setting an upper limit of 0.8 ppb from the J=4-3 line and ∼ 2 ppb for the J=2-1 +line. However, subsequent reanalysis of the SOFIA data found that the calibration +stage that sets an absolute flux scale adds noise and artefacts to the resulting spectra. +This calibration stage is not needed if we are only interested in the line-to-continuum +ratio, as is the case when measuring an absorption line. By purely analysing the line- +to-continuum ratios [53], phosphine at a level of ∼1-2 ppb is found, averaged over +altitudes from 75-110 km, with 6.5σ significance. +These other observations in search of phosphine absorption using different ap- +5Sadly such observations can no longer be performed since the SOFIA observatory was decommissioned and +retired at the end of September 2022. +16 + +Figure 8. +The SOFIA observatory, which consists of a 2.5m telescope and instruments mounted inside a +747 jumbo jet, and capable of flying above much of the far-IR absorption in the Earth’s atmosphere. Credit: +NASA/DLR +proaches, whether from the ground or from Venus Express, have produced a number +of conflicting results. They need to be carefully interpreted since the different wave- +lengths and observational approaches are in fact probing the presence of phosphine at +different altitudes and times, as we shall see below. None has yet definitively disproved +the original JCMT and ALMA results of [6], and the SOFIA observations may in fact +have provided some level of confirmation, depending on which analysis approach is +used. +6.3.5. In Situ Confirmation +The ideal way to determine the presence and amount of phosphine in the atmosphere +of Venus would be to send a space probe directly into the atmosphere equipped with +instrumentation that can detect and measure the presence of the gas in situ. This +would avoid all the difficulties of observing phosphine remotely, as well as all issues +of interpretation. At this point, as we will see below, we are some years away from +any such future mission. However, past missions to Venus did send probes into the +planet’s atmosphere. Principal among these, for our current purposes, is the Pioneer +Venus Multiprobe (also known as Pioneer Venus 2 or Pioneer 13) [54] (see Figure 9) +which, among many other instruments, sent a mass spectrometer into the atmosphere +of Venus on its largest entry probe. +Data from the Pioneer Venus Large Probe’s Neutral Mass Spectrometer (LNMS) +were reanalysed in 2021 [40] subsequent to the announcement of the discovery of +phosphine by [6]. This reanalysis of data taken during its descent into the atmosphere +of Venus on 9 December 1978, was the first to look for trace or minor constituents +17 + +DLR +SOFIA +N747NFigure 9. +The Pioneer Venus Probe Spacecraft. The mission consisted of an orbiter and four probes that +were sent into the atmosphere of Venus, seen here in artists conception. Credit: NASA +of the atmosphere beyond methane and water. The LNMS takes gas in from the +atmosphere through inlet tubes. Molecules in the gas are then ionised by an electron +source, accelerated by an electric field and then passed through a magnetic field which +deflects the ions by an amount that depends on their mass. These ions are subsequently +detected, allowing their mass and abundance to be determined. For more information +see [55]. +The data reanalysed in search of phosphine came from within the clouds, at an alti- +tude of 51.3 km above the surface, part of the atmosphere that is largely inaccessible +to ground or space based observations, but which is of critical importance to searches +for possible life in the clouds, as this is where that life might actually live. The detailed +analysis found evidence for phosphine at 0.1 to 2 parts per million (ppm) levels in the +clouds themselves, a much higher abundance than is seen in the JCMT or ALMA ob- +servations. It also found evidence for other species such as nitrite, nitrate, nitrogen and +possibly ammonia. Taken together these molecules indicate that unexpected chemical +processes are underway in the clouds and suggest chemical disequilibrium. Whether +this disequilibrium is due to biological or some other as-yet unknown chemical process +is yet to be determined. +6.4. Where and When is the Phosphine Seen? +The forgoing sections, describing the various observations in search of phosphine in +the atmosphere of Venus, can seem confusing and mutually contradictory. This is at +least partly because observational constraints mean that they sample the atmosphere +of Venus at different altitudes and, because they encompass datasets that span over +40 years, at different times. We already know that some species in Venus’ atmosphere +18 + +are highly variable - SO2 levels, for example, can vary by large factors on timescales +of both years and days at various altitudes [56] - so this may also apply to phosphine. +Spatial variations across the disk of the planet are also possible, but this is difficult to +assess for phosphine since many of the observations to date have been of the average +phosphine level across the planetary disk. +Of particular importance is the amount of phosphine in the atmosphere as a func- +tion of altitude. While the in situ observations of the LNMS and Venus Express have a +clearly determined altitude, this is harder to extract for the observations from Earth. +In principle, the effect of pressure broadening on the absorption lines can be used to +determine the vertical abundance profile of an absorbing molecule. Pressure broaden- +ing of an absorption line occurs when the absorbing molecules interact collisionally +with other molecules in the atmosphere. These interactions shorten the characteristic +time of the absorption process, in accordance with Heisenberg’s uncertainty principle, +increasing the uncertainty of the absorption frequency and thus broadening the line +(see eg. [57]). The overall effect is to make the line shape a Lorentzian function, which +has much broader wings than the usually assumed Gaussian shape. The exact width of +the Lorentzian line depends on the pressure, temperature and nature of the molecules +that are interacting. The higher the pressure, the broader the wings, so a full analysis +of the shape of the phosphine absorption line can reveal its vertical abundance profile +in the atmosphere of Venus. +There are, however, a number of problems with a full pressure broadening analysis +of the phosphine line seen in the atmosphere of Venus. Firstly, the pressure broadening +coefficient for phosphine in CO2, the dominant constituent of Venus’ atmosphere, is +not currently known. Analyses have so far used either a modification of the phosphine +broadening coefficient in air [50] or have used the CO2 pressure broadening coefficient +for NH3 as an analog to that of phosphine [6]. Secondly, and more significantly for +the immediate understanding of phosphine in Venus, the data reduction techniques +used to date to extract the absorption line remove any broad line wings as part of +the process that removes baseline ripples. This just leaves narrow line cores, meaning +that the observations are insensitive to any significantly broadened lines, and thus are +only sensitive to phosphine at altitudes of 75 to 80 km. Most recently, an experimental +data processing approach applied to new observations of Venus from the JCMT-Venus +project (PI: D.L. Clements) seems to be able to recover the broad line wings of the +J=1-0 phosphine line, suggesting an abundance of phosphine at the ppm level inside +the clouds at an altitude of about 60 km, consistent with the high levels seen by the +LNMS. +The other factor to consider is the timing of the observations. While we do not yet +have enough observations to allow us to monitor any changes in the abundance of +phosphine with time or in relation to other species such as HDO or SO2, we can see +if there are any correlations between the amounts of phosphine seen and the timing +of the observations relative to the illumination of Venus’ atmosphere by the Sun. +This may well be an important factor since photolysis by sunlight is a significant +destruction route for phosphine in the Earth’s atmosphere [22]. If we combine all the +phosphine observations - detections and non-detections - together with information +about whether the Sun is rising or setting on the atmosphere at the time of observation +we perhaps begin to see a pattern (see Figure 10). +19 + +Figure 10. +The trend of phosphine abundance by altitude from the currently available data. Shading indi- +cates cloud (orange, centred at ∼ 60 km) and haze (grey, centred at ∼ 80 km and 40 km) layers of Venus’ +atmosphere. Superposed symbols indicate candidate detections plus upper limits for phosphine abundance. +Rising arrows indicate observations made where the atmosphere was rising into sunlight and falling arrows +indicate observations made when the atmosphere was descending towards the nightside. Abundances are, from +top: 20, 25 ppb from J=1-0 data [6]; ∼1 ppb or < 0.8 ppb from J=4-3 data [52,53]; < 7 ppb at 62 km from +4 µm spectra [51]; < 5ppb at 60 km from 10 µm spectra [50]; ∼2 ppm at 60 km from initial JCMT-Venus +processing; high ppb to 2 ppm at 51 km from Pioneer-Venus in situ sampling [40]. As can be seen all the +significant detections of phosphine take place as the atmosphere is moving out of night and into sunlight, while +the non-detections take place as the atmosphere is moving from sunlight into night. If sunlight destroys phos- +phine at high altitudes during daylight, as is the case on Earth, this would explain the apparent contradictions +between some of the observations. From: Greaves et al. in prep, by permission. +20 + +altitude (km) +100 +detection (gas entering sunlight) +detection (gas departing sunlight) +80 +upper limit (gas departing sunlight) +60 +40 +20 +- +log-abundance(ppb) +0.1 +1 +10 +100 +1000Figure 11. +Potential chemical pathways for the synthesis of phosphine in the atmosphere of Venus, and +their derived production vs. destruction rate. There are stages where, for all possible pathways, the rate of +destruction of phosphine exceeds its formation by many orders of magnitude, as shown in red/purple. As can +be seen, there is no route to produce phosphine by these processes that can account for the amounts observed. +From [58] where more details can be found. +7. The (Im)Possible Origins of Phosphine on Venus +The presence of phosphine in the atmosphere of Venus is a surprise since a compound +of phosphorous with hydrogen should not naturally appear in the atmosphere of a +planet, such as Venus, which has an oxidised atmosphere. On Earth, phosphine does +not occur through normal chemical processes and is produced only by anaerobic life or +through human industrial activity. While this is the expectation, Venus is a complex +environment with a wide range of chemical and physical processes underway from the +surface to the top of the atmosphere. A detailed analysis is thus necessary to see if +there are any possible routes through which the levels of phosphine seen might occur +through normal chemical processes. Such an analysis was conducted in [58] where a +wide range of chemical processes were examined to see if there is any potential source +of phosphine in sufficient abundance to explain the observations with processes that we +know are underway on the planet. The processes examined included gas reactions, geo- +chemical reactions, photochemistry, volcanism (see also [59]), lightning and impactors. +An example of the kind of chemical reaction network considered is shown in Figure +11, where the reaction rate and the destruction rates are compared. Only segments of +this reaction network where the ratio of the production rate over the destruction rate +is ≥ 1 can produce an accumulation of the relevant chemical. For phosphine to be pro- +duced in significant amounts the whole reaction network must have this ratio ≥ 1 but, +as can be seen, critical segments of the network have ratios orders of magnitudes less +than this. More generally, it was found that the lifetime of free phosphine at various +altitudes in the atmosphere of Venus ranged from < 1 second to perhaps a century +[59], making it highly unlikely that a significant amount of phosphine can accumulate +from any hypothetical source. +The most obvious conclusion that can be drawn from this analysis is that we do +not know how phosphine came to be in the atmosphere of Venus. There may be +geochemical or photochemical processes that can produce it in sufficient amounts, but +these are currently not known to us. The alternative, that, by analogy with Earth, +phosphine is being produced by anaerobic biological processes, is another potential +explanation. However, before we can make this particular leap, and claim that we +21 + +>1 +1 - 10-3 +Rateofforward reaction +Ratio +10-3- 10-6 +Destructionrate +10-6- 10-9 +<10-9 +H. +H2 +H20 +H. +H20 +OH. +H2 +H. +H. +H. +H20 +A +H2PO3 +HPO, +PO +H.PO +PH +OH +0 +H. +H.o +OH +02 +H. +H +H, +H2Q +H. +OH.: +H. +H. +H20 +H2 +H. +H. +H. +OH: +0. +H,PO4 +HPO3 +PO2 +PH2 +HPO +H. +H,0 +H. +H. +0. +H20 +OHhave found evidence for life in the clouds of Venus, we must first exclude all other +possible origins, and also explain how life is able to survive in the extremely acidic +environment of Venusian cloud droplets. One possible solution to the latter problem is +that ammonia, if present, is able to buffer the sulphuric acid in these droplets to some +extent [34]. The possible detection of ammonia in the clouds of Venus by the LNMS +[40] and in preliminary analysis of data from the Green Bank Telescope (Greaves et +al., private communication), is thus rather interesting. +8. The Next Steps +As has become clear in the previous section, studies of phosphine, and the search for +life on Venus, are very much works in progress. While the current results are intriguing, +there are no solid conclusions that can yet be determined. Much more work needs to +be done, and it will be the work of many years before we can have a definitive answer +to the question of whether there is life in the clouds of Venus. This will require not only +observations from Earth, but also in situ probes and, ideally, missions that can return +samples from Venus to Earth. In this section we look at some of the projects that are +planned or already underway to improve our knowledge of the clouds of Venus. +8.1. Earth-based Studies +Observations from Earth were responsible for the first detections of phosphine, and +these are continuing to both monitor phosphine and to search for other molecules +that may have a bearing on the chemistry, or biochemistry, underway in the clouds of +Venus. +The largest of the projects currently underway is JCMT-Venus (PI: D.L. Clements). +This uses the ’¯U’¯u receiver, the replacement for RxA3, together with the ACSIS system +to obtain whole disk spectra for Venus. The new receiver has a wider bandwidth than +RxA3 so we can simultaneously observe phosphine, HD and SO2, and search for other +molecules such as SO and PO2 which have spectral features in the band covered by +’¯U’¯u. By simultaneously monitoring phosphine, HDO and SO2 we can see how these +different species vary in relation to each other. This should provide indications as to +the chemical processes behind the presence of phosphine. If, for example, phosphine is +produced by reducing processes in the upper atmosphere, the proportions of reduced +compounds, like phosphine, and oxidized compounds, like HDO and SO2, will be +anticorrelated. The JCMT-Venus project is a long term programme at the JCMT and +has been awarded 200 hours of time over a period of three years. The visibility of +Venus means that observations will be possible in three tranches, including Feb 2022, +July 2023 and September 2023. The first of these observing campaigns has already +taken place, with Venus observed over a period of 20 consecutive mornings. The data +obtained already contains 140 times as much information as in the original JCMT +observations, so is taking some time to process and analyse, especially since ’¯U’¯u +has its own difficulties dealing with the brightness of Venus and thus an interesting +new set of baseline drifts and ripples to be removed. Nevertheless the analysis is well +underway and initial results, some of which have been briefly discussed above, have +already emerged, including further confirmation of the presence of phosphine. When +complete, JCMT-Venus will provide a major new database of observations of Venus in +the mm band, including phosphine and other important molecules which will provide +significant new insights into the origin of phosphine. +22 + +Further ALMA observations have yet to be approved, but these hold the promise +of providing further information about the distribution of phosphine across the face of +the planet. The original ALMA observations provided some hints that the distribution +is not uniform, but a full map of the abundance of phosphine across the planetary disk +could not be made because of excess ripples affecting the signal over significant portions +of the disk. Additional observations in the mid-IR from the IRTF and elsewhere will +also be helpful. Sadly, further observations with SOFIA are not possible since the +observatory has been decommissioned. +Studies related to the search for potential signs of life on Venus are also underway +that do not directly target phosphine. These include observations with the Green +Bank Telescope (GBT) at radio wavelengths to look for ammonia (NH3) in absorption. +This is important since detection of ammonia would indicate the presence of another +reduced molecule that should not be expected in the oxidised atmosphere of Venus. +Ammonia is also important because of its buffering effect against the high acidity in +the liquid droplets in Venus’ clouds [34]. Analysis of archival data from the 1970s as +well as an initial set of observations with the GBT suggest the presence of ammonia +(Greaves et al., private communication), as do the in situ measurements of the LNMS, +but more data is needed to confirm this. +Laboratory studies also have a role to play since they can validate and test the vari- +ous assumptions that went into the analysis of [34], and allow more accurate predictions +for the formation and destruction of phosphine and other hydrogen-rich compounds +in Venus-like conditions. Such studies are already being planned. +8.2. Space-based studies +Venus is also being studied from much closer quarters by space probes. These missions +take many years to prepare and so have largely not been designed to examine the +possibilities of unusual chemistry, or even life, in the clouds of Venus. Nevertheless, +existing missions do have useful capabilities for these purposes and future missions are +being planned that can respond to recent discoveries. +The Japanese mission AKATSUKI (see eg. Figure 1) is currently operating in orbit +around Venus. While it does not have any instruments that are directly relevant to +the search for phosphine, its UV imaging instruments are monitoring the unidentified +UV absorber, the origin of which is one of the outstanding mysteries of the Venusian +atmosphere. Comparing AKATSUKI’s results with future data from the ground, es- +pecially any future imaging observations with ALMA, may be able to see if there is a +link between the presence of phosphine and the presence of the UV absorber. +The next potentially important space mission to go to Venus, from the point of view +of phosphine observations, is not in fact a specific mission to Venus, but the JUICE +mission to the moons of Jupiter [5]. The JUICE spacecraft is scheduled to be launched +in the second quarter of 2023. It will then perform a series of flybys of planets as gravity +assists on its journey to Jupiter. Of particular interest here is the flyby of Venus in +August 2025 where an observational campaign is possible. Of particular importance in +the context of phosphine on Venus is the Submillimetre Wave Instrument (SWI) which +will be able to observe higher J transitions of phosphine, including those observed from +the Earth by SOFIA. Whether JUICE will be able to undertake an observing campaign +at Venus will be up to the JUICE mission directors, and no decision will be made until +after launch. +In the 2030s three missions directly targeted at Venus are due to be launched. +23 + +These include the European Space Agency’s EnVISION mission [60], and NASA’s +VERITAS (Venus Emissivity, Radio Science, InSAR, Topography, and Spectroscopy) +[61] and DAVINCI (Deep Atmosphere Venus Investigation of Noble gases, Chemistry, +and Imaging) [62] missions. VERITAS and EnVISION are primarily concerned with +the surface and interior of Venus, studying the history and role of volcanism on the +planet. While they will doubtless reveal much that is of interest, they are unlikely to +have much to say about phosphine and the processes underway in the clouds of Venus +unless they uncover volcanic activity vastly in excess of our current understanding +[59]. DAVINCI, however, is a much more interesting prospect. +The goal of DAVINCI is to study the atmosphere of Venus. To do this its primary +set of instruments are on board a descent stage that will fly through the clouds of +Venus, sampling the atmosphere as it goes. It will be the first NASA mission to enter +the atmosphere of Venus since Pioneer Venus Probe in 1978. Among the instruments +on the descent stage is a mass spectrometer that will be able to significantly improve +on the results of the LNMS. This will be able to detect phosphine and other trace gas +species and see how their abundance changes with altitude and other conditions. Other +instruments include a tuneable laser spectrometer which is able to measure even small +amounts of specific gases. Altogether, the four instruments on the DAVINCI probe, +combined with imagers on the orbiting mothership, will provide a vast improvement in +our in situ knowledge of Venus’ atmosphere. It will provide the ground truth against +which observations of the planet from Earth can be compared. +National and international space agencies are not the only organisations looking to +send probes to Venus. Private companies now have the capability to send missions +to other planets independently of governments, and they are also interested in the +possibility of life on Venus. One company in particular, Rocket Lab, is taking special +interest in Venus and has set up a team to develop a series of Venus Life Finder (VLF) +missions [63]. The first of these missions, which may launch as soon as mid-2023, is +intended to look for organic molecules using an ultraviolet autofluorescence technique. +Further missions are planned including a balloon borne laboratory that will be able to +float in the clouds for an extended period. Amongst the planned instruments for this +payload are not only mass spectrometers but also a microscope that will search cloud +droplets for evidence of biological cells. +Perhaps the most ambitious mission planned by the VLF team is a sample return +mission that will use a balloon to collect samples of cloud droplets and gas, and return +these to Earth for detailed laboratory study. If there is in fact life in the clouds of +Venus, a mission like this will be necessary to answer fundamental questions about its +origin and how it operates. It is perhaps the dream mission in the search for evidence +of life on Venus. +9. Conclusions +The discovery of phosphine in the atmosphere of Venus has caused some controversy +and has renewed discussions about the possibility of life in the planet’s clouds. The +observational evidence for phosphine has been challenged and examined in detail. The +JCMT and ALMA results have so far survived these challenges, and there has been +independent in situ confirmation of the presence of phosphine from the Pioneer Venus +LNMS instrument. Observations from other telescopes in search of phosphine have pro- +duced rather more mixed results, with several upper limits and one possible detection. +However, the apparent disagreement between these different sets of observations may +24 + +soon be understood in the context of day-night variations in the amount of phosphine +above the clouds thanks to photolysis by sunlight. +While the presence of phosphine in the atmosphere of Venus is becoming more secure +with the arrival of new and improved datasets such as JCMT-Venus, an understanding +of its origin still eludes us. It is clear that no conventional chemical process can produce +phosphine in the amounts observed, but it is still far from clear whether biological +processes are involved, or if there is some as-yet unknown non-biological source. +More data is clearly necessary for us to understand what is really going on in the +atmosphere of Venus, and this is being sought by a number of different ground and +space-based approaches. Over the next several years our understanding of the origin +of phosphine on Venus will certainly improve, and we will hopefully reach a point at +which the question of life in the clouds of Venus moves from being something that +we can only speculate about, to something about which we have clear and decisive +knowledge. Whatever conclusion we finally reach, we will have learnt a lot more +about our nearest neighbour planet, and this knowledge will help guide our search for +possible biospheres on planets orbiting other stars. Confirmation that there is in fact +life in the clouds of Venus would be a truly epoch making discovery, but we are still +a very long way from drawing that conclusion. +Acknowledgements It is a pleasure to thank Jane Greaves, Janusz Petkowski, +Anita Richards, and Wei Tang for many useful comments. It is also a pleasure to +thank all members of the phosphine team for their enthusiasm and expertise in what +has already been quite an exciting, and unexpected, adventure, which, for me, started +in a bar in Hilo. +References +[1] National Academies of Sciences, Engineering, and Medicine 2021. Pathways to Discovery +in Astronomy and Astrophysics for the 2020s. Washington, DC: The National Academies +Press. doi.org/10.17226/26141. +[2] Mayor, M, & Queloz, D., A Jupiter-mass companion to a solar-type star, 1995, Nature, +378, 355 +[3] Vago J., Witasse O., Svedhem H., Baglioni P., Haldemann A., Gianfiglio G., Blancquaert +T., et al., ESA ExoMars program: The next step in exploring Mars, 2015, Solar System +Research, Volume 49, 518 +[4] Farley, Keneth A, et al., Mars 2020 Mission Overview, 2020, SSRv, 215, 142 +[5] Grasset, O., et al., 2013, JUpiter ICy moons Explorer (JUICE): An ESA mission to orbit +Ganymede and to characterise the Jupiter system, Planetary and Space Science, 78, 1 +[6] Greaves, Jane S., et al., Phosphine in the cloud decks of Venus, 2021, NatAs, 5, 655 +[7] Morowitz, H. & Sagan, C. Life in the clouds of Venus? 1967, Nature 215, 1259 +[8] Limaye, S. S. et al. Venus spectral signatures and the potential for life in the clouds, 2018, +Astrobiology 18, 1181 +[9] Benner SA, Defining Life, 2010, Astrobiology 10, 1021 +[10] Vitas, M. and Doboviek, A., Towards a General Definition of Life, Origins of Life and +Evolution of the Biosphere, 2019, OLEB, 49, 77 +[11] Lutz, RA. and Kennish, MJ, Ecology of deep-sea hydrothermal vent communities: A +review, 1993, RvGeo, 31, 211 +[12] Dodd, M., Papineau, D., Grenne, T. et al., 2017, Evidence for early life in Earth’s oldest +hydrothermal vent precipitates. Nature 543, 60?64 +[13] Hand, K.P., Sotin, C., Hayes, A. et al., 2020, On the Habitability and Future Exploration +of Ocean Worlds. Space Sci Rev 216, 95 +25 + +[14] Edwards, KJ., Becker, K., and Colwell, F., 2012, The Deep, Dark Energy Biosphere: +Intraterrestrial Life on Earth, AnnRev Earth, 40, 551 +[15] Petkowski, JJ, Bains, W., Seager, S., 2020, On the Potential of Silicon as a Building Block +for Life, Life, 10, 84 +[16] Bains, W., 2004, Many Chemistries could be used to build living systems, Astrobiology, +4, 137 +[17] Lyons, T., Reinhard, C. and Planavsky, N. 2014, The rise of oxygen in Earth’s early ocean +and atmosphere. Nature 506, 307 +[18] Fulii, Y., and Matsuo, T., 2021, Detecting Atmospheric Molecules of Nontransiting Tem- +perate Terrestrial Exoplanets Using High-resolution Spectroscopy in the Mid-infrared +Domain, AJ, 161, 180 +[19] Janin,E., 2021, Exoplanet biosignatures, Astronomy & Geophysics, 62, 6.26 +[20] Seager, S., Bains, W., Petkowski, J.J., 2016, Toward a List of Molecules as Potential +Biosignature Gases for the Search for Life on Exoplanets and Applications to Terrestrial +Biochemistry, Astrobiology, 16, 465 +[21] Meadows, V., et al., 2018, Exoplanet Biosignatures: Understanding Oxygen as a Biosig- +nature in the Context of Its Environment, Astrobiology, 18, 630 +[22] Sousa-Silva, C., et al., 2020, Phosphine as a Biosignature Gas in Exoplanet Atmospheres, +Astrobiology, 20, 235 +[23] Bregman, J. D., Lester, D. F. & Rank, D. M., 1975, Observation of the ν2 band of PH3 +in the atmosphere of Saturn. ApJ. 202, L55 +[24] Lewis, John S., 2004, Physics and Chemistry of the Solar System (2nd ed.). Academic +Press, ISBN 978-0-12-446744-6 +[25] Titov, D.V, et al., 2016, Clouds and hazes of Venus. Space Sci. Rev. 214, 1 +[26] Krasnopolsky, V.A., 2013, Photochemistry of the Atmospheres of Mars and Venus +(Springer Science & Business Media, 2013) +[27] Morowitz, H., Sagan, C., 1967, Life in the clouds of Venus? Nature 215, 1259 +[28] Schulze-Makuch, D., 2021, The case (or not) for life in the Venusian clouds. Life (Basel) +11, 255 +[29] Seager, S. et al., 2021, The Venusian lower atmosphere haze as a depot for desiccated +microbial life: A proposed life cycle for persistence of the Venusian aerial biosphere. As- +trobiology 21, 1206 +[30] Donahue, T. M., J. J. Hoffman, R. R. Hodges Jr., and A. J. Watson, 1982, Venus was +wet: A measurement of the ratio of deuterium to hydrogen, Science, 216, 630. +[31] Gillmann, C, et al., 2022, The Long-Term Evolution of the Atmosphere of Venus: Processes +and Feedback Mechanisms, Space Science Reviews, 218, 56 +[32] Way, M.J., et al., 2016, Was Venus the first habitable world of our solar system? Geo- +physical Research Letters 43, 8376 +[33] Turbet, M., et al., 2021, Day-night cloud asymmetry prevents early oceans on Venus but +not on Earth. Nature 598, 276 +[34] Bains, W, Rimmer, P.B., Seager, S., 2021, Production of ammonia makes Venusian clouds +habitable and explains observed cloud-level chemical anomalies, PNAS, 118, e2110889118 +[35] Krasnopolsky, V.A., 2006, Chemical composition of Venus atmosphere and clouds: Some +unsolved problems. Planet. Space Sci. 54, 1352 +[36] Oyama V.I., et al., 1980, Pioneer Venus gas chromatography of the lower atmosphere of +Venus. J. Geophys. Res. Space Phys. 85, 7891 +[37] Mukhin, L.M., et al., 1982, VENERA-13 and VENERA-14 gas chromatography analysis +of the Venus atmosphere composition. Sov. Astron. Lett. 8, 216 +[38] Ross F.E., 1928, Photographs of Venus. ApJ 68, 57 +[39] Knollenberg, R.G., & Hunten, D.M., 1980, The microphysics of the clouds of Venus: +Results of the Pioneer Venus Particle Size Spectrometer Experiment, J. Geophys. Res. +Space Phys, 85, 8039 +[40] Mogul, R., Limaye, S.S., Way, M.J., Cordova, J.A., 2021, Venus’ Mass Spectra Show Signs +of Disequilibria in the Middle Clouds, Geophys. Res. Lett. 48, e2020GL091327 +26 + +[41] Wilson, T.L., Rohlfs, K., Huttermeister, S., 2013, Tools of Radio Astronomy, Springer +[42] Greaves, J.S., et al., 2022, Low levels of sulphur dioxide contamination of Venusian phos- +phine spectra, MNRAS, 514, 2994 +[43] Villanueva, G.L., et al., 2021, No evidence of phosphine in the atmosphere of Venus from +independent analyses, Nat Astron 5, 631 +[44] Greaves, J.S., et al., 2021, Reply to: No evidence of phosphine in the atmosphere of Venus +from independent analyses. Nat Astron 5, 636 +[45] Snellen, I. A. G., et al., 2020, Re-analysis of the 267 GHz ALMA observations of Venus: +no statistically significant detection of phosphine. A&A 644, L2 +[46] Thompson, M. A, 2021, The statistical reliability of 267-GHz JCMT observations of +Venus: no significant evidence for phosphine absorption. MNRAS, 501, L18 +[47] Greaves, J.S., et al., 2021, Addendum: Phosphine gas in the cloud deck of Venus. Nat +Astron 5, 726 +[48] Akins, A.B., et al., 2021, Complications in the ALMA Detection of Phosphine at Venus, +ApJ, 907, L27 +[49] Lincowski, A.P., et al., 2021, Claimed Detection of PH3 in the Clouds of Venus Is Con- +sistent with Mesospheric SO2, ApJ., 908, L44 +[50] Encrenanz, T., et al., 2020, A stringent upper limit of the PH3 abundance at the cloud +top of Venus, A&A, 643, L5 +[51] Trompet, L., et al., 2021, Phosphine in Venus? atmosphere: Detection attempts and upper +limits above the cloud top assessed from the SOIR/VEx spectra, A&A, 645, L1 +[52] Cordiner, M.A., et al., 2022, Phosphine in the Venusian Atmosphere: A Strict Upper +Limit From SOFIA GREAT Observations, GeoRL, 49, e2022GL101055 +[53] Greaves, J.S., et a., 2023, Recovery of Phosphine in Venus? Atmosphere from SOFIA +Observations, GeoRL, submitted +[54] Donahue, T.M., 1979, Pioneer Venus Results: An Overview, Science, 205, 41 +[55] Hoffman, J.H., Hiodges, R.R., Duerksen, K.D., 1978, Pioneer Venus large probe neutral +mass spectrometer, Journal of Vacuum Science and Technology 16, 692 +[56] Vandaele A.C., et al., 2017, Sulfur dioxide in the Venus Atmosphere: II. Spatial and +temporal variability, Icarus, 295, 1 +[57] Peach, G., 1981, Theory of the pressure broadening and shift of spectral lines, Advances +in Physics, 30, 367 +[58] Bains, W., et al., 2021, Phosphine on Venus Cannot Be Explained by Conventional Pro- +cesses, Astrobiology, 21, 1277 +[59] Bains, W., et al., 2022, Constraints on the Production of Phosphine by Venusian Volca- +noes, Universe, 8, 54 +[60] Ghail, R., et al., 2021, EnVision Understanding why Earth?s closest neighbour is so +different: Mission Assessment Report, ESA/SCI(2021)1, European Space Agency +[61] Smrekar, S., et al., 2021, Veritas (Venus Emissivity, Radio Science, Insar, Topography, +and Spectroscopy): A Proposed Discovery Mission, LPSC 51 1449 +[62] Garvin, J.B., 2022, Revealing the Mysteries of Venus: The DAVINCI Mission, Planet Sci +J,3, 117 +[63] Seager, S., et al., 2022, Venus Life Finder Missions Motivation and Summary, Aerospace, +9, 385 +27 + diff --git a/R9E4T4oBgHgl3EQflg05/content/tmp_files/load_file.txt b/R9E4T4oBgHgl3EQflg05/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8c4c0b2922ae6dfcb84e94fa3d8a2c7b60763f85 --- /dev/null +++ b/R9E4T4oBgHgl3EQflg05/content/tmp_files/load_file.txt @@ -0,0 +1,841 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf,len=840 +page_content='Venus, Phosphine and the Possibility of Life David L Clementsa aDepartment of Physics, Imperial College, Prince Consort Road, London, SW7 2AZ, UK ARTICLE HISTORY Compiled January 13, 2023 ABSTRACT The search for life elsewhere in the universe is one of the central aims of science in the 21st century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While most of this work is aimed at planets orbiting other stars, the search for life in our own Solar System is an important part of this endeavour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Venus is often thought to have too harsh an environment for life, but it may have been a more hospitable place in the distant past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' If life evolved there in the past then the cloud decks of Venus are the only remaining niche where life as we know it might survive today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The discovery of the molecule phosphine, PH3, in these clouds has reinvigorated research looking into the possibility of life in the clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' In this review we examine the background to studies of the possibility of life on Venus, discuss the discovery of phosphine, review conflicting and confirming observations and analyses, and then look forward to future observations and space missions that will hopefully provide definitive answers as to the origin of phosphine on Venus and to the question of whether life might exist there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' KEYWORDS Venus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' astrobiology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' search for life;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Introduction The search for life elsewhere in the universe is one of the major driving forces for astronomy and astrophysics in the 21st century [1] with extremely large telescopes on the ground and in space being designed and built to this end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The main goal of these facilities is to study the atmospheres of rocky, terrestrial planets in orbit around other stars - so-called exo-planets - and follows the explosive development of exoplanet studies since their first discovery in 1995 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' We now know of over 5000 exoplanets, and more are announced all the time (for the latest information on exoplanet discoveries see www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='eu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' However, while convincing evidence for life on exoplanets may arrive in the next 20 years, there are many questions that such observations will not be able to answer, including the biochemical processes exoplanet life uses, how and when it originated, and how it evolved into whatever form it has today - a form that will also be unknowable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' If we want answers to these further questions the only places they may be answered is in our own Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While still very distant, planets such as Mars, the Jovian moon Europa, and Saturn’s moons Enceladus and Titan, which might harbour signs of life, are accessible to direct in situ studies that can answer these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' And any answers inevitably lead back to the question of our own origins on Earth, as well as the more general issue of the prevalence of life in the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' CONTACT David L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Clements Email: d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='clements@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='uk arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='05160v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='EP] 12 Jan 2023 The search for signs of life on Mars is well underway, with orbiting spacecraft look- ing at its atmosphere (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' the ExoMars Trace Gas Orbiter (TGO) [3]) and an ever- increasing number of rovers scouring its surface [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Some of these are already preparing samples of rock to be returned to Earth for laboratory analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Further out in the Solar System, the first stages of the exploration of Jupiter’s moons Ganymede and Europa, in part for signs of habitable environments, are already under development with the European Space Agency’s (ESA’s) JUpiter ICy moons Explorer (JUICE) [5], aimed primarily at Ganymede, due for launch in 2023, and NASA’s Europa Express, aimed squarely at Europa, due for launch in 2024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Plans are at an earlier stage for the exploration of the moons of Saturn that might harbour life, including Titan with its thick atmosphere and Enceladus with its plumes of water vapour spewing into space, but it is clear that these too will be visited sometime in the next few decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Until very recently this list of targets - Mars, Europa, Ganymede, Titan and Ence- ladus - would have been considered the most likely places to find signs of life in the Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' It was thus rather surprising to find Venus added to this list in late 2020 with the discovery of an unusual gas, phosphine, chemical formula PH3, in its atmosphere [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Venus, as we will see below, has a surface that is completely hostile to life and so had been largely discounted from these considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' But, as we will also see, there have long been thoughts that there might be niches in the atmosphere of this planet that might be more favourable to the existence of life than its deeply unpleasant surface [7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' In this article we discuss how life is sought using atmospheric observations, why phosphine is a potential signature of biological activity, how it was detected on Venus, and what its discovery might mean for our understanding of the history of Venus and of life in the Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' We also look at the prospects for future studies of Venus in search of further possible signs of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' What is Life?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The formal definition of life adopted by NASA for searches for life elsewhere is that ‘Life is a self-sustaining chemical system capable of Darwinian evolution’ [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This definition clearly applies to most things that we would consider alive on Earth, but it does leave some things out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Viruses, for example, cannot reproduce on their own, but instead require a host cell, whose reproductive machinery they take over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Alternative definitions are available (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' [10]), but the NASA definition seems a useful starting point to begin any discussion of where life might exist in the Solar System or elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Given this definition we can start to examine what the requirements might be for life to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' From a physicist’s perspective the key thing that life needs to be able to support itself is a source of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' For most of the life we are familiar with on the Earth that source of energy is the Sun - plants photosynthesise using sunlight, while animals and other organisms consume plants in various ways, including eating things that have eaten plants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' However, sunlight is not the only source of energy used by life on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' In the depths of Earth’s oceans, where sunlight never shines, there are thriving communities of organisms surrounding and fed by hydrothermal vents [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' These arise where ocean water can enter the Earth’s crust and be heated by magma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The heated water dissolves minerals from the rocks and circulates back into the ocean through the vents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The primary producers of energy in these vents are chemosynthetic bacteria that use a variety of processes to derive energy from the chemicals emerging from the vents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' These then support a diverse community of other organisms around the vents, 2 including giant tube worms that can be up to 3 metres long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Hydrothermal vents in the young Earth are a possible site for the first emergence of life on our planet [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Similar hydrothermal vent structures are thought to exist beneath the ice-covered surfaces of moons like Europa and Enceladus [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' However, life without light on Earth is not limited to hydrothermal vents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' It has even been suggested that most ecosystems on Earth exist in the dark, deriving their energy from chemical processes separate from, and independent of, photosynthesis [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This clearly has implications for the search for life elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' What requirements for life are there beyond a source of energy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Revisiting NASA’s definition of life we see that it is defined as a chemical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The chemical basis for all the life we know on Earth is the element carbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This element is exceptional in the Periodic Table as the lightest element in Group IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' It thus has a half-filled electron shell giving it a valence of 4 so it can donate or receive up to 4 electrons, allowing it to bind with itself to form chains, and with numerous other elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Some of the commonest are hydrogen, oxygen and nitrogen, contributing to the wide variety of complex organic compounds found in living organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The next heaviest Group IV element is silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' It has similar high valence and has been proposed as an alternative building block for life [15], but there are a number of issues which mean that carbon is a better choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Given a chemical basis for life, there must be a way for the necessary chemical reactions to take place so that the processes of life can operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The best way to achieve this is for the reacting chemicals to be dissolved by some solvent so that they can easily combine and interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' On Earth the solvent behind all biology is water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While other solvents have been suggested, especially in environments too hot or too cold for liquid water to be available [16], water remains the only solvent which we know is associated with life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The bulk of our searches for life elsewhere have thus been focussed on places where liquid water might, or is known to, exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Our consideration of the nature of life thus leads us to the conclusion that we should be looking for life on an astronomical body where there is a ready supply of energy, where complex carbon-based chemistry can take place, and where liquid water can exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' In our own Solar System this leads us to focus on the planet Mars and the icy moons of gas giants, such as Europa and Ganymede orbiting Jupiter, and Titan and Enceladus orbiting Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Venus does not appear on this list because, as we will see below, its current surface conditions are far too hostile for life as we know it, but, as we will also see, this may not always have been the case, and there may be loopholes that could allow any life that might have emerged on the surface of this planet to persist today in the dense clouds above its surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Looking for Life Life is easy to find on Earth - we are surrounded by it - but looking for life on astro- nomical bodies, even our closest neighbours, is a rather more difficult task, especially since we do not expect to find something as obviously alive as a tree or a cow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' In the absence of complex macroscopic life we have to look for signs of activity from microbial life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' What might these be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Using Earth as an example, the clearest sign that life exists is the presence of oxygen in our atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This is generated by photosynthesising organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Today we would associate this with plants, but in the ancient history of our planet, the first oxygenating organisms were single-celled microbes known as blue- green algae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' These were responsible for the Great Oxygenation Event which took place 3 about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='5 billion years ago [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Before this event oxygen was not a major constituent of Earth’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' After it, the Earth had an atmosphere much closer to what we see today, with abundant free oxygen available across the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' In principle, the presence of oxygen in the atmosphere of Earthlike exoplanets will be detectable by future instruments through the absorption of ozone, O3, at mid-infrared wavelengths [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Molecules that potentially indicate the presence of life, such as oxygen and ozone, are known as biosignatures (see [19] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' They include oxygen, ozone, methane (CH4), N2O, C2H6 CH3Cl, CH3SH and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' A wider variety of biosignatures than just oxygen and ozone are needed to cope with potentially dif- ferent biospheres than the one we currently inhabit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' In fact, for much of the history of life on Earth, there would not have been sufficient oxygen or ozone for life to be detectable since the Earth was dominated by anaerobic life (ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' life that does not re- quire or produce oxygen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Searches for other small molecules that might be additional biosignatures are also underway [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The common factor among all of these biosigna- ture molecules is that they should not exist in the abundances seen unless biological processes are maintaining their abundance ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' their abundance is out of equilibrium with their environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' However, biological processes are not the only way of main- taining an out-of-equlibrium abundance of some of these biosignatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' For example, the splitting of water into hydrogen and oxygen by stellar ultraviolet radiation, and the subsequent escape of hydrogen from the planet’s atmosphere, can produce a sig- nificant partial pressure of oxygen on an abiotic (ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' lifeless) planet given certain other conditions [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Care must therefore be taken in interpreting any unusual abundance of a single molecule as an unambiguous biosignature without a good understanding of the broader environment in which it is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Phosphine, PH3, is one of the small molecules suggested as a potential biosigna- ture by the team investigating novel biosignature molecules [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' On Earth it is ex- clusively associated with anaerobic ecosystems, or with human industrial chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Sources have been found associated with anaerobic environments in ponds, marshes and sludges, and specifically with piles of penguin guano and in the faeces of European badgers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While it is clearly associated with anaerobic biology, the specific biochemical pathway for phosphine production in anaerobic systems remains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Its use as a biosignature was originally envisioned in the context of a significant concentration of the gas within the atmosphere of a terrestrial exoplanet with a large anaerobic bio- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While phosphine has been detected in the vast reducing (ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' hydrogen rich) atmosphere of the gas giant Jupiter [23], where it is produced by normal chemical processes in the very dense, hot inner regions then brought to the surface by convec- tion, its presence in the oxidised atmosphere of a terrestrial planet would be difficult to explain by equilibrium chemistry, making it a good candidate biosignature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Obser- vational facilities able to find phosphine in the atmosphere of a distant exoplanet are some way in the future, but, as we see below, a surprise was waiting for us much closer to home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Venus - an unlikely candidate for astrobiology 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Venus Today Venus has often been described as Earth’s evil twin since the two planets have very similar sizes and masses, with Venus having a radius about 95%, and a mass about 4 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Left: Venus as seen in the optical by the Messenger mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' In the optical the planet appears almost featureless because of the highly reflective clouds that cover the entire planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Credit: NASA/Johns Hopkins University Applied Physics Laboratory/Carnegie Institution of Washington Right: Venus as seen in the ultraviolet by the Japanese space mission AKATSUKI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This image is produced by observations in two ultraviolet bands, at 365 and 283 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The colours are the result of unexplained ultraviolet absorption by small particles in the cloud layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Credit: JAXA/ISAS/DARTS/Kevin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Gill 82% of the Earth’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' It is also a little under 30% closer to the Sun than the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Viewed from a telescope, Venus appears as a featureless pale disk because it has a thick atmosphere containing a permanent cloud layer, opaque to optical observations, that reflects over 70% of the sunlight that falls on it (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Venus’ orbit closer to the Sun, combined with the effects of these reflective clouds, might naively suggest a surface temperature not too different from the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Because of this, in the early 20th century it was thought that the clouds might be water vapour and that the surface of Venus could be habitable, inspiring visions of steamy tropical jungles filled with alien life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Once more detailed observations became available, and space probes started to visit in the 1960s, it became apparent that Venus was very different from these initial speculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Rather than having an Earthlike atmosphere, early 1960s space probes like NASA’s Mariner 2 and the Soviet Union’s Venera 4 found that Venus has an extremely dense atmosphere dominated by carbon dioxide, CO2, with a small amount of nitrogen (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='5%) and traces of other gases such as sulphur dioxide (SO2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The surface atmospheric pressure on Venus is about 93 times higher than sea level pressure on the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This huge blanket of CO2 absorbs thermal radiation from the surface of the planet which would otherwise be radiated away into space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The Sun’s radiation is thus trapped beneath the clouds, leading to a runaway greenhouse effect and surface temperatures of about 735 K (462 C), making it hotter than the maximum surface temperature of Mercury [24]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' To make things even more unpleasant, the thick clouds are largely made up of sulphuric acid as a result of atmospheric SO2 dissolving into droplets of water that condense at altitudes of about 55 km above the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' All of this combines to make the surface of Venus today an utterly inhospitable place for anything we might consider a biological system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The true nature of the surface of Venus in fact led leading 1It is worth noting that the discovery of the runaway greenhouse effect on Venus prompted early discussions about the dangers of CO2 emissions from fossil fuel use on Earth and their possible impact on climate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' A 360 degree panorama of the surface of Venus captured by the Venera 9 probe during its 53 minutes of operation on the harsh surface of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This is the first image ever obtained from the surface of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Part of the probe can be seen at the bottom of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' astronomer Carl Sagan to describe the planet as hell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Spacecraft continued to visit Venus after the early missions, including a long series of Venera probes launched by the Soviet Union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Several of these, as well as the Pioneer Venus mission from NASA, sent probes into the atmosphere of Venus to better under- stand its chemistry and constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The Venera and Vega projects also attempted to land probes on the hostile surface of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' After a number of failures they were eventually successful and conducted a number of studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' None of the landers lasted very long, with the longest lived surviving only about 2 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Nevertheless, some of these landers managed to send back images of the Venusian surface, one of which can be seen in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While the surface of Venus is undoubtedly hostile to life at the present time, there is a potential niche for biological processes in the clouds that obscure the planet since they are cool enough for liquid water to be present, and at an atmospheric pressure that matches that of the Earth at sea level [25,26] (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Speculation about the possibility of life in the clouds of Venus dates back to the very earliest days of our understanding of the true conditions on the planet [27] and continues to the present day (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' [28,29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Venus in the past While the surface of Venus is undoubtedly hostile today, this might not have been the case in the distant past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' If the surface of Venus was warm and wet - in the sense that liquid water could flow on the surface - then it is possible that life might have emerged there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' When conditions later become increasingly hostile to life on the surface it might have evolved to seek sanctuary in clouds, the last habitable ecological niche on the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' But is there any evidence to suggest that Venus was ever less hostile than it is today?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Sadly, we do not have direct access to information about the conditions on Venus billions of years ago, but there are hints that suggest that Venus may once have had much more water on its surface, and in its atmosphere, than it does today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Principal among this evidence is the deuterium to hydrogen ratio (D/H ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Venus currently has a D/H ratio that is 150±30 times that of the Earth [30] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This suggests that Venus has lost a substantial fraction of its water through the escape of hydrogen, which is favoured over the escape of deuterium since the latter has a higher mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Hydrogen is driven off Venus through interactions with the solar wind which can directly interact with the upper layers of its atmosphere since, unlike the Earth, Venus lacks a protective magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' However, it is possible that Venus may have retained 2Earth’s D/H ratio is ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='6 × 10−4 6 3 Figure 3: Left: Image of Venus from the AKATSUKI UV imager, using two channels, 3650˚A in or- ange, which tracks SO2, and 2830˚A in blue, which tracks the unknown absorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Note the highly complex structure of the unknown absorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Right: A schematic diagram of the vertical structure of the atmosphere of Venus (from Seager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' of Venus so significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' PH3 has previously been detected in the atmospheres of three other planets Jupiter, Saturn and Earth (see Weisstein & Serabyn (1996) and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' In gas giants, phosphine is produced in the deepest atmospheric layers where the temperature and pressure is high enough for the formation to be thermodynamically favoured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The PH3 seen in Jupiter and Saturn is thought to account for nearly the entire phosphorous content of these atmospheres (Visscher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Some of this material is then dredged up to the upper atmosphere by convection currents, where it can be detected (Noll & Marley 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This is not the situation on Earth, where phosphine production is highly disfavoured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' In fact, life is the sole source of phosphine on Earth, whether as a result of industrial chemistry processes or through as-yet poorly understood biological processes in a number of di↵erent anaerobic environments (Sousa-Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Phosphine production on rocky terrestrial planets is in fact so disfavoured that it has been suggested as a biosignature gas for exoplanets (Sousa-Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2020 and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' However, the discovery of phosphine in the atmosphere of Venus cannot, on its own be taken as evidence for life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Before that can happen we must eliminate all other potential sources of PH3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Extensive analysis of non-biological routes to the formation of phosphine on Venus was undertaken in Bains et al (2020) which analysed potential thermodynamic, photochemical, surface, and sub-surface routes to its production, as well as lightning, volcanism and injection from impactors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' They concluded that all potential sources of phosphine fell short of the observed levels by many orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This leaves us in the uncomfortable position of having no way to explain the presence of phosphine without either invoking unknown chemical processes or the presence of life, though the life hypothesis itself has its own problems since it would require life to exist in a profoundly hostile environment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' As a first step in addressing this problem, we here propose a long term programme of monitoring observations of phosphine and several other species: HDO, SO, HCO+ and SO2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 5 and Tech Case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' We already know that the abundance of several of these species in the atmosphere of Venus varies with time, but we do not know the degree to which phosphine varies with time, or if any variation is correlated with other species or with other factors such as orbital phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Correlations, or anticorrelations, between other species and the abundance of phosphine could provide clues to the chemical, or biochemical, processes that produce phosphine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' These observations will also provide useful legacy information on the variability of other molecular species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The same dataset will also provide higher S/N observations of the line wings of phosphine, thanks to the improved stability of the ’¯U’¯u receiver over RxA and our multiple observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This will Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The atmospheric structure of Venus, showing how temperature and pressure vary with height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The cloud layer at around 55km altitude has temperature and atmospheric pressure levels that are comparable to those on the surface of the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' [29] a magnetic field, and thus the possibility of liquid water oceans, for several billion years after its formation [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Computer simulations have been used to asses whether a warm wet Venus in the first billion years of the Solar System would have a stable climate that could be conducive to the emergence of life [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While the conclusions of this work are still not agreed some suggest that even a wet Venus would never have been able to condense its water into oceans, leading to a so-called ‘steam Earth’ scenario [33] - it is intriguing to consider the possibility that Venus may in fact have been the first habitable planet in the Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' A warm wet Venus (see Figure 4) might have remained habitable until as recently as about 700 million years ago, allowing substantial time for life to evolve and propagate across its surface given that life seems to have emerged on Earth somewhere between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='7 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='3 billion years ago [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' If this is correct, it may only be in the most recent 15% of the age of the Solar System that massive volcanic activity on Venus established a runaway greenhouse effect on the planet, leading to water stripping and the hot, dry, hostile surface that we see today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Atmospheric mysteries As well as uncertainty about the role of water in Venus’ past, there are also some significant mysteries about the planet’s atmosphere today even before we get to the detection of phosphine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Long-standing issues include ([34] and references therein): The variation of the abundance of water vapour and SO2 with altitude in and above the cloud layers [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Water, H2O, persists throughout the atmosphere but SO2 levels drop from parts per million abundances below the clouds to parts per billion above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This is not what is expected given that both gases are thought 7 Altitude [km] Temperature[°C] Pressure[bar 100 10-4 100 10-3 80 Upperhaze 10-2 45 Upper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='cloud 10-1 60 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='Temperatezone Middle&Lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='cloud 60 Y7 40 Lower-haze 220 10 20 Surface haze 45 0Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' An artist conception of what a warm, wet Venus might have looked like during earlier stages in the evolution of the Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Credit: NASA to be released by volcanism at the surface and to be well mixed throughout the atmosphere until both are destroyed by solar UV at altitudes of 70 km or higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The apparent depletion of SO2 in the clouds is currently not understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Oxygen, O2, is present in the clouds of Venus where it was detected by gas chromatographs on board Pioneer Venus Probe [36] and the Venera 13 and 14 descent modules [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' There is currently no known process by which oxygen can be formed in the cloud layers so its origin is something of a mystery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Observations of the clouds of Venus in the ultraviolet reveal complex spatial and temporal changes in absorption and reflection [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This is in contrast to observations in the optical and near-infrared where the clouds of Venus appear nearly featureless on the dayside (see eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Significant cloud contrasts are only seen in reflected sunlight at wavelengths shorter than about 400 nm, and at near-infrared wavelengths (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='7 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='4 µm) in emission on the nightside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' These variations in ultraviolet absorption were first observed in photographic observations in 1928 [38], but, despite all the ground-based observations, space- craft observations from orbit, and descent probes sampling the atmosphere, the chemical and physical origin of the absorber responsible remains unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The clouds of Venus contain a variety of constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Based on size analysis from the Pioneer Venus Probe particle size spectrometer [39] they can be divided up into three particle sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' These correspond to aerosols, of size ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='4 µm, droplets of size ∼2 µm, and larger particles of size ∼7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The larger particles are present only in the middle and lower cloud layers, at altitudes from 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='5 to 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='5 km above the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The nature of these largest particles, which may have a substantial solid component, and be non-spherical in shape, is currently unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 8 Recent reanalysis of the chemistry of Venus’ atmosphere based on measurements by mass spectrometers on descent probes suggest the possibility of chemical disequilibrium in the middle cloud layers [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This result is based on the presence of several species in the mass spectrometer data, including hydrogen sulphide, nitrous, nitric & hydrochloric acids, carbon monoxide, ethane, and hydrogen cyanide as well as phosphine (this reanalysis was conducted after the detection of phosphine by ground based observations, of which more later) and possibly ammonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This chemical mix indicates that reducing chemistry is taking place in the clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' If so, then the processes behind this activity would be out of equilibrium with the oxidising chemistry of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' In the context of the search for life elsewhere, chemical disequilibrium is a potential biosignature, making these results very interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' More recently still, it seems that ammonia, NH3, may have been independently detected in Venus’ atmosphere by ground based observations (Greaves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', private communication), confirming the tentative results from the Pioneer Venus Probe mass spectrometer reanalysis, and adding an extra piece of evidence in favour of chemical disequilibrium in the clouds of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' These problems with our understanding of the atmosphere of Venus, and specifically its clouds, make the case for renewed interest in the planet as a possible astrobiology target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Even if biological processes do not provide the explanation for these poorly understood phenomena, it is clear that there are chemical and physical processes underway in the clouds of Venus that we do not currently understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Further obser- vations to explore the chemistry of Venus and its atmosphere are thus needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' It is in this context that a team of astronomers proposed to look for phosphine, PH3, on Venus3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The Search for Phosphine Phosphine, as we have seen above, has been proposed as a potential biosignature gas which might be present in significant quantities on inhabited planets orbiting other stars [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Current observational facilities, however, are not yet able to make phosphine observations of terrestrial exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While we know that phosphine is present in the Earth’s atmosphere in small amounts, thanks to industrial processes and anaerobic organisms, there were no limits on the amount of phosphine present on other Solar System terrestrial planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Mercury has essentially no atmosphere so is an inappropriate target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Mars has a very thin atmosphere so is unlikely to have much phosphine even if there is biological activity underway there, and any phosphine present would be rapidly destroyed by solar UV radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This leaves Venus as the only reasonable terrestrial planet in the Solar System where some test observations in search of phosphine might be conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Therefore in 2016 a team of astronomers led by Prof Jane Greaves put together a proposal to the James Clerk Maxwell Telescope (JCMT) to conduct test observations of Venus’ atmosphere to look for absorption from the J=1-0 rotational transition of phosphine, which would produce an absorption line at a wavelength of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='123 mm (∼ 267 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' There are other transitions of phosphine at other wavelengths, notably in the far-IR and in the mid-IR, but this particular transition has some advantages, Firstly, 3The author of the current paper was part of this team and is part of the ongoing work to study phosphine on Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 9 observations can be conducted from the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Observations of the next highest rotational transition, J=2-1, would require observations from the stratosphere (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Mid-IR observations of other transitions can be conducted from the ground, of which more later, but the Greaves team did not have easy access to the necessary mid-IR facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The initial idea for the observations was to acquire a few hours of data to better understand the observational issues with looking for a weak absorption line against a very bright continuum source, Venus, with the eventual intent to propose a longer series of observations to set a stringent upper limit, since phosphine was not expected to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' That is not, however, how things turned out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' JCMT Observations The JCMT is a 15 m diameter mm/submm telescope on the mountain Mauna Kea on the Big Island of Hawaii, at an altitude of about 4000 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Mauna Kea is an ideal site for mm/submm observations since it is both high and dry, and so avoids much of the water vapour in the atmosphere of the Earth that would otherwise absorb and contaminate observations at these wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' It is equipped with an array of instruments that operate both as continuum imagers and as high resolution spectrometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' For the first set of JCMT phosphine observations [6] an instrument called Receiver A3 (RxA3) was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' RxA3 was one of the early instruments used on the JCMT and was delivered to the telescope in 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' It was retired not long after the phosphine detection observations reported in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Like many mm/submm spectroscopy receivers, RxA3 uses a heterodyne approach, whereby the incoming astronomical signal, in this case at frequencies of around 267 GHz, is multiplied by a pure sine wave signal at a nearby frequency, the so-called local oscillator frequency, using a device called a mixer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This results in the production of a signal at a frequency that corresponds to the difference in frequencies between the received signal and the local oscillator frequency, and allows signals at frequencies outside the frequency range of interest to be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This lower frequency signal, at what is called the intermediate frequency, or IF, is then measured and dealt with by later stages of processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The technology used in most mm/submm receivers relies on SIS mixers (superconductor-insulator-superconductor) to mix the astronomical and local oscillator frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' For more information on how these operate, and on much else in radio astronomy, see [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The IF signal from RxA3 is then processed by the Auto-Correlation Spectral Imag- ing System (ACSIS - this system is used by all spectral receivers at the JCMT, includ- ing both RxA3 and its replacement ’¯U’¯u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This digitises the input IF signal, calculates the autocorrelation of the signal with itself - essentially multiplying the signal by a time delayed version of itself - and then calculates the Fourier transform of the auto- correlated signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' According to the Wiener-Khinchin theorem [41], the autocorrelation of a signal is the Fourier transform of the signal’s power spectral density ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' the amount power received as a function of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Fourier transforming the autocorrelation of the IF signal thus gives us what we want - the spectrum of the source in the frequency range of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Venus was observed by the JCMT using RxA3 in search of phosphine on five morn- ings in June 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' These dates were chosen so that Venus appeared large enough to fill the telescope beam, minimising any effects due to errors in pointing the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Venus is a strong continuum emitter at millimetre wavelengths, so phosphine would 10 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The James Clerk Maxwell Telescope, a 15 m diameter mm/submm telescope on Mauna Kea in Hawaii, currently operated by the East Asian Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The 15 m primary mirror is protected from wind during observations by a large gortex screen which is why it cannot be seen directly even when the telescope is taking observations, as in this picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Credit: William Montgomerie/EAO/JCMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' be detected as a weak absorption line against this strong continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This strong continuum, however, leads to a number of problems with the quality of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' A number of effects, including reflections from the floor or roof of the telescope dome, or in the receiver cabin itself, entering the beam, lead to strong, time varying baselines in the output spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' These have to be detected and removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' For the initial JCMT detection of phosphine [6] these effects were removed by the usual method of fitting polynomial functions to the data, excluding the region of the spectrum where phos- phine might lie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Once this process was applied to each of the 140 spectra that made up the observations, and despite the original assumption that only an upper limit would be found, an absorption line ascribed to phosphine was detected, corresponding to an abundance of about 20 to 25 parts per billion (ppb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The JCMT spectrum of phosphine can be seen on the right hand side in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' ALMA Observations Following the rather surprising detection of phosphine at the JCMT some further observations in search of independent confirmation of this discovery were needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' To this end, observing time was granted on the Atacama Large Millimetre/Submillimetre Array (ALMA) in March 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Despite operating at similar mm/submm wavelengths, ALMA is a rather different telescope to the JCMT because it is an interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' It is made up of 66 separate antennae, mostly 12m in diameter, the signals of which are combined together to produce the final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 43 of the 12 m antennae were used for the Venus phosphine observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 11 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The Phosphine 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='123mm J=1-0 line as detected by ALMA (left) [42] and JCMT with RxA3 (right) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The black lines indicate the level of SO2 absorption derived from simultaneous (ALMA) and near- simultaneous (JCMT) observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' As can be seen the PH3 detections are clear and the SO2 contamination is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' These spectra are continuum subtracted, so zero on the y-axis represents the continuum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' We use the standard astrophysics approach for presenting high resolution spectra in this Figure, where the spectrum is centred on the line of interest at zero velocity and frequencies are indicated by the doppler velocity in km/s needed to shift from this central value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 12 d compound coordinate system .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='0002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='0001 ine:continuum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='0002 50-40-30-20 -10 0 10 20 30 40 50 Venus-framevelocity(km/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='0001 line:continuum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='0003 50 -40 -30 -20 -10 0 10 20 30 40 50 Venus-frame velocity (km/s)Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Some of the 64 antennae that make up the ALMA telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Credit: ESO/C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Malin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While the signals received by each ALMA antenna are dealt with in a manner similar to RxA3 on the JCMT, using a heterodyne SIS mixer and local oscillator in the receiver, these signals are then cross-correlated pairwise with those from each of the other antennae in the array (where each pair of antennae forms a ‘baseline’) to produce an interferometric map of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Interferometry allows angular resolutions to be achieved that correspond to a telescope whose diameter equals the longest baseline separating individual antennae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The cross-correlation of signals detected by each pair of antennae produces a series of ‘visibilities’ which are a measure of the two-dimensional Fourier transform of the sky distribution of brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The visibilities at each observed frequency are then Fourier transformed to produce a series of images at successive frequencies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' a spectral cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' However, since only a finite number of antennae pairs are available, even for an array with as many antennae as ALMA, the Fourier plane is like a telescope with lots of holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Various methods are used in a process called ‘cleaning’ to derive the actual image from the limited sampling in the Fourier plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' For more information on interferometry see [41] or the ALMA Primer4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Processing interferometric data involves different challenges to those encountered at the JCMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' For example, the angular size of Venus was so great that even the shortest ALMA baselines could not provide good images on the scale of the whole disc and the imperfect sampling led to strong ripples so the data from the affected short baselines, all less than 33 m in length, were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' There were also strong spectral ripples on some parts of the planet, such as the poles, which had to be excluded from further analysis otherwise they would add noise to the spectra, reducing the sensitivity of the final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Further analysis of the ALMA data processing also found some errors in 4https://almascience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='org/documents-and-tools/cycle9/alma-science-primer 13 the standard reduction script used, see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='1, which improved on the initial detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The end result of the ALMA observations once all these various effects are taken into account is shown in Figure 6 - a good detection of phosphine absorption at a level of ∼20 ppb that matches what was seen by the JCMT but with somewhat higher signal-to-noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The Detection of Phosphine from the Ground The detection of phosphine in the atmosphere of Venus was, to say the least, a surprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The observations from JCMT and ALMA thus prompted a considerable amount of debate and further observations using other facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' In this section we look at these various discussions, their conclusions, and counter-arguments to the suggestion that phosphine has not been detected or that whatever has been detected was not phos- phine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Reanalysis of the ALMA Data One of the first responses to the Greaves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' detection paper, [6], was a reanalysis of the ALMA data by a separate group [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This analysis did not reproduce the phos- phine detection of [6], and instead found an upper limit to the phosphine abundance of about 1 ppb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' They identified some processes used in the standard ALMA calibration scripts which were not adequate for a very bright, time-varying, beam-filling target or indeed for the correspondingly bright calibrator sources used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This led to reprocessing of the raw data by the ALMA observatory and European Southern Observatory (ESO) staff (independently of any of the research groups), who provided new scripts taking these and additional problems into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The reprocessing simplified the basic re- moval of instrumental bandpass ripples using the moon Callisto as a calibrator, and also avoided the chance of spectral averaging producing sharp edges which could mimic an absorption line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The new scripts also accounted for the non-linear instrumental re- sponse to the high intensity of Venus (the brightest source in the sky after the Sun at these wavelengths) and its large angular size, although, since this exceeds the extent of accurate models of the response of individual ALMA dishes, this is thought to be a source of residual error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Greaves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' responded to this reanalysis [42,44] by employing the improved ob- servatory scripts and updating their own processing, using three different independent methods to obtain final images and spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The first step after observatory calibration is to remove the shortest baselines as explained above, and then to make a simple, linear spectral fit to the visibility data to remove the contribution of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Next, residual spectral ripples can be corrected either in the visibility data or after Fourier transforming to make an image cube, and before or after cleaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Spectra were ex- tracted over different portions of the planet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' small residual errors meant that only those spectra extracted from regions symmetric about the planet centre were consid- ered reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' A range of parameters allowed the continued recovery of a phosphine signal using all the updated methods, optimised at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='7σ significance by excluding the planetary poles [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' They attributed the non-recovery of the phosphine signal by [43] as a result of including baselines shorter than 33 m in most of their analyses, as well as including parts of the image of the planet that had significant spectral artefacts that raise the noise in the final combined spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' They concluded that the phosphine detection in the ALMA data remained robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Was it a real line?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' A common feature of both the original ALMA and JCMT data analyses in [6] was the use of fairly high order polynomials to allow the removal of varying baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' In doing this, it is necessary to mask out the region of the spectrum around a suspected line otherwise the polynomial fitting method might fit and remove a real line, mistaking it for a small scale baseline ripple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Several authors suggested that this process can instead lead to the creation of fake lines, and that this was in fact the origin of the claimed phosphine detection [43,45,46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' There are two counter-arguments to this suggestion that the detection is essentially a statistical false positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The argument that the claimed phosphine detection is a false positive is that when you take the ripple-contaminated spectrum, block out a portion of it where there might be a line, and use a sixth or higher order polynomial to fit the baseline, then some noise spikes or contaminating ripples in the blocked out section may end up looking like a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This is in fact correct, and blind searches for line candidates at random locations in the spectrum would indeed suffer from this effect, significantly reducing confidence that any detections are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' However, the detection of phosphine in [6] did not solely rely on measuring the depth of an absorption line at a random position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Instead, it also relied on the wavelength of the line seen coinciding with that of the line being searched for, phosphine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This significantly reduces the chance of a noise spike or residual masquerading as a phosphine detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Analysis in [47] shows that adding the additional constraint that a fake line must be at a specific frequency reduces the chance of a false positive for line detection to < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Furthermore, if the line was in fact a false positive then there would be no reason for any such noise-generated feature to lie at exactly the same frequency in both the ALMA and JCMT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' As pointed out in [6], the only feature at matching wavelengths in both the ALMA and JCMT data lies at the expected frequency of phosphine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This further bolsters our confidence that the detected phosphine line is real, and not a statistical artefact resulting from the data processing approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Is it really phosphine?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The foregoing analysis suggests that the line discovered is in fact real and not a statistical false positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' However, can we be sure that it is in fact phosphine and not some other molecular species that happens to have an absorption feature at a similar frequency?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Sulphur dioxide, SO2, a known constituent of Venus’ atmosphere, has a transition due to the (J = 309,21 − 318,24) transition at 266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='943329 GHz, a frequency shift from the PH3 J = 1-0 line at 266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='944513 GHz that corresponds to a velocity difference of just 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='3 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The possibility that the claimed phosphine line is actually a misidentification of this SO2 line was first suggested by [43] and has been further explored by others [48,49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While they have concluded that SO2 contamination or misidentification is a possibility, a number of problems with this interpretation have been pointed out by [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Firstly, while the line centres of PH3 J=1-0 and SO2 J = 309,21 − 318,24 are close, they are still 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='3 km/s apart, leading to a ∼ 3 σ discrepancy between the measured line centre and that expected for the SO2 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Furthermore, simultaneous (in the case of ALMA) and near-simultaneous (in the case of JCMT) observations of a different and stronger SO2 line [42] provide predictions of the relative strength of the SO2 transition that might contaminate the phosphine line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' They find that the level of contamination of the phosphine line by SO2 is ∼ 10% for the JCMT data and < 2% for the ALMA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This level of contamination by SO2 is shown as a black line in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' On this basis it seems likely that the detected line is indeed 15 phosphine, and that any contamination by the neighbouring SO2 line is insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Other Observations The phosphine J=1-0 line at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='123 mm is not the only line of this molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' However, many of the other transitions are at wavelengths that are more difficult to observe from the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Nevertheless, observations have been attempted of other lines in search of independent confirmation of the presence of phosphine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The first of these used archival data from the TEXES (Texas Echelon Cross Echelle Spectrograph) instrument, a 5 to 25µm high resolution mid-infrared spectrometer, on the NASA Infrared Telescope Facility (IRTF) on Mauna Kea in Hawaii [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' These observations were part of a long term project to monitor SO2 and H2O in the cloud tops of Venus, and involved observations at a range of frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' One of these datasets, obtained in March 2015, fortuitously included a range of wavelengths where there are some relatively strong phosphine transitions, at a wavelength around 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='471 µm (corresponding to a frequency of 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='65 THz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' No phosphine absorption is detected, indicting an upper limit of about 5 ppb, which is substantially lower than the claimed millimetre wave phosphine detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Further infrared data, this time from the Venus Express spacecraft, were analysed, looking for absorption from phosphine lines at wavelengths around 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='125 µm above the cloud layers [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This data was taken at various times from June 2006 to December 2014, and measured absorption against the light of the Sun as it rises or sets, rather than against the emission of Venus itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This means that only a small part of the atmosphere is studied rather than the entire planetary disk as is the case, for example, for the JCMT or TEXES observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' These Venus Express observations also failed to find any phosphine absorption, setting limits on its abundance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='2 to 20 ppb depending on the specific observations and the assumed altitude of the absorption, ranging from 60 to 95 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' A third approach to confirm the detection of phosphine is to search for absorption lines in the far-infrared, at frequencies around 534 and 1067 GHz [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While the Earth’s atmosphere is completely opaque at these frequencies at sea level and even on tall mountains like Mauna Kea, the SOFIA observatory (Stratospheric Observatory For Infrared Astronomy) - essentially a 747 Jumbo Jet with a hole cut in the fuselage with a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='5m telescope pointing out (see Figure 8) - can perform these observations since it flies at an altitude of about 13 km, above much of the water vapour that absorbs far-IR radiation in the Earth’s atmosphere5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' SOFIA observations of Venus in search of phosphine were carried out in November 2021 [52] using the GREAT (German REceiver At Terraherz frequencies) instrument, a receiver similar to the JCMT receivers but operating at much higher frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Data reduction and analysis by the original authors failed to find any sign of phosphine, setting an upper limit of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='8 ppb from the J=4-3 line and ∼ 2 ppb for the J=2-1 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' However, subsequent reanalysis of the SOFIA data found that the calibration stage that sets an absolute flux scale adds noise and artefacts to the resulting spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This calibration stage is not needed if we are only interested in the line-to-continuum ratio, as is the case when measuring an absorption line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' By purely analysing the line- to-continuum ratios [53], phosphine at a level of ∼1-2 ppb is found, averaged over altitudes from 75-110 km, with 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='5σ significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' These other observations in search of phosphine absorption using different ap- 5Sadly such observations can no longer be performed since the SOFIA observatory was decommissioned and retired at the end of September 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 16 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The SOFIA observatory, which consists of a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='5m telescope and instruments mounted inside a 747 jumbo jet, and capable of flying above much of the far-IR absorption in the Earth’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Credit: NASA/DLR proaches, whether from the ground or from Venus Express, have produced a number of conflicting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' They need to be carefully interpreted since the different wave- lengths and observational approaches are in fact probing the presence of phosphine at different altitudes and times, as we shall see below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' None has yet definitively disproved the original JCMT and ALMA results of [6], and the SOFIA observations may in fact have provided some level of confirmation, depending on which analysis approach is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' In Situ Confirmation The ideal way to determine the presence and amount of phosphine in the atmosphere of Venus would be to send a space probe directly into the atmosphere equipped with instrumentation that can detect and measure the presence of the gas in situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This would avoid all the difficulties of observing phosphine remotely, as well as all issues of interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' At this point, as we will see below, we are some years away from any such future mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' However, past missions to Venus did send probes into the planet’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Principal among these, for our current purposes, is the Pioneer Venus Multiprobe (also known as Pioneer Venus 2 or Pioneer 13) [54] (see Figure 9) which, among many other instruments, sent a mass spectrometer into the atmosphere of Venus on its largest entry probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Data from the Pioneer Venus Large Probe’s Neutral Mass Spectrometer (LNMS) were reanalysed in 2021 [40] subsequent to the announcement of the discovery of phosphine by [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This reanalysis of data taken during its descent into the atmosphere of Venus on 9 December 1978, was the first to look for trace or minor constituents 17 DLR SOFIA N747NFigure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The Pioneer Venus Probe Spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The mission consisted of an orbiter and four probes that were sent into the atmosphere of Venus, seen here in artists conception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Credit: NASA of the atmosphere beyond methane and water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The LNMS takes gas in from the atmosphere through inlet tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Molecules in the gas are then ionised by an electron source, accelerated by an electric field and then passed through a magnetic field which deflects the ions by an amount that depends on their mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' These ions are subsequently detected, allowing their mass and abundance to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' For more information see [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The data reanalysed in search of phosphine came from within the clouds, at an alti- tude of 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='3 km above the surface, part of the atmosphere that is largely inaccessible to ground or space based observations, but which is of critical importance to searches for possible life in the clouds, as this is where that life might actually live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The detailed analysis found evidence for phosphine at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='1 to 2 parts per million (ppm) levels in the clouds themselves, a much higher abundance than is seen in the JCMT or ALMA ob- servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' It also found evidence for other species such as nitrite, nitrate, nitrogen and possibly ammonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Taken together these molecules indicate that unexpected chemical processes are underway in the clouds and suggest chemical disequilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Whether this disequilibrium is due to biological or some other as-yet unknown chemical process is yet to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Where and When is the Phosphine Seen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The forgoing sections, describing the various observations in search of phosphine in the atmosphere of Venus, can seem confusing and mutually contradictory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This is at least partly because observational constraints mean that they sample the atmosphere of Venus at different altitudes and, because they encompass datasets that span over 40 years, at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' We already know that some species in Venus’ atmosphere 18 are highly variable - SO2 levels, for example, can vary by large factors on timescales of both years and days at various altitudes [56] - so this may also apply to phosphine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Spatial variations across the disk of the planet are also possible, but this is difficult to assess for phosphine since many of the observations to date have been of the average phosphine level across the planetary disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Of particular importance is the amount of phosphine in the atmosphere as a func- tion of altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While the in situ observations of the LNMS and Venus Express have a clearly determined altitude, this is harder to extract for the observations from Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' In principle, the effect of pressure broadening on the absorption lines can be used to determine the vertical abundance profile of an absorbing molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Pressure broaden- ing of an absorption line occurs when the absorbing molecules interact collisionally with other molecules in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' These interactions shorten the characteristic time of the absorption process, in accordance with Heisenberg’s uncertainty principle, increasing the uncertainty of the absorption frequency and thus broadening the line (see eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' [57]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The overall effect is to make the line shape a Lorentzian function, which has much broader wings than the usually assumed Gaussian shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The exact width of the Lorentzian line depends on the pressure, temperature and nature of the molecules that are interacting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The higher the pressure, the broader the wings, so a full analysis of the shape of the phosphine absorption line can reveal its vertical abundance profile in the atmosphere of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' There are, however, a number of problems with a full pressure broadening analysis of the phosphine line seen in the atmosphere of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Firstly, the pressure broadening coefficient for phosphine in CO2, the dominant constituent of Venus’ atmosphere, is not currently known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Analyses have so far used either a modification of the phosphine broadening coefficient in air [50] or have used the CO2 pressure broadening coefficient for NH3 as an analog to that of phosphine [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Secondly, and more significantly for the immediate understanding of phosphine in Venus, the data reduction techniques used to date to extract the absorption line remove any broad line wings as part of the process that removes baseline ripples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This just leaves narrow line cores, meaning that the observations are insensitive to any significantly broadened lines, and thus are only sensitive to phosphine at altitudes of 75 to 80 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Most recently, an experimental data processing approach applied to new observations of Venus from the JCMT-Venus project (PI: D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Clements) seems to be able to recover the broad line wings of the J=1-0 phosphine line, suggesting an abundance of phosphine at the ppm level inside the clouds at an altitude of about 60 km, consistent with the high levels seen by the LNMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The other factor to consider is the timing of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While we do not yet have enough observations to allow us to monitor any changes in the abundance of phosphine with time or in relation to other species such as HDO or SO2, we can see if there are any correlations between the amounts of phosphine seen and the timing of the observations relative to the illumination of Venus’ atmosphere by the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This may well be an important factor since photolysis by sunlight is a significant destruction route for phosphine in the Earth’s atmosphere [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' If we combine all the phosphine observations - detections and non-detections - together with information about whether the Sun is rising or setting on the atmosphere at the time of observation we perhaps begin to see a pattern (see Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 19 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The trend of phosphine abundance by altitude from the currently available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Shading indi- cates cloud (orange, centred at ∼ 60 km) and haze (grey, centred at ∼ 80 km and 40 km) layers of Venus’ atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Superposed symbols indicate candidate detections plus upper limits for phosphine abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Rising arrows indicate observations made where the atmosphere was rising into sunlight and falling arrows indicate observations made when the atmosphere was descending towards the nightside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Abundances are, from top: 20, 25 ppb from J=1-0 data [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' ∼1 ppb or < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='8 ppb from J=4-3 data [52,53];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' < 7 ppb at 62 km from 4 µm spectra [51];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' < 5ppb at 60 km from 10 µm spectra [50];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' ∼2 ppm at 60 km from initial JCMT-Venus processing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' high ppb to 2 ppm at 51 km from Pioneer-Venus in situ sampling [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' As can be seen all the significant detections of phosphine take place as the atmosphere is moving out of night and into sunlight, while the non-detections take place as the atmosphere is moving from sunlight into night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' If sunlight destroys phos- phine at high altitudes during daylight, as is the case on Earth, this would explain the apparent contradictions between some of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' From: Greaves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' in prep, by permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 20 altitude (km) 100 detection (gas entering sunlight) detection (gas departing sunlight) 80 upper limit (gas departing sunlight) 60 40 20 log-abundance(ppb) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='1 1 10 100 1000Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Potential chemical pathways for the synthesis of phosphine in the atmosphere of Venus, and their derived production vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' destruction rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' There are stages where, for all possible pathways, the rate of destruction of phosphine exceeds its formation by many orders of magnitude, as shown in red/purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' As can be seen, there is no route to produce phosphine by these processes that can account for the amounts observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' From [58] where more details can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The (Im)Possible Origins of Phosphine on Venus The presence of phosphine in the atmosphere of Venus is a surprise since a compound of phosphorous with hydrogen should not naturally appear in the atmosphere of a planet, such as Venus, which has an oxidised atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' On Earth, phosphine does not occur through normal chemical processes and is produced only by anaerobic life or through human industrial activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While this is the expectation, Venus is a complex environment with a wide range of chemical and physical processes underway from the surface to the top of the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' A detailed analysis is thus necessary to see if there are any possible routes through which the levels of phosphine seen might occur through normal chemical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Such an analysis was conducted in [58] where a wide range of chemical processes were examined to see if there is any potential source of phosphine in sufficient abundance to explain the observations with processes that we know are underway on the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The processes examined included gas reactions, geo- chemical reactions, photochemistry, volcanism (see also [59]), lightning and impactors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' An example of the kind of chemical reaction network considered is shown in Figure 11, where the reaction rate and the destruction rates are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Only segments of this reaction network where the ratio of the production rate over the destruction rate is ≥ 1 can produce an accumulation of the relevant chemical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' For phosphine to be pro- duced in significant amounts the whole reaction network must have this ratio ≥ 1 but, as can be seen, critical segments of the network have ratios orders of magnitudes less than this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' More generally, it was found that the lifetime of free phosphine at various altitudes in the atmosphere of Venus ranged from < 1 second to perhaps a century [59], making it highly unlikely that a significant amount of phosphine can accumulate from any hypothetical source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The most obvious conclusion that can be drawn from this analysis is that we do not know how phosphine came to be in the atmosphere of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' There may be geochemical or photochemical processes that can produce it in sufficient amounts, but these are currently not known to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The alternative, that, by analogy with Earth, phosphine is being produced by anaerobic biological processes, is another potential explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' However, before we can make this particular leap, and claim that we 21 >1 1 - 10-3 Rateofforward reaction Ratio 10-3- 10-6 Destructionrate 10-6- 10-9 <10-9 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' H2 H20 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' H20 OH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' H2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' H20 A H2PO3 HPO, PO H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='PO PH OH 0 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='o OH 02 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' H H, H2Q H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' OH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' : H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' H20 H2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' OH: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' H,PO4 HPO3 PO2 PH2 HPO H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' H,0 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' H20 OHhave found evidence for life in the clouds of Venus, we must first exclude all other possible origins, and also explain how life is able to survive in the extremely acidic environment of Venusian cloud droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' One possible solution to the latter problem is that ammonia, if present, is able to buffer the sulphuric acid in these droplets to some extent [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The possible detection of ammonia in the clouds of Venus by the LNMS [40] and in preliminary analysis of data from the Green Bank Telescope (Greaves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', private communication), is thus rather interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The Next Steps As has become clear in the previous section, studies of phosphine, and the search for life on Venus, are very much works in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While the current results are intriguing, there are no solid conclusions that can yet be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Much more work needs to be done, and it will be the work of many years before we can have a definitive answer to the question of whether there is life in the clouds of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This will require not only observations from Earth, but also in situ probes and, ideally, missions that can return samples from Venus to Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' In this section we look at some of the projects that are planned or already underway to improve our knowledge of the clouds of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Earth-based Studies Observations from Earth were responsible for the first detections of phosphine, and these are continuing to both monitor phosphine and to search for other molecules that may have a bearing on the chemistry, or biochemistry, underway in the clouds of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The largest of the projects currently underway is JCMT-Venus (PI: D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Clements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This uses the ’¯U’¯u receiver, the replacement for RxA3, together with the ACSIS system to obtain whole disk spectra for Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The new receiver has a wider bandwidth than RxA3 so we can simultaneously observe phosphine, HD and SO2, and search for other molecules such as SO and PO2 which have spectral features in the band covered by ’¯U’¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' By simultaneously monitoring phosphine, HDO and SO2 we can see how these different species vary in relation to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This should provide indications as to the chemical processes behind the presence of phosphine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' If, for example, phosphine is produced by reducing processes in the upper atmosphere, the proportions of reduced compounds, like phosphine, and oxidized compounds, like HDO and SO2, will be anticorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The JCMT-Venus project is a long term programme at the JCMT and has been awarded 200 hours of time over a period of three years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The visibility of Venus means that observations will be possible in three tranches, including Feb 2022, July 2023 and September 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The first of these observing campaigns has already taken place, with Venus observed over a period of 20 consecutive mornings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The data obtained already contains 140 times as much information as in the original JCMT observations, so is taking some time to process and analyse, especially since ’¯U’¯u has its own difficulties dealing with the brightness of Venus and thus an interesting new set of baseline drifts and ripples to be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Nevertheless the analysis is well underway and initial results, some of which have been briefly discussed above, have already emerged, including further confirmation of the presence of phosphine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' When complete, JCMT-Venus will provide a major new database of observations of Venus in the mm band, including phosphine and other important molecules which will provide significant new insights into the origin of phosphine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 22 Further ALMA observations have yet to be approved, but these hold the promise of providing further information about the distribution of phosphine across the face of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The original ALMA observations provided some hints that the distribution is not uniform, but a full map of the abundance of phosphine across the planetary disk could not be made because of excess ripples affecting the signal over significant portions of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Additional observations in the mid-IR from the IRTF and elsewhere will also be helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Sadly, further observations with SOFIA are not possible since the observatory has been decommissioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Studies related to the search for potential signs of life on Venus are also underway that do not directly target phosphine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' These include observations with the Green Bank Telescope (GBT) at radio wavelengths to look for ammonia (NH3) in absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This is important since detection of ammonia would indicate the presence of another reduced molecule that should not be expected in the oxidised atmosphere of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Ammonia is also important because of its buffering effect against the high acidity in the liquid droplets in Venus’ clouds [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Analysis of archival data from the 1970s as well as an initial set of observations with the GBT suggest the presence of ammonia (Greaves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', private communication), as do the in situ measurements of the LNMS, but more data is needed to confirm this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Laboratory studies also have a role to play since they can validate and test the vari- ous assumptions that went into the analysis of [34], and allow more accurate predictions for the formation and destruction of phosphine and other hydrogen-rich compounds in Venus-like conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Such studies are already being planned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Space-based studies Venus is also being studied from much closer quarters by space probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' These missions take many years to prepare and so have largely not been designed to examine the possibilities of unusual chemistry, or even life, in the clouds of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Nevertheless, existing missions do have useful capabilities for these purposes and future missions are being planned that can respond to recent discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The Japanese mission AKATSUKI (see eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Figure 1) is currently operating in orbit around Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While it does not have any instruments that are directly relevant to the search for phosphine, its UV imaging instruments are monitoring the unidentified UV absorber, the origin of which is one of the outstanding mysteries of the Venusian atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Comparing AKATSUKI’s results with future data from the ground, es- pecially any future imaging observations with ALMA, may be able to see if there is a link between the presence of phosphine and the presence of the UV absorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The next potentially important space mission to go to Venus, from the point of view of phosphine observations, is not in fact a specific mission to Venus, but the JUICE mission to the moons of Jupiter [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The JUICE spacecraft is scheduled to be launched in the second quarter of 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' It will then perform a series of flybys of planets as gravity assists on its journey to Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Of particular interest here is the flyby of Venus in August 2025 where an observational campaign is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Of particular importance in the context of phosphine on Venus is the Submillimetre Wave Instrument (SWI) which will be able to observe higher J transitions of phosphine, including those observed from the Earth by SOFIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Whether JUICE will be able to undertake an observing campaign at Venus will be up to the JUICE mission directors, and no decision will be made until after launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' In the 2030s three missions directly targeted at Venus are due to be launched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 23 These include the European Space Agency’s EnVISION mission [60], and NASA’s VERITAS (Venus Emissivity, Radio Science, InSAR, Topography, and Spectroscopy) [61] and DAVINCI (Deep Atmosphere Venus Investigation of Noble gases, Chemistry, and Imaging) [62] missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' VERITAS and EnVISION are primarily concerned with the surface and interior of Venus, studying the history and role of volcanism on the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While they will doubtless reveal much that is of interest, they are unlikely to have much to say about phosphine and the processes underway in the clouds of Venus unless they uncover volcanic activity vastly in excess of our current understanding [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' DAVINCI, however, is a much more interesting prospect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The goal of DAVINCI is to study the atmosphere of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' To do this its primary set of instruments are on board a descent stage that will fly through the clouds of Venus, sampling the atmosphere as it goes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' It will be the first NASA mission to enter the atmosphere of Venus since Pioneer Venus Probe in 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Among the instruments on the descent stage is a mass spectrometer that will be able to significantly improve on the results of the LNMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' This will be able to detect phosphine and other trace gas species and see how their abundance changes with altitude and other conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Other instruments include a tuneable laser spectrometer which is able to measure even small amounts of specific gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Altogether, the four instruments on the DAVINCI probe, combined with imagers on the orbiting mothership, will provide a vast improvement in our in situ knowledge of Venus’ atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' It will provide the ground truth against which observations of the planet from Earth can be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' National and international space agencies are not the only organisations looking to send probes to Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Private companies now have the capability to send missions to other planets independently of governments, and they are also interested in the possibility of life on Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' One company in particular, Rocket Lab, is taking special interest in Venus and has set up a team to develop a series of Venus Life Finder (VLF) missions [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The first of these missions, which may launch as soon as mid-2023, is intended to look for organic molecules using an ultraviolet autofluorescence technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Further missions are planned including a balloon borne laboratory that will be able to float in the clouds for an extended period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Amongst the planned instruments for this payload are not only mass spectrometers but also a microscope that will search cloud droplets for evidence of biological cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Perhaps the most ambitious mission planned by the VLF team is a sample return mission that will use a balloon to collect samples of cloud droplets and gas, and return these to Earth for detailed laboratory study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' If there is in fact life in the clouds of Venus, a mission like this will be necessary to answer fundamental questions about its origin and how it operates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' It is perhaps the dream mission in the search for evidence of life on Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Conclusions The discovery of phosphine in the atmosphere of Venus has caused some controversy and has renewed discussions about the possibility of life in the planet’s clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The observational evidence for phosphine has been challenged and examined in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' The JCMT and ALMA results have so far survived these challenges, and there has been independent in situ confirmation of the presence of phosphine from the Pioneer Venus LNMS instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Observations from other telescopes in search of phosphine have pro- duced rather more mixed results, with several upper limits and one possible detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' However, the apparent disagreement between these different sets of observations may 24 soon be understood in the context of day-night variations in the amount of phosphine above the clouds thanks to photolysis by sunlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' While the presence of phosphine in the atmosphere of Venus is becoming more secure with the arrival of new and improved datasets such as JCMT-Venus, an understanding of its origin still eludes us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' It is clear that no conventional chemical process can produce phosphine in the amounts observed, but it is still far from clear whether biological processes are involved, or if there is some as-yet unknown non-biological source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' More data is clearly necessary for us to understand what is really going on in the atmosphere of Venus, and this is being sought by a number of different ground and space-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Over the next several years our understanding of the origin of phosphine on Venus will certainly improve, and we will hopefully reach a point at which the question of life in the clouds of Venus moves from being something that we can only speculate about, to something about which we have clear and decisive knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Whatever conclusion we finally reach, we will have learnt a lot more about our nearest neighbour planet, and this knowledge will help guide our search for possible biospheres on planets orbiting other stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Confirmation that there is in fact life in the clouds of Venus would be a truly epoch making discovery, but we are still a very long way from drawing that conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Acknowledgements It is a pleasure to thank Jane Greaves, Janusz Petkowski, Anita Richards, and Wei Tang for many useful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' It is also a pleasure to thank all members of the phosphine team for their enthusiasm and expertise in what has already been quite an exciting, and unexpected, adventure, which, for me, started in a bar in Hilo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' References [1] National Academies of Sciences, Engineering, and Medicine 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Pathways to Discovery in Astronomy and Astrophysics for the 2020s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Washington, DC: The National Academies Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='17226/26141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' [2] Mayor, M, & Queloz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', A Jupiter-mass companion to a solar-type star, 1995, Nature, 378, 355 [3] Vago J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Witasse O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Svedhem H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Baglioni P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Haldemann A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Gianfiglio G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Blancquaert T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', ESA ExoMars program: The next step in exploring Mars, 2015, Solar System Research, Volume 49, 518 [4] Farley, Keneth A, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Mars 2020 Mission Overview, 2020, SSRv, 215, 142 [5] Grasset, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2013, JUpiter ICy moons Explorer (JUICE): An ESA mission to orbit Ganymede and to characterise the Jupiter system, Planetary and Space Science, 78, 1 [6] Greaves, Jane S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Phosphine in the cloud decks of Venus, 2021, NatAs, 5, 655 [7] Morowitz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' & Sagan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Life in the clouds of Venus?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 1967, Nature 215, 1259 [8] Limaye, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Venus spectral signatures and the potential for life in the clouds, 2018, Astrobiology 18, 1181 [9] Benner SA, Defining Life, 2010, Astrobiology 10, 1021 [10] Vitas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' and Doboviek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Towards a General Definition of Life, Origins of Life and Evolution of the Biosphere, 2019, OLEB, 49, 77 [11] Lutz, RA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' and Kennish, MJ, Ecology of deep-sea hydrothermal vent communities: A review, 1993, RvGeo, 31, 211 [12] Dodd, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Papineau, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Grenne, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2017, Evidence for early life in Earth’s oldest hydrothermal vent precipitates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Nature 543, 60?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='64 [13] Hand, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Sotin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Hayes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2020, On the Habitability and Future Exploration of Ocean Worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Space Sci Rev 216, 95 25 [14] Edwards, KJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Becker, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', and Colwell, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2012, The Deep, Dark Energy Biosphere: Intraterrestrial Life on Earth, AnnRev Earth, 40, 551 [15] Petkowski, JJ, Bains, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Seager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2020, On the Potential of Silicon as a Building Block for Life, Life, 10, 84 [16] Bains, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2004, Many Chemistries could be used to build living systems, Astrobiology, 4, 137 [17] Lyons, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Reinhard, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' and Planavsky, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 2014, The rise of oxygen in Earth’s early ocean and atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Nature 506, 307 [18] Fulii, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', and Matsuo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2021, Detecting Atmospheric Molecules of Nontransiting Tem- perate Terrestrial Exoplanets Using High-resolution Spectroscopy in the Mid-infrared Domain, AJ, 161, 180 [19] Janin,E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2021, Exoplanet biosignatures, Astronomy & Geophysics, 62, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='26 [20] Seager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Bains, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Petkowski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2016, Toward a List of Molecules as Potential Biosignature Gases for the Search for Life on Exoplanets and Applications to Terrestrial Biochemistry, Astrobiology, 16, 465 [21] Meadows, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2018, Exoplanet Biosignatures: Understanding Oxygen as a Biosig- nature in the Context of Its Environment, Astrobiology, 18, 630 [22] Sousa-Silva, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2020, Phosphine as a Biosignature Gas in Exoplanet Atmospheres, Astrobiology, 20, 235 [23] Bregman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Lester, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' & Rank, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 1975, Observation of the ν2 band of PH3 in the atmosphere of Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' ApJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 202, L55 [24] Lewis, John S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2004, Physics and Chemistry of the Solar System (2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Academic Press, ISBN 978-0-12-446744-6 [25] Titov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2016, Clouds and hazes of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 214, 1 [26] Krasnopolsky, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2013, Photochemistry of the Atmospheres of Mars and Venus (Springer Science & Business Media, 2013) [27] Morowitz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Sagan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 1967, Life in the clouds of Venus?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Nature 215, 1259 [28] Schulze-Makuch, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2021, The case (or not) for life in the Venusian clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Life (Basel) 11, 255 [29] Seager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2021, The Venusian lower atmosphere haze as a depot for desiccated microbial life: A proposed life cycle for persistence of the Venusian aerial biosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' As- trobiology 21, 1206 [30] Donahue, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Hoffman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Hodges Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Watson, 1982, Venus was wet: A measurement of the ratio of deuterium to hydrogen, Science, 216, 630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' [31] Gillmann, C, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2022, The Long-Term Evolution of the Atmosphere of Venus: Processes and Feedback Mechanisms, Space Science Reviews, 218, 56 [32] Way, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2016, Was Venus the first habitable world of our solar system?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Geo- physical Research Letters 43, 8376 [33] Turbet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2021, Day-night cloud asymmetry prevents early oceans on Venus but not on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Nature 598, 276 [34] Bains, W, Rimmer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Seager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2021, Production of ammonia makes Venusian clouds habitable and explains observed cloud-level chemical anomalies, PNAS, 118, e2110889118 [35] Krasnopolsky, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2006, Chemical composition of Venus atmosphere and clouds: Some unsolved problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 54, 1352 [36] Oyama V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 1980, Pioneer Venus gas chromatography of the lower atmosphere of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Space Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 85, 7891 [37] Mukhin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 1982, VENERA-13 and VENERA-14 gas chromatography analysis of the Venus atmosphere composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 8, 216 [38] Ross F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 1928, Photographs of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' ApJ 68, 57 [39] Knollenberg, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', & Hunten, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 1980, The microphysics of the clouds of Venus: Results of the Pioneer Venus Particle Size Spectrometer Experiment, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Space Phys, 85, 8039 [40] Mogul, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Limaye, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Way, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Cordova, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2021, Venus’ Mass Spectra Show Signs of Disequilibria in the Middle Clouds, Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' 48, e2020GL091327 26 [41] Wilson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Rohlfs, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Huttermeister, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2013, Tools of Radio Astronomy, Springer [42] Greaves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2022, Low levels of sulphur dioxide contamination of Venusian phos- phine spectra, MNRAS, 514, 2994 [43] Villanueva, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2021, No evidence of phosphine in the atmosphere of Venus from independent analyses, Nat Astron 5, 631 [44] Greaves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2021, Reply to: No evidence of phosphine in the atmosphere of Venus from independent analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Nat Astron 5, 636 [45] Snellen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2020, Re-analysis of the 267 GHz ALMA observations of Venus: no statistically significant detection of phosphine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' A&A 644, L2 [46] Thompson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' A, 2021, The statistical reliability of 267-GHz JCMT observations of Venus: no significant evidence for phosphine absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' MNRAS, 501, L18 [47] Greaves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2021, Addendum: Phosphine gas in the cloud deck of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Nat Astron 5, 726 [48] Akins, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2021, Complications in the ALMA Detection of Phosphine at Venus, ApJ, 907, L27 [49] Lincowski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2021, Claimed Detection of PH3 in the Clouds of Venus Is Con- sistent with Mesospheric SO2, ApJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 908, L44 [50] Encrenanz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2020, A stringent upper limit of the PH3 abundance at the cloud top of Venus, A&A, 643, L5 [51] Trompet, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2021, Phosphine in Venus?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' atmosphere: Detection attempts and upper limits above the cloud top assessed from the SOIR/VEx spectra, A&A, 645, L1 [52] Cordiner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2022, Phosphine in the Venusian Atmosphere: A Strict Upper Limit From SOFIA GREAT Observations, GeoRL, 49, e2022GL101055 [53] Greaves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2023, Recovery of Phosphine in Venus?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Atmosphere from SOFIA Observations, GeoRL, submitted [54] Donahue, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 1979, Pioneer Venus Results: An Overview, Science, 205, 41 [55] Hoffman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Hiodges, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', Duerksen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 1978, Pioneer Venus large probe neutral mass spectrometer, Journal of Vacuum Science and Technology 16, 692 [56] Vandaele A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2017, Sulfur dioxide in the Venus Atmosphere: II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=' Spatial and temporal variability, Icarus, 295, 1 [57] Peach, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 1981, Theory of the pressure broadening and shift of spectral lines, Advances in Physics, 30, 367 [58] Bains, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2021, Phosphine on Venus Cannot Be Explained by Conventional Pro- cesses, Astrobiology, 21, 1277 [59] Bains, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2022, Constraints on the Production of Phosphine by Venusian Volca- noes, Universe, 8, 54 [60] Ghail, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2021, EnVision Understanding why Earth?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='s closest neighbour is so different: Mission Assessment Report, ESA/SCI(2021)1, European Space Agency [61] Smrekar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2021, Veritas (Venus Emissivity, Radio Science, Insar, Topography, and Spectroscopy): A Proposed Discovery Mission, LPSC 51 1449 [62] Garvin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2022, Revealing the Mysteries of Venus: The DAVINCI Mission, Planet Sci J,3, 117 [63] Seager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} +page_content=', 2022, Venus Life Finder Missions Motivation and Summary, Aerospace, 9, 385 27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQflg05/content/2301.05160v1.pdf'} diff --git a/RNAzT4oBgHgl3EQfXPwH/content/tmp_files/2301.01313v1.pdf.txt b/RNAzT4oBgHgl3EQfXPwH/content/tmp_files/2301.01313v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..574dd7c92888de7a9f29ad64b16afee17fabcf07 --- /dev/null +++ b/RNAzT4oBgHgl3EQfXPwH/content/tmp_files/2301.01313v1.pdf.txt @@ -0,0 +1,2199 @@ +DECENTRALIZED GRADIENT TRACKING WITH LOCAL STEPS +Yue Liu +University of Toronto +Canada +Tao Lin +Westlake University +China +Anastasia Koloskova +EPFL +Switzerland +Sebastian U. Stich +CISPA Helmholtz Center for Information Security +Germany +ABSTRACT +Gradient tracking (GT) is an algorithm designed for solving decentralized optimization problems over +a network (such as training a machine learning model). A key feature of GT is a tracking mechanism +that allows to overcome data heterogeneity between nodes. +We develop a novel decentralized tracking mechanism, K-GT, that enables communication-efficient +local updates in GT while inheriting the data-independence property of GT. We prove a convergence +rate for K-GT on smooth non-convex functions and prove that it reduces the communication overhead +asymptotically by a linear factor K, where K denotes the number of local steps. We illustrate the +robustness and effectiveness of this heterogeneity correction on convex and non-convex benchmark +problems and on a non-convex neural network training task with the MNIST dataset. +1 +Introduction +We consider distributed optimization problems, where the objective function f(x) on model x ∈ Rd is defined as the +average of n different components {f1(x), ..., fn(x)}, i.e., +f(x) = 1 +n +n +� +i=1 +fi(x). +In distributed applications, different contributors (or ‘clients’) take part in the training. Such clients can be, for example, +mobile edge devices, or computing nodes. Typically, each component fi(x) is only available to a single client (for +instance, when fi(x) is defined over the training data available only locally on the client). This makes distributed +optimization problems more difficult to solve than centralized problems. +Besides convergence rate in terms of iterations, communication efficiency is one of the most important metrics +in distributed algorithm design. For illustration we consider the calculation of a gradient of the global function, +∇f(x) = 1 +n +� +i ∇fi(x), that forms be basis for general first-order methods. Since each client only has the ability +to evaluate the local gradient ∇fi(x), it is further necessary to calculate the average of these local gradients. +Centralized algorithms [14, 23] realize such global aggregation by a central controller, e.g., with a parameter sever [16]. +However, this approach requires all clients to communicate with the central server simultaneously, resulting in a +communication bottleneck at this hot point and a slowdown in clock time. Instead of the exact averaging, decentralized +algorithms [10, 18, 31] require only partial communication through gossip averaging and reduce communication +overhead by allowing a node to communicate with fewer nodes, e.g., only its neighbors, thus avoiding having the +busiest point. How nodes are connected between each other makes up the network topology. +One of the most challenging aspects in decentralized optimization is data-heterogeneity, that is when the training data is +not identically and independently (non-i.i.d.) distributed across the nodes. Such non-i.i.d. distributions often arise in +practical applications, since, for example, training data originating from cell phones, sensors, or hospitals can have +regional differences [8]. In this case, the local empirical losses on each client are different. This can slow down the +arXiv:2301.01313v1 [math.OC] 3 Jan 2023 + +convergence [12, 25] or even yield local overfitting (often termed client-drift) as the clients may drift away from the +global optimum in the course of the optimization process [6, 9, 12, 17]. +There are several decentralized algorithms that have been shown to mitigate heterogeneity. However, most of them are +proven to converge for only strongly convex functions [1, 15, 24, 28, 32], or are proven for smooth non-convex functions +but have a strict constraint on network topologies [such as e.g. 30]. Stochastic gradient tracking (GT) [11, 22, 24, 27, 35] +algorithms have been proposed to address data-heterogeneity for arbitrary networks for smooth non-convex functions. +Its convergence rate only depends on the data heterogeneity at the initial point, which can be completely removed with +proper initialization. However, the clients are required to communicate with all their neighbors in the network after +every single model update. These methods are still therefore associated with high communication overheads. +In order to further reduce communication overhead within distributed training, various engineering techniques have been +proposed, such as using large batch [4, 20, 33], model/gradient compression [13] or asynchronized communication [19]. +In this work, we focus on local updates to reduce communication frequency, which is often efficient in practice but +remains challenging in the theoretical analysis [5, 14, 23, 29]. However, performing a large number of local steps can +exacerbate the client-drift. The resulting optimization difficulties can negate the communication savings [9, 12]. The +analysis of incorporating local steps while heterogeneity independence in the decentralized optimization is still seldom +investigated. +Integrating local updates into GT is non-trivial. For instance, simply skipping communication rounds in GT (and thereby +performing a number of local updates in-between) does not work well in practice1. A concurrent work LU-GT[26] +analyzed the performance of GT periodically skipping the communication but only in the deterministic setting.2 +As a solution, we carefully design a novel tracking mechanism that enables to combine GT with local steps.3 The +resulting algorithm—K-GT, where K denotes the number of local steps—is a novel decentralized method that provides +communication-efficient tracking with local updates. We prove that the convergence of K-GT depends only on the +data heterogeneity at the starting point and that this weak data dependence can be completely circumvented with +an additional round of global communication. As long as K-GT uses the same initialization as GT, K-GT inherits +the heterogeneity independence property of GT. We prove that K-GT (Algorithm 1) achieves asymptotically linear +speed-up in terms of communication round w.r.t. local steps K and number of clients n, and that it converges in +O +� +σ2 +nKϵ2 +� +rounds to an ϵ-approximate stationary point. The number of communication rounds is asymptotically +reduced by a factor of K compared to GT. We further show that the convergence rate (including higher order terms) +does not depend on the data-heterogenity if with proper initialization, opposed as e.g. for decentralized stochastic +gradient descent (D-SGD) without tracking. +The outline of this paper is as follows: In Section 2, we give the precise formulation of the distributed optimization +problem setting. In Section 3, we introduce the algorithm design of K-GT and demonstrate how it helps to correct +for heterogeneity. Here, our main result state its convergence rate, see Theorem 3.2. In Section 4, we generalize the +gradient tracking framework and discuss about the drawbacks of other GT alternatives that could also be stemmed from +the same framework. We in addition contribute their convergence results and give a comparison to show that K-GT is +the most communication efficiency theorectically. In Section 5, we compare the GT-variants with baseline D-SGD with +numerical examples in detail. +Contributions. +We summarize our main results below. +• We develop a novel gradient tracking algorithm for distributed optimization and analyze its convergence properties. +We prove that K-GT enjoys heterogeneity-independent complexity estimates (with proper initialization) and prove +that it converges asymptotically in O +� +σ2 +nKϵ2 +� +rounds, where n denotes client number, K the number of local steps, +σ2 the stochastic noise level and ϵ the accuracy. This improves by a factor of K over the GT baseline. +• We provide additional theoretical insights, by studying (i) the convergence of the naïve local extension of GT, periodic +GT, explaining that it performs worse than K-GT when the stochastic noise is large, and (ii) a computationally +inefficient variant, large-batch GT that matches the iteration, but not the computation complexity of K-GT. +• We empirically verify the theoretical results on strongly convex and non-convex functions and explain the impact of +noise, local steps and data-heterogeneity on the convergence. K-GT is robust against the data-heterogeneity while +improving the communication efficiency and improves generalization performance over baseline algorithms. +1We evaluate this variant (termed periodical GT) below in Section 5, see e.g. Figure 2. +2This concurrent work was independently developed while we were finalizing this manuscript. We will add a more detailed +comparison to the next version of this manuscript. +3Partial results of this paper were previously presented in YL’s master thesis [21] +2 + +Table 1: A comparison under different working conditions. ∆ ≤ n denotes the maximum degree of the communication graph. +K-GT is the first fully-decentralized tracking algorithm with local steps. +Algorithm +Settings +Communication cost at the busiest point +Local steps +heterogeneity-robustnessa +SCAFFOLD [9] +O(n) + + +GOSSIP-PGA [2] +O(∆) + + +D-SGD [12] + + +GT [24] + + +D2 [30] + + +K-GT [ours] + + +aThe +data +heterogeneity +does +not +impact +the +worst-case +convergence +rate +(but +might +require +special +initialization). +2 +Problem setting +We introduce the notation and setup in this section. +2.1 +Decentralized Optimization Problem +We consider the optimization problems as the summation from n-client loss functions, +min +x∈Rdf(x) := 1 +n +n +� +i=1 +[fi(x) := Eξi∼DiFi(x; ξi)] , +(1) +where n denotes the number of clients within the system, ξi is a random sample from Di and Di denotes the local +distribution only available on node i ∈ [n]. Di could be arbitrary and different among clients considering the applications. +This setup models both empirical risk minimization and the online optimization setting. +In this work, we consider general smooth non-convex functions and bounded stochastic noise. +Assumption 1 (Smoothness). Each function fi(x) : Rd → R, ∀i ∈ [n] is differentiable and there exists a constant +L > 0 such that for each x, y ∈ Rd, +fi(y) ≤ fi(x) + ∇fi(x)T (y − x) + L +2 ||x − y||2 +2 . +Assumption 2 (Bounded variance). Each client variance is uniformly bounded, +∀i ∈ [n], ∀x ∈ Rd, Eξ∼Di||∇Fi(x; ξ) − ∇fi(x)||2 +2 ≤ σ2 . +2.2 +Communication graph +The training is implemented over a decentralized network, and its topology is modelled as an undirected graph: (V, E), +where V := {1, 2, . . . , n} is the node set and E ⊆ V × V is the edge set. Node (or client) represents a computing node, +and clients communicate only along the edges e ∈ E. We denote the adjacency matrix W ∈ Rn×n, where wij = 0 +means node i and j are not connected, i.e., eij = (i, j) /∈ E. +Assumption 3 (Mixing rate). Given the symmetric and doubly stochastic mixing matrix W ∈ Rn×n of nonnegative +real numbers, i.e., ∀i, j ∈ [n], wij ≥ 0, �n +i=1 wij = �n +j=1 wij = 1, the consensus distance decreases linearly after +averaging step, i.e. there exists a 1 ≥ p > 0 such that +||XW − ¯X||2 +F ≤ (1 − p)||X − ¯X||2 +F , ∀X ∈ Rd×n . +Note that if the commonly used network parameter ρ := ||W − 1n1T +n +n +|| [2] is strictly less than 1, then 1 ≥ p > 0 [see +e.g. 11, 12]. The mixing rate describes the connectivity of the network. The larger value of p means the communication +graph is better connected. p = 1 for a complete graph W = 1 +n11T , and p = 0 for a disconnected graph W = In. +2.3 +Data heterogeneity and correction +When the local distributions {Di} are identical on each client, the local functions {fi(x)} are identical to each other, i.e., +fi(x) ≡ f(x). Otherwise, heterogeneous local distributions result in heterogeneous local functions. And heterogeneity +is usually measured by the discrepancy between local gradients {∇fi(x)} and global gradient ∇f(x) [9, 12] as follows. +3 + +Assumption 4 (Data-heterogeneity). There exists constants ζ2 > 0 and B ≥ 1 such that +∀x ∈ Rd, 1 +n +n +� +i=1 +||∇fi(x)||2 ≤ ¯ζ2 + B2||∇f(x)||2 , +where both ¯ζ2 and B2 represent the degree of heterogeneity within the system. +The baseline Decentralized SGD (D-SGD) uses naïve gradient w.r.t local model, the convergence of which inevitably +are influenced by both ¯ζ2 and B [25]. +2.3.1 +Notations +Gradient tracking algorithm mainly manipulates between two variables, model iterate x ∈ Rd and tracking variable +z ∈ Rd. More precisely, we denote vector y ∈ {x, z} as y(t)+k +i +on node i in local step k at communication round t, +and denote its average by ¯y = 1 +n +� +i yi. +The collection of vectors yi for all i ∈ [n] in matrix form is denoted by a capital letter with columns yi, i.e., +Y = [y1, . . . , yn] ∈ Rd×n, +¯Y = [¯y, . . . , ¯y] = 1 +nY1n1T +n ∈ Rd×n . +Also, we extend this matrix definition to both gradient and stochastic gradient of (1) w.r.t model X on sample +ξ = [ξ1, . . . , ξn], where ξi ∼ Di, +∇F(X; ξ) = [∇F1(x1; ξ1), . . . , ∇Fn(xn; ξn)] ∈ Rd×n, +∇f(X) = E(ξ1,...,ξn)∇F(X; ξ) = [∇f1(x1), . . . , ∇fn(xn)] ∈ Rd×n . +2.3.2 +Gradient tracking +Gradient tracking algorithm (GT) [27] is defined by the following update equations: +X(t+1) = (X(t) − ηZ(t))W +Z(t+1) = Z(t)W + G(t+1) − G(t), +(2) +in matrix format. Here G(t) = ∇F(X(t); ξ(t)) and η > 0 denotes the stepsize. +When data is heterogeneous among different nodes, {∇Fi(x; ξi), ∀i} are different. But GT uses bias-correction to +compensate heterogeneous gradient at each node. This correction is governed by the tracking variable Z that replaces +the naïve gradient: +Z(t+1) = ∇F(X(t+1); ξ(t+1)) + Z(t)W − G(t) +� +�� +� +correction +(3) +Since the update (2) simultaneously updates both the model X and the tracking variable Z, there is no need to take +extra consideration on the heterogeneous local gradient. GT is proven to converge regardless of data heterogeneity [27]. +3 +K-GT: Gradient Sum Tracking algorithm +In this section, we present our new decentralized stochastic algorithm K-GT with its convergence analysis for general +non-convex functions. +3.1 +Algorithm +In the K-GT algorithm we allow each client to perform K ≥ 1 local steps between each communication round. To +compensate to the data-heterogeneity, we use a similar correction as in (3) on top of the stochastic gradient. We denote +the correction as ci on node i. Then each node repeats the following updating rule, i ∈ [n]: +1. Compute a local stochastic gradient ∇Fi(xi; ξi) by sampling ξi from distribution Di; +2. Update the local model x(t)+k+1 +i += x(t)+k +i +− ηc +� +∇Fi(x(t)+k +i +; ξ(t)+k +i +) + c(t) +i +� +using the stochastic gradients +at (t) + k-th iteration and correction c(t) +i +in t-th communication; +4 + +3. Repeat step (1)-(2) K times, then obtain the tracking throughout local steps, z(t) +i += +1 +Kηc +� +x(t) +i +− x(t)+K +i +� +. +Exchange {xi, ci} with neighbors: (in matrix format): +X(t+1) = +� +X(t) − ηs(X(t) − X(t)+K) +� +W , +C(t+1) = C(t) + Z(t)(W − I) . +(4) +The complete algorithm is summarized in Algorithm 1. +Proposition 3.1 (Gradient Sum Tracking). Define Z(t) = +1 +Kηc +� +X(t) − X(t)+K� +as the tracking variable during +communication round t. The update rule for both models X(t) and tracking variables Z(t) at communication in K-GT +can be rewritten as (η = ηsηc): +X(t+1) = +� +X(t) − KηZ(t)� +W , +Z(t+1) = Z(t+1)W + G(t+1) − G(t) , +(5) +where G(t) = 1 +K +� +k ∇F(X(t)+k; ξ(t)+k) denotes the mean update over the local steps. +The detailed proof is included in Appendix B.1. +Remark 1. If K = 1 in (5), K-GT is equivalent to Gradient Tracking [27] with η = ηsηc. +K-GT essentially runs SGD if communication is the most sufficient. +To understand the intuition behind K-GT, +let us consider the global average ¯X at each iterate, which gets updated just like the standard stochastic gradient descent: +¯X(t)+k+1 = +� +X(t)+k − ηc(∇F(X(t)+k; ξ(t)+k) + C(t)) +� +11T +n += ¯X(t) − ηc +� +∇F(X(t)+k; ξ(t)+k) + ¯C(t)� +. +If initialized to be C(0) = ∇F(X(0); ξ(0)) +� 11T +n +− I +� +, the average of correction satisfies +¯C(t+1) = ¯C(t) + Z(t)(W − I) 11T +n += ¯C(t) , +¯C(0) = ∇F(X(0); ξ(0)) +� +11T +n +− I +� 11T +n +≡ 0 . +Then the average of model iterate satisfies +¯X(t)+k+1 = ¯X(t) − ηc∇F(X(t)+k; ξ(t)+k), +which updates model with averaged stochastic gradient. +How does this correction improves D-SGD? +We consider applying the similar analysis from [30] to illustrate the +effectiveness of K-GT. Assume that X(t) has achieved an optimum X⋆ := x⋆1T with all local models equal to the +optimum x⋆. Based on our analysis in appendix (Lemma C.8), the correction will be equal to +c⋆ +i := −∇Fi(x⋆; ξ) + 1 +n +� +j +∇Fj(x⋆ +j; ξ) . +Then the next local update for K-GT would be +X(t)+1 = X⋆ − ηc(∇F(X⋆; ξ) + C⋆) += X⋆ − ηc∇F(X⋆; ξ) 11T +n . +This illustration shows that for K-GT, the convergence when we approach a solution with only local update relies on +the magnitude of E||∇F(X⋆; ξ) 11T +n ||2 +F , which is bounded by O(σ2). +However, consider the same situation for D-SGD, +X(t)+1 = X⋆ − η∇F(X⋆; ξ). +On different nodes, ∇Fi(X⋆; ξ) deviates from each other due to data heterogeneity, and the deviation can only be +characterized by ζ2 as suggested in Assumption 4. Then the upper bound for D-SGD of the same magnitude of +convergence when in the neighborhood of solution is O(σ2 + ¯ζ2) [30], which is obviously worse than that for K-GT. +The additional O(¯ζ2) in D-SGD from the data heterogeneity can never be improved if always using the sole stochastic +gradient [25]. +5 + +Algorithm 1 K-GT: Gradient Sum Tracking +1: parameters: T: number of communication; K: number of local steps; ηc, ηs: local, communication stepsize; W: given +topology. +2: Initialize: ∀i, j ∈ [n], x(0) +i += x(0) +j ; c(0) +i += −∇Fi(x(0); ξi) + 1 +n +� +j ∇Fj(x(0); ξj)a. +3: for client i ∈ {1, . . . , n} parallel do +4: +for communication: t ← 0 to T − 1 do +5: +for local step: k ← 0 to K − 1 do +6: +x(t)+k+1 +i += x(t)+k +i +− ηc(∇Fi(x(t)+k +i +; ξ(t)+k +i +) + c(t) +i ) +7: +end for +8: +z(t) +i += +1 +Kηc (x(t) +i +− x(t)+K +i +) +9: +c(t+1) +i += c(t) +i +− z(t) +i ++ � +j wijz(t) +j +▷ update tracking variable +10: +x(t+1) +i += � +j wij(x(t) +j +− Kηsηcz(t) +j ) +▷ update model parameters +11: +end for +12: end for +aThis initialization in correction c(0) +i +is required for heterogeneity-independent analysis in theory. We demonstrate later with experiment +that simply choosing c(0) +i += 0 works well in practice. +3.2 +Main theorem: data-independent convergence on non-convex functions +In this section, we present the convergence rate of K-GT. Note that p is the network parameter defined in Assumption 3. +Theorem 3.2 (K-GT convergence). For schemes as in Algorithm 1 with mixing matrices such as in Assumption 3 and +arbitrary error ϵ > 0, there exists a constant stepsize ηc = O( p +KL) and ηs = O(p) such that under Assumption 1 and 2 +for L-smooth, (possibly non-convex) functions, it holds +1 +T +1 +� +t E||∇f(¯x(t))||2 ≤ ϵ after +O +� σ2 +Knϵ2 + +σ +p2√ +Kϵ +3 +2 + 1 +p2ϵ +� +· L +communication rounds. +4 +Discussion +In this section, we are going to introduce and compare with other possible ways of introducing local steps to GT that +has the similar communication pattern as K-GT. +4.1 +Other GT alternatives +4.1.1 +Gradient Tracking with Periodical Communication (Periodical GT). +There is another way to incorporate local steps into above framework (2). Instead of communication via fixed topology +W, the communication graph changes along with time denoted by W(t). Note that W(t) = I, which means there is +actually no communication. If W(t) periodically alternates between {W, I}, it also reduces communication frequency. +The full detail is concluded in Algorithm 2 (Appendix A.1). +K-GT suffers from less noise than Periodical GT. +It is possible to reformulate local steps of Periodical GT as +corrected SGD, same as that for K-GT. But Periodical GT has different update for correction at communication with +C(t+1) = C(t)W + ∇F(X(t)+K−1; ξ(t)+K−1)(W − I) . +(6) +The equivalence of reformulation is proven in Appendix B.2.1. +However, if we simply reformulate equation (4), we obtain that K-GT uses the average of K stochastic gradient, i.e., +C(t+1) = C(t)W + 1 +K +� +k ∇F(X(t)+k; ξ(t)+k)(W − I) , +which can reduce stochastic noise by K. Periodical GT uses only one stochastic gradient, thus would suffer more from +stochastic noise. +Using more random samples on stochastic gradient can reduce noise in Periodical GT. +It is trivial to reduce the +stochastic noise in (6) if using more random samples {ξ(t),s|s = 0, . . . , K − 1} to replace ∇F(X(t)+K−1; ξ(t)+K−1) +6 + +Table 2: The comparison of communication rounds needed to reach target accuracy ϵ on non-convex functions. Our results on both +K-GT, Periodical GT improve the rate of D-SGD in terms of heterogeneity parameter ¯ζ2(defined in Ass. 4) when using local steps, +and accelerates the rate of GT. +Local steps +Algorithm +Communication rounds +K = 1 +GT [11] +O +� +σ2 +nϵ2 + +σ +p +3 +2 ϵ +3 +2 + +1 +p2ϵ +� +K > 1 +D-SGD [12] +O +� +σ2 +Knϵ2 + ( +¯ζ +p + +σ +√pK ) 1 +ϵ +3 +2 + 1 +pϵ +� +K-GT [ours] +O +� +σ2 +Knϵ2 + +σ +p2√ +Kϵ +3 +2 + +1 +p2ϵ +� +Periodical GT [ours] +O +� +σ2 +Knϵ2 + +σ +p2ϵ +3 +2 + +1 +p2ϵ +� +Periodical GT w/ full gradient [ours] +O +� +σ2 +Knϵ2 + +σ +p2√ +Kϵ +3 +2 + +1 +p2ϵ +� +with +∇F(X(t)+K−1; ξ(t)+K−1) = 1 +K +� +s +∇F(X(t)+K−1; ξ(t)+K−1,s) , +then the correction C(t) has the same level of stochastic noise as K-GT. However, using more sample to calculate SGD +requires a lot more extra computation than K-GT. +Theorem 4.1 (Periodical GT convergence). For schemes as in Algorithm 2 (Appendix A.1) with mixing matrices such +as in Assumption 3 and arbitrary error ϵ > 0, there exists a constant stepsize η = O( p2 +KL) such that under Assumption +1 and 2 for L-smooth, (possibly non-convex) functions, it holds +1 +T +1 +� +t E||∇f(¯x(t))||2 ≤ ϵ after +O +� σ2 +Knϵ2 + +σ +p2ϵ +3 +2 + 1 +p2ϵ +� +· L +communication rounds. Conversely, if we consider using the full-batch tracking Algorithm 3 (Appendix A.1), then the +convergence rate can be improved to +O +� σ2 +Knϵ2 + +σ +p2√ +Kϵ +3 +2 + 1 +p2ϵ +� +· L . +Note that the latter result refers to full batch tracking which comes at additional computation cost each communication +round (in contrast to K-GT). +4.1.2 +Gradient Tracking with Large Batch (Large-batch GT) +Apart from local training, large-batch training is also popular to achieve acceleration in distributed setting. Similar +to Large-batch SGD, we calculate G(t) = � +k ∇F(X(t); ξ(t),k) in (2) with K i.i.d. random samples, {ξ(t),k|k = +0, . . . , K − 1} and make W(t) = W. It is theoretically workable to improve the asymptotical communication rounds +needed to reach the desired accuracy ϵ from O +� σ2 +nϵ2 +� +[11] to O +� +σ2 +nKϵ2 +� +, while remains heterogeneity-independent. +We empirically show that Large-batch GT remains heterogeneity-independent and has the same communication +performance as K-GT (Figure 1, 3). +4.2 +Convergence comparison +We summarized the convergence rate for the related decentralized algorithms in Table 2. In order to analyze the +convergence for other methods that depend on data heterogeneity, there is an addition assumption to measure data +heterogeneity [9, 12]. +K-GT achieves acceleration by local steps in high-noise regime. +When ϵ is sufficiently small, the noise dominates +the convergence rate (σ > 0) and it is not affected by graph parameter p for GT, Periodical GT and K-GT. Then after +enough transient time, Periodical GT and K-GT with O( +σ2 +nKϵ2 ) achieves linear speedup by K compared to GT with +rate O( σ2 +nϵ2 ). In addition, the transient time for K-GT also decreases with O( +1 +√ +K ) comparing to GT baseline. +GT methods are in general more sensitive to the network parameter than diffusion methods [34], e.g., D-SGD, in the +non-asymptotical regime. In our analysis of K-GT, the dependency on the network parameter p is worse than for vanilla +7 + +GT. Combining our analysis with the tighter analysis of GT presented in concurrent work [11] would be an interesting +future direction—however in this work we focused on the aspect of equipping GT with local steps. +The impact of data heterogeneity is removable for K-GT . +K-GT does not completely solve data heterogeneity +in general, and depends on the data heterogeneity at the initial point, which is the same case in GT. It has been proven +for GT that in the non-asymptotic regime a weaker dependence on the data heterogeneity at the initial point actually +remains [11]. However, with a single round of global communication for the initial iterates (e.g. in Alg. 1), we +can remove the heterogeneity from the complexity estimates for GT, K-GT and Periodical GT. Table 2 removes the +initialization terms from the rate to simplify the presentation. On the contrary, heterogeneity ¯ζ2 under no circumstance +can be eliminated for D-SGD and slows down its convergence. +Periodical GT suffers more from noise comparing to K-GT. +In asymptotical regime, the transient time for K-GT +O( +σ +√ +K ) decrease with local steps while Periodical GT O(σ) does not. But this noise term can be improved as discussed +in section 4.1.1. If we consider a full gradient in equation (6), Periodical GT performs similar to K-GT at the expense +of extra computation. +5 +Experimental results +We evaluate the effectiveness of K-GT by comparing it with D-SGD and periodical GT. +5.1 +Setting +We conduct experiments in two settings. +1. SYNTHETIC DATASETS: We first construct the distributed least squares objective with fi(x) = 1 +2||Aix−bi||2 +with fixed Hessian A2 +i = i2 +n · Id, and sample each bi ∼ N(0, +¯ζ2 +i2 · Id) for each client i ∈ [n], where ¯ζ2 can +control the deviation between local objectives [12]. Stochastic noise is controlled by adding Gaussian noise +with σ2 = 1. +2. REAL-WORLD DATASET, MNIST [3]: We test the case that all clients collaboratively train a convolutional +neural network (CNN)4 on real-world dataset, MNIST. In total this dataset contains 60,000 images of size +28×28 and 10 labels. +We use a ring topology for both sets of experiments. For simplicity, instead of using the initialization in Alg. 1, we +initialize ci = 0 for all experiments.For data partition on MNIST, we consider both homogeneous and heterogeneous +cases. The homogeneous dataset is first shuffled and then uniformly partitioned among all the clients. We call this +the ‘random’ setting. The heterogeneous datasets is created when each client only has exclusive access to subset of +classes. We call this the ‘sorted’ case, and the data variation across clients is maximized at this time. We use n = 5 and +n = 10 clients and each client has access to one and two classes case accordingly, and the case n = 10 has more severe +heterogeneity condition than the case n = 5. +Parameter tuning. +For SYNTHETIC DATASETS, we use the same learning rate ηs=1 and η=1e-3. For MNIST, we use +the best constant learning rate tuned from {0.5, 0.1, 0.05, 0.01, 0.005, 0.001} for algorithms and batch size 128 on +each client. Note that even though our algorithm is purposed with constant learning rate, using more sophisticated and +time-varying learning rate scheduler would definitely bring much better performance. +Comparison. +We mainly illustrate the acceleration and robustness in convergence rate of the K-GT compared to +baseline D-SGD. We also consider the performance of several GT-variants that supports local steps discussed in +Section 4. +5.2 +Numerical results +K-GT is the most robust against heterogeneity. +In the convex case, client drift only happens for D-SGD suggested +by Figure 1 in which the larger value ¯ζ2 ̸= 0 gets, the poorer model quality D-SGD ends up with. However, K-GT, +Periodical GT (w/ and w/o full grad) and Large-batch GT do not suffer from ’client-drift’ and ultimately reach the +consistent level of model quality regardless of increasing of ¯ζ and K (number of either local steps or random samples). +In the non-convex case, since it’s known the optimality condition and optimization trajectory is more complex than +the convex case, generalization performance of all methods in Figure 2 cannot fully recover the baseline performance +when data partition is random. However, K-GT could always outperform when data partition is non-i.i.d. and the +improvement is more significant when the degree of heterogeneity is increasing from Figure 2b. +4Here we only consider a very simple network without Batch Norm layers [7] for simplicity, since it inherently assumes that the +data distribution is uniform across different batches, which is not the case that we are interested in. The detailed network structure is +listed in Appendix D. +8 + +0 +2000 +4000 +10 +5 +10 +3 +10 +1 +101 +103 +|| f(x)||2 +K = 1 +2 = 0 +0 +100 +200 +communication rounds +10 +5 +10 +3 +10 +1 +101 +103 +|| f(x)||2 +K = 20 +0 +2000 +4000 +10 +5 +10 +3 +10 +1 +101 +103 +2 = 1 +0 +100 +200 +communication rounds +10 +5 +10 +3 +10 +1 +101 +103 +0 +2000 +4000 +10 +5 +10 +3 +10 +1 +101 +103 +2 = 10 +0 +100 +200 +communication rounds +10 +5 +10 +3 +10 +1 +101 +103 +D-SGD +K-GT +Periodical GT +Periodical GT, w/ full grad +Large-batch GT +Large-batch SGD +Figure 1: Training SYNTHETIC convex functions over ring by 10 clients with noise σ2 = 1. In total 5000 communication rounds +for K = 1 (top row) while only 250 rounds for K = 20 (bottom row). All uses the same learning rate and are averaged by three +repetitions. The client-drift for D-SGD is even more severe with increasing heterogeneity (larger ¯ζ) as well as K. K-GT, GT, and +GT w/ full grad are consistent for different ¯ζ2 while achieving communication reduction when K > 1. +0 +40 +80 +120 +160 +200 +60 +70 +80 +90 +100 +top-1 test acc(%) +n = 5 +0 +40 +80 +120 +160 +200 +computation epochs +60 +70 +80 +90 +100 +top-1 test acc(%) +n = 10 +DSGD, random +DSGD, sorted +K-GT, random +K-GT, sorted +(a) K = 1, and data is partitioned either random or sorted. +0 +20 +40 +60 +80 +100 +60 +70 +80 +90 +100 +top-1 test acc(%) +n = 5 +0 +40 +80 +120 +160 +200 +communication rounds +60 +70 +80 +90 +100 +top-1 test acc(%) +n = 10 +DSGD +K-GT +Periodical GT +Periodical GT w/ full grad +(b) K = 1 +n epoch, and data is partitioned sorted. +Figure 2: Generalization performance on MNIST among D-SGD (blue), K-GT(red), GT w/o (orange) and w/ (green) full gradient +for n = 5 (top row) and n = 10 (bottom row) clients. The x-axis corresponds to (a) the number of pass over overall dataset(epoch), +and (b) the number of communication rounds. In (a), when K = 1, K-GT and Periodical GT are identical to GT baseline (remark 1). +Learning rates are tuned to be the best. Note that 1 epoch of passing over global dataset is equivalent to 470 times computation on +SGD when mini-batch sized 128. And for (b), since nK = 470 is fixed for both n = 5 (top right) and n = 10 (top left), the number +of local steps between communication rounds is Kn=5 = 2Kn=10. +9 + +0 +20 +40 +60 +80 +computation epochs +0 +20 +40 +60 +80 +100 +top-1 test acc(%) +K = 10, Bloc = 128 +D-SGD, B = Bloc +Large-batch SGD, B = KBloc +K-GT, B = Bloc +Large-batch GT, B = KBloc +Figure 3: Generalization performance comparison on MNIST between large-batch training and training with local steps, where +large-batch training uses KBloc local batch size and communicates every update, while training with local steps uses Bloc local +batch size and communicates periodically every K local update. +Local step reduces communication. +From K = 1 to K = 20 in Figure 1, K-GT and other GT alternatives reach +the same target after 2000 rounds to only 100 rounds, achieving linear reduction in communication with the help of +local steps. However, more local steps makes D-SGD suffer even more in model quality. At the same time, introducing +local steps into the training of non-convex functions would still achieve communication reduction but not by a linear +factor of K as in the convex case. Within Figure 2b, we fixe nK = 1 epoch over the data such that for no matter +which n = 5 or n = 10 client communicates once after 1 epoch of computation. Compared to K = 1 in Figure 1, the +acceleration when K > 1 is still by a huge amount. However, note that introducing more local steps when data partition +is heterogeneous would result in more severe quality loss, but K-GT still outperforms. +Large-batch GT has similar performance to tracking with local steps. +From both convex (Figure 1) and non- +convex (Figure 3) functions, either K-GT or Large-batch GT, after the same number of communication rounds while the +simultaneously the same number of computation epochs, reaches the similar level of accuracy. But for D-SGD, training +with local steps could be even more stable and generalizes better than the Large-batch, which has been empirically +investigated in [20]. +6 +Conclusion +Decentralized learning is a promising building block for the democratization of Deep Learning. Especially in Edge +AI applications, users’ data does not follow a uniform distribution. This requires robustness of decentralized learning +algorithms to data heterogeneity. +We propose a novel decentralized optimization algorithm (K-GT) supports communication efficient local update steps +and overcomes data dissimilarity. The tracking mechanism uses the accumulated gradient sum, akin to momentum, +thereby reducing variance across local updates without the need of large batch sizes. We demonstrated the superiority +of K-GT with both convergence guarantees and empirical evaluations. +10 + +References +[1] S. Alghunaim and A. Sayed, Linear convergence of primal–dual gradient methods and their performance in +distributed optimization, Automatica 117 (2020), p. 109003. +[2] Chen et al., Accelerating gossip SGD with periodic global averaging, in ICML. 2021. +[3] L. Deng, The mnist database of handwritten digit images for machine learning research, IEEE Signal Processing +Magazine 29 (2012), pp. 141–142. +[4] P. Goyal, P. Dollár, R.B. Girshick, P. Noordhuis, L. Wesolowski, A. Kyrola, A. Tulloch, Y. Jia, and K. He, Accurate, +large minibatch SGD: training ImageNet in 1 hour, arXiv preprint arXiv:1706.02677 (2017). +[5] F. Haddadpour, M.M. Kamani, M. Mahdavi, and V. Cadambe, Local SGD with Periodic Averaging: Tighter +Analysis and Adaptive Synchronization, in NeurIPS. 2019. +[6] K. Hsieh, A. Phanishayee, O. Mutlu, and P. Gibbons, The Non-IID Data Quagmire of Decentralized Machine +Learning, in ICML. 2020. +[7] S. I. and C. S., Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift, +in ICML. 2015. +[8] P. Kairouz, H.B. McMahan, B. Avent, A. Bellet, M. Bennis, A.N. Bhagoji, K. Bonawitz, Z. Charles, G. Cormode, +R. Cummings, R.G.L. D’Oliveira, H. Eichner, S.E. Rouayheb, D. Evans, J. Gardner, Z. Garrett, A. Gascón, B. +Ghazi, P.B. Gibbons, M. Gruteser, Z. Harchaoui, C. He, L. He, Z. Huo, B. Hutchinson, J. Hsu, M. Jaggi, T. Javidi, +G. Joshi, M. Khodak, J. Koneˇcný, A. Korolova, F. Koushanfar, S. Koyejo, T. Lepoint, Y. Liu, P. Mittal, M. Mohri, +R. Nock, A. Özgür, R. Pagh, M. Raykova, H. Qi, D. Ramage, R. Raskar, D. Song, W. Song, S.U. Stich, Z. Sun, +A.T. Suresh, F. Tramèr, P. Vepakomma, J. Wang, L. Xiong, Z. Xu, Q. Yang, F.X. Yu, H. Yu, and S. Zhao, Advances +and open problems in federated learning, Foundations and Trends® in Machine Learning 14 (2021), pp. 1–210. +[9] S.P. Karimireddy, S. Kale, M. Mohri, S. Reddi, S.U. Stich, and A.T. Suresh, SCAFFOLD: Stochastic Controlled +Averaging for Federated Learning, in ICML. 2020. +[10] A. Koloskova, T. Lin, S.U. Stich, and M. Jaggi, Decentralized deep learning with arbitrary communication +compression, ICLR (2020). +[11] A. Koloskova, T. Lin, and S. Stich, An improved analysis of gradient tracking for decentralized machine learning, +NeurIPS (2021). +[12] A. Koloskova, N. Loizou, S. Boreiri, M. Jaggi, and S. Stich, A Unified Theory of Decentralized SGD with Changing +Topology and Local Updates, in ICML. 2020. +[13] A. Koloskova, S. Stich, and M. Jaggi, Decentralized Stochastic Optimization and Gossip Algorithms with +Compressed Communication, in ICML. 2019. +[14] J. Koneˇcn`y, H.B. McMahan, F.X. Yu, P. Richtárik, A.T. Suresh, and D. Bacon, Federated learning: Strategies for +improving communication efficiency, arXiv preprint arXiv:1610.05492 (2016). +[15] B. Li, S. Cen, Y. Chen, and Y. Chi, Communication-Efficient Distributed Optimization in Networks with Gradient +Tracking and Variance Reduction, in AISTATS. 2020. +[16] M. Li, D.G. Andersen, A.J. Smola, and K. Yu, Communication Efficient Distributed Machine Learning with the +Parameter Server, in NeurIPS. 2014. +[17] T. Li, A.K. Sahu, M. Zaheer, M. Sanjabi, A. Talwalkar, and V. Smith, Federated Optimization in Heterogeneous +Networks, in MLSys. 2020. +[18] X. Lian, C. Zhang, H. Zhang, C.J. Hsieh, W. Zhang, and J. Liu, Can Decentralized Algorithms Outperform +Centralized Algorithms? A Case Study for Decentralized Parallel Stochastic Gradient Descent, in NeurIPS. 2017. +[19] X. Lian, W. Zhang, C. Zhang, and J. Liu, Asynchronous Decentralized Parallel Stochastic Gradient Descent, in +ICML. 2018. +[20] T. Lin, S.U. Stich, K.K. Patel, and M. Jaggi, Don’t Use Large Mini-batches, Use Local SGD, in ICLR. 2020. +[21] Y. Liu, Variance reduction on decentralized training over heterogeneous data, Master’s thesis, ETH Zürich, 2021. +Available at https://pub.tik.ee.ethz.ch/students/2020-HS/MA-2020-32.pdf. +11 + +[22] P.D. Lorenzo and G. Scutari, NEXT: In-network nonconvex optimization, IEEE Trans. Signal and Information +Processing over Networks 2 (2016), pp. 120–136. +[23] B. McMahan, E. Moore, D. Ramage, S. Hampson, and B.A. y Arcas, Communication-efficient learning of deep +networks from decentralized data, in AISTATS. PMLR, 2017, pp. 1273–1282. +[24] A. Nedi´c, A. Olshevsky, and W. Shi, Achieving geometric convergence for distributed optimization over time- +varying graphs, SIAM Journal on Optimization 27 (2017), pp. 2597–2633. +[25] A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro, Robust stochastic approximation approach to stochastic +programming, SIAM Journal on Optimization 19 (2009), pp. 1574–1609. +[26] E.D.H. Nguyen, S.A. Alghunaim, K. Yuan, and C.A. Uribe, On the performance of gradient tracking with local +updates (2022). Available at https://arxiv.org/abs/2210.04757. +[27] S. Pu and A. Nedi´c, A Distributed Stochastic Gradient Tracking Method, in ICDC. 2018. +[28] W. Shi, Q. Ling, G. Wu, and W. Yin, EXTRA: An exact first-order algorithm for decentralized consensus +optimization, SIAM Journal on Optimization 25 (2015), pp. 944–966. +[29] S.U. Stich, Local SGD Converges Fast and Communicates Little, in ICLR. 2019. +[30] H. Tang, X. Lian, M. Yan, C. Zhang, and J. Liu, D2: Decentralized Training over Decentralized Data, in ICML. +2018. +[31] J.N. Tsitsiklis, Problems in decentralized decision making and computation., Tech. Rep., Massachusetts Inst of +Tech Cambridge Lab for Information and Decision Systems, 1984. +[32] R. Xin, U.A. Khan, and S. Kar, Variance-reduced decentralized stochastic optimization with accelerated conver- +gence, IEEE Trans. Signal Process 68 (2020), pp. 6255–6271. +[33] Y. You, Z. Zhang, C.J. Hsieh, J. Demmel, and K. Keutzer, ImageNet Training in Minutes, in ICPP. 2018. +[34] K. Yuan, W. Xu, and Q. Ling, Can primal methods outperform primal-dual methods in decentralized dynamic +optimization?, IEEE Trans. Signal Process 68 (2020), pp. 4466–4480. +[35] J. Zhang and K. You, Decentralized stochastic gradient tracking for non-convex empirical risk minimization, +arXiv preprint arXiv:1909.02712 (2019). +12 + +A +Algorithm +A.1 +Periodical Algorithm +Algorithm 2 Periodical GT: GT with periodical communication +1: parameters: +2: T: number of communication; K: number of local steps; ηs, ηc: communication, local stepsize; W: given topology. +3: Initialize: x0 +i = x0 +j, z0 +i = 1 +n +� +i Fi(x0; ξi) = z0 +j, ∀ i, j ∈ [n]a +4: for node i ∈ {1, ..., n} parallel do +5: +for communication: t ← 0 to T − 1 do +6: +for local steps: k ← 0 to K − 2 do +7: +x(t)+k+1 +i += x(t)+k +i +− ηcz(t)+k +i +8: +z(t)+k+1 +i += z(t)+k +i ++ ∇Fi(x(t)+k+1 +i +; ξ(t)+k+1 +i +) − ∇Fi(x(t)+k +i +; ξ(t)+k +i +) +9: +end for +10: +x(t)+K +j += x(t)+K−1 +i +− ηcz(t)+K−1 +i +11: +x(t+1) +i += � +j wij +� +x(t) +j +− ηs(x(t) +j +− x(t)+K +j +) +� +12: +z(t+1) +i += � +j wijz(t)+K−1 +j ++ ∇Fi(x(t+1) +i +; ξ(t+1) +i +) − ∇Fi(x(t)+K−1 +i +; ξ(t)+K−1 +i +) +13: +end for +14: end for +aThis initialization in tracking variable z(0) +i +is required for heterogeneity-independent analysis in theory. In fact, we show later +with experiment that z(0) +i += ∇Fi(x0; ξ) works well in practice. +Algorithm 3 Periodical GT with full-batch gradient +1: parameters: +2: T: number of communication; K: number of local steps; ηs, ηc: communication, local stepsize; W: given topology. +3: Initialize: x0 +i = x0 +j, c0 +i = −∇Fi(x0; ξi) + 1 +n +� +j ∇Fj(x0; ξj)a +4: for node i ∈ {1, ..., n} parallel do +5: +for communication: t ← 0 to T − 1 do +6: +for local steps: k ← 0 to K − 2 do +7: +z(t)+k +i += ∇Fi(x(t)+k +i +; ξ(t)+k +i +) + c(t) +i +8: +x(t)+k+1 +i += x(t)+k +i +− ηcz(t)+k +i +9: +end for +10: +Compute full gradient on x(t)+K−1 +i +, gi = ∇fi(x(t)+K−1 +i +). +11: +x(t)+K +j += x(t)+K−1 +i +− ηc(gi + c(t) +i ) +12: +x(t+1) +i += � +j wij +� +x(t)+K−1 +j +− ηs(x(t) +j +− x(t)+K +j +) +� +13: +c(t+1) +i += � +j wijc(t) +j ++ � +j wijgj − gi +14: +end for +15: end for +aThis initialization in correction c(0) +i +is required for heterogeneity-independent analysis in theory. In fact, we show later with +experiment that c(0) +i += 0 works well in practice. +B +Proof of proposition +In this section, we will prove the propositions previously discussed. +B.1 +Tracking Property of K-GT +Proposition B.1 (Gradient Sum Tracking). Define Z(t) = +1 +Kηc +� +X(t) − X(t)+K� +as the tracking variable during +communication round t. The update rule for both models X(t) and tracking variables Z(t) at communication in K-GT +1 + +can be rewritten as (η = ηsηc): +X(t+1) = +� +X(t) − KηZ(t)� +W , +Z(t+1) = Z(t+1)W + G(t+1) − G(t) , +(5) +where G(t) = 1 +K +� +k ∇F(X(t)+k; ξ(t)+k) denotes the mean update over the local steps. +Proof. The updating schemes of the model for K-GT are shown in equation (4), then if we define Z(t) = +1 +Kηc +� +X(t) − +X(t)+K� +, and η = ηsηc, then with simply reformulating we could derive the set of equations shown above. +B.2 +Periodical Gradient Tracking reformulation +The periodical GT is actually time-varying GT with skipping communication. That is W(t) = W in equation (2) when +mod(t, K)=0, otherwise W(t) = I no communication and local step. +Then we adopt the notation for K-GT that we denote the model at k-th local step after t-th communication round as +X(t)+k. And the same principle is applied to tracking variable Z(t)+k. In the following sections, we will first show that +Periodical GT can be equivalently reformulated and corrected SGD with constant correction throughout local steps, and +then provide the update scheme for both correction and model. +B.2.1 +Corrected SGD +Claim B.2. The local tracking variable during local steps can be equivalently rewritten as corrected SGD with +correction, i.e., +Z(t)+k+1 = ∇F(X(t)+k+1; ξ(t)+k+1) + Z(t)+k − ∇F(X(t)+k; ξ(t)+k) +� +�� +� +C(t)+k +. +And the correction C remains unchanged throughout local steps, i.e., C(t)+k+1 = C(t)+k, +∀k ∈ {0, ..., K − 1}, +and is updated only at each time of communication. +Proof. We know that local model is updated with Z instead of ∇F(X; ξ). We define the deviation of Z from the SGD +as C. By contradiction we assume that deviation is different for each local iterate (t) + k. That’s C(t)+k+1 ̸= C(t)+k. +Then for each local update, we have +Z(t)+k+1 = Z(t)+k + ∇F(X(t)+k+1; ξ(t)+k+1) − ∇F(X(t)+k; ξ(t)+k) +Z(t)+k+1 − ∇F(X(t)+k+1; ξ(t)+k+1) = Z(t)+k − ∇F(X(t)+k; ξ(t)+k) +C(t)+k+1 = C(t)+k, +∀k ∈ {0, ..., K − 1} +which contradicts the assumed fact that C(t)+k+1 ̸= C(t)+k. +B.2.2 +Updating scheme reformulation +Proposition B.3. If we additionally consider separate step sizes for local steps and communication, we can equivalently +rewrite Periodical GT as follows, +• Local steps. We consider local steps as corrected SGD. The correction C ∈ Rd×n captures the difference +between local update and communication update. For local steps, i.e, ∀k ∈ {0, ..., K − 1}, X(t)+0 ≡ X(t), +X(t)+k+1 = X(t)+k − ηc(∇F(X(t)+k; ξ(t)+k) + C(t)), +(7) +where C(t) is constant for all local steps. +• Communication. Then it synchronizes both X and C, +X(t+1) = +� +X(t) − ηs(X(t) − X(t)+K) +� +W +C(t+1) = C(t)W + ∇F(X(t)+K−1; ξ(t)+K−1)(W − I) +(8) +Proof. By Claim B.2, the local update is equivalent to corrected SGD. Note that different stepsizes ηc and ηs are used +for model update of local step and communication. +2 + +The correction C is constant during local steps by Claim B.2, then consider its update during communication. +Z(t+1) = Z(t)+K−1W + ∇F(X(t+1); ξ(t+1)) − ∇F(X(t)+K−1; ξ(t)+K−1) +⇔ (∇F(X(t+1); ξ(t+1)) + C(t+1)) = +� +∇F(X(t)+K−1; ξ(t)+K−1) + C(t)� +W + ∇F(X(t+1); ξ(t+1)) − ∇F(X(t)+K−1; ξ(t)+K−1) +⇔ C(t+1) = C(t)W + ∇F(X(t)+K−1; ξ(t)+K−1)(W − I) +C +Proof of theorem +C.1 +Technical tools +In this section, we mainly introduce some analytical tools that help in convergence analysis. +Proposition C.1 (Implications of the smoothness Assumption 1). Assumption 1 implies ∀i and ∀x, y ∈ Rd, +||∇fi(x) − ∇fi(y)|| ≤ L||x − y||. +Lemma C.2. For arbitrary set of n vectors {ai}n +i=1, ai ∈ Rd, || 1 +n +�n +i ai||2 ≤ 1 +n +�n +i ||ai||2. +Lemma C.3. For given two vectors a, b ∈ Rd, 2⟨a, b⟩ ≤ α||a||2 + 1 +α||b||2, α > 0, which is equivalent to ||a + b||2 ≤ +(1 + α)||a||2 + (1 + 1 +α)||b||2. +Remark 2. Above inequality also holds for matrix in Frobenius norm. For A, B ∈ Rd×n, ||AB||F ≤ ||A||F ||B||2. +Lemma C.4 (Variance upperbound). If there exist n zero-mean random variables {ξi}n +i=1 that may not be independent +of each other, but all have variance smaller than σ2, then the variance of sum is upperbounded by E|| � +i ξi||2 ≤ nσ2. +Proof. E|| � +i ξi||2 ≤ E +� +n � +i ||ξi||2� +≤ nσ2. +Lemma C.5 (Unrolling recursion [12]). For any parameters r0 ≥ 0, b ≥ 0, e ≥ 0, u ≥ 0 there exists constant stepsize +η ≤ 1 +u such that +ΨT := +r0 +T + 1 +1 +η + bη + eη2 ≤ 2( br0 +T + 1) +1 +2 + 2e +1 +3 ( +r0 +T + 1) +2 +3 + +ur0 +T + 1 +Additional definitions +Before proceeding with the proof of the convergence theorem, we need some addition set of definitions of the various +errors we track. For simplicity, we define the special matrix J = 1n1T +n +n +as it could be used to calculate the averaged +matrix, XJ = ¯X = [¯x +¯x +... +¯x] . +We define the client variance (or consensus distance) to be how much each node deviates from their averaged model: +Ξt = 1 +n +�n +i E||x(t) +i +− ¯x(t)||2. +Since we are doing local steps between communication, we define the local progress to be how much each node moves +from the globally averaged starting point as client-drift: +• at k-th local step: ek,t := 1 +n +�n +i E||x(t)+k +i +− ¯x(t)||2 +• accumulation of local steps: Et := �K−1 +k=0 ek,t = �K−1 +k=0 +1 +n +�n +i E||x(t)+k +i +− ¯x(t)||2 +Because we update model with correction, the corrected gradient will be aligned with the direction of the global update +instead of the local update. The correction is updated every communication, and remains constant during local steps. +We define the quality of this correction to be how much it approximates the true deviation between global update and +local update, γt = +1 +nL2 E||C(t) + ∇f( ¯X(t)) − ∇f( ¯X(t))J||2 +F , where J = 1 +n11T . +C.2 +Convergence analysis +This section we will show the proof of Theorem 3.2 and Theorem 4.1. Since from the previous analysis that K-GT and +periodical GT are equivalent to corrected SGD for local step, and have similar pattern during communication. We can +analyze them within the same prove framework. +In order to prove the theorems, we first provide the recursion for client-drift, consensus distance and qualify of correction +in following sections. +3 + +Bounding the client drift +We will next consider the progress made within local steps. That’s the accumulated model update before next +communication. +Lemma C.6. Suppose the local step-size for node ηc ≤ +1 +8KL, and for arbitrary communication step size ηs ≥ 0, we +could bound the drift as +Et ≤ 3(KΞt) + 12K2η2 +cL2(Kγt) + 6K2ηc +2(KE||∇f(¯x(t))||2) + 3K2ηc +2σ2 +Proof. First, observe that K = 1, Et = 1 +nE||X(t) − ¯X(t)||2 +F , +0 ≤ 2 +nE||X(t) − ¯X(t)||2 +F + 6ηc +2E||∇f(¯x(t))||2 + 3ηc +2σ2, +the inequality will always hold since RHS is always positive. Then the lemma is trivially proven for K = 1. +Then we consider the case for K ≥ 2, and +nek,t := E||X(t)+k − ¯X(t)||2 +F += E||X(t)+k−1 − ηc +� +∇F(X(t)+k−1; ξ(t)+k) + C(t)� +− ¯X(t)||2 +F +≤ (1 + +1 +K − 1)E||X(t)+k−1 − ¯X(t)||2 +F + nηc +2σ2 ++ Kηc +2E||∇f(X(t)+k−1) − ∇f( ¯X(t)) + C(t) + ∇f( ¯X(t))(I − J) + ∇f( ¯X(t))J||2 +F +≤ (1 + +1 +K − 1 + 4Kηc +2L2) +� +�� +� +:=C +E||X(t)+k−1 − ¯X(t)||2 +F + 4Kηc +2L2nγt + 2Kηc +2nE||∇f(¯x(t))||2 + nηc +2σ2 +≤ CkE||X(t) − ¯X(t)||2 +F + +k−1 +� +r=0 +Cr� +4Kηc +2L2nγt + 2Kηc +2nE||∇f(¯x(t))||2 + nηc +2σ2� +If ηc ≤ +1 +8KL, then 4K(ηcL)2 ≤ +1 +16K < +1 +16(K−1). Since C > 1, then Ck ≤ CK ≤ (1+ +1 +K−1+ +1 +16(K−1))K ≤ e1+ 1 +16 ≤ 3, +and �k−1 +r +Cr ≤ KCK ≤ 3K. We could rewrite the bound on client drift at kth local step, +nek,t ≤ 3Ξt + 3K +� +4Kηc +2L2nγt + 2Kηc +2nE||∇f(¯x(t))||2 + nηc +2σ2� +(9) +Clearly, in inequality (9), the RHS is independent of time step k ∈ [0, K). Then the accumulated progress within local +steps Et could be formulated by +Et := +K−1 +� +k=0 +ek,t ≤ 3(KΞt) + 12K2ηc +2L2(Kγt) + 6K2ηc +2(KE||∇f(¯x(t))||2) + 3K2ηc +2σ2 +Consensus distance +We then consider how the consensus distance for communicated model is developed between communications after +local training. +Lemma C.7. For any effective step-size η = ηsηc, we have the descent lemma for Ξt as +Ξt+1 ≤ (1 − p +2)Ξt + 6Kη2L2 +p +Et + 6K2η2L2 +p +γt + Kη2σ2. +Proof. We know that the update between two communication round is as follows, +X(t+1) = +� +X(t) − KηZ(t)� +W, +4 + +where Z(t) = 1 +K +� +k +� +∇F(X(t)+k; ξ(t)+k) + C(t)� +. Then consensus distance at time (t + 1) can be measured by +nΞt+1 = E||X(t+1) − ¯X(t+1)||2 +F += E|| +� +X(t) − η +K−1 +� +k=0 +(∇F(X(t)+k; ξ(t)+k) + C(t)) +� +(W − J)||2 +F +≤ (1 − p)E|| +� +X(t) − η +K−1 +� +k=0 +(∇f(X(t)+k) + C(t)) +� +(I − J)||2 +F + nKη2σ2 +≤ nKη2σ2 + (1 + α)(1 − p)E||X(t)(I − J)||2 +F ++ (1 + 1 +α)η2E|| +K−1 +� +k=0 +∇f(X(t)+k)(I − J) ± K∇f( ¯X(t))(I − J) + KC(t)||2 +F +≤ +α= p +2 , 1 +p ≤1 +nKη2σ2 + (1 − p +2)E||X(t) − ¯X(t)||2 +F + 6 +p +� +Kη2L2||I − J||2 +K−1 +� +k=0 +E||X(t)+k − ¯X(t)||2 +F ++ K2η2 L2 +L2 E||f( ¯X(t))(I − J) + C(t)||2 +F +� +≤ (1 − p +2)nΞt + 6Kη2L2 +p +nEt + 6K2η2L2 +p +nγt + nKη2σ2 +Quality measure of correction +We now bound the quality measure of correction. The correction is thought to depict the deviation of local and global +gradient of the ideally averaged model ¯X at the time of communication. That is, quality measure of correction is defined +to be γt = +1 +nL2 E||C(t) + ∇f( ¯X(t)) − ∇f( ¯X(t))J||2 +F , where J = 1 +n11T . +How to estimate correction, K-GT and periodical GT have different options, which is carefully discussed in section 4.1. +Lemma C.8. For any effective step-size η = ηsηc ≤ +√p +√ +6KL, we have the descent lemma for γ in periodical GT as +follow, +Kγt+1 ≤ (1 − p +2)Kγt + 24 +p (KeK−1,t) + 2 +pEt + 12K2η +p +KηE||∇f(¯x(t))||2 + 2Kσ2 +L2 +, +and if we instead of using the average of local steps for K-GT in correction, we have the descent lemma for γ as follow, +Kγt+1 ≤ (1 − p +2)Kγt + 30 +p Et + 12K2η2 +p +KE||∇f(¯x(t))||2 + 2σ2 +L2 . +Proof. The averaged correction between two consecutive communication round satisfies +C(t+1)J = C(t)J + +1 +Kηc +(X(t) − X(t)+K)(W − I)J = C(t)J +for K-GT. We assume that the correction is initialized with arbitrary value as long as its globally average always equals +to zero, i.e., C(t)J = C(0)J = 0. Recall the definition of γt, note that +(C(t) + ∇f( ¯X(t)) − ∇f( ¯X(t)J)J = C(t)J + ∇f( ¯X(t))(J − J) = 0. +It’s easy to check that Periodical GT has the same property. +5 + +Then for K-GT, we have the recursion of quality measure can be formulated as follows, +nL2γt+1 := E||C(t+1) + ∇f( ¯X(t+1))(I − J)||2 +F += E||C(t)W + 1 +K +K−1 +� +k=0 +∇F(X(t)+k; ξ(t)+k)(W − I) + ∇f( ¯X(t+1))(I − J)||2 +F += E|| +� +C(t) + ∇f( ¯X(t))(I − J) +� +W ++ +� 1 +K +K−1 +� +k=0 +∇f(X(t)+k) − ∇f( ¯X(t)) +� +(W − I) ++ +� +∇f( ¯X(t+1)) − ∇f( ¯X(t)) +� +(I − J)||2 +F + nσ2 +K +≤ (1 + α)(1 − p)nL2γt + 2(1 + 1 +α) +� +||W − I||2 1 +K +K−1 +� +k=0 +E||∇f(X(t)+k) − ∇f( ¯X(t))||2 +F ++ ||I − J||2E||∇f( ¯X(t+1)) − ∇f( ¯X(t))||2 +F +� ++ nσ2 +K +(due to ||W − I|| ≤ 2, ||I − J|| ≤ 1) +≤ +α= p +2 , 1 +p ≤1 +(1 − p +2)nL2γt + 6 +p +� 4 +K +K−1 +� +k=0 +L2E||X(t)+k − ¯X(t)||2 +F + nL2E||¯x(t+1) − ¯x(t)||2� ++ nσ2 +K +≤ (1 − p +2)nγt + 6L2 +pK +� +4nEt + 2K2η2L2nEt + 2K2η2n(KE||∇f(¯x(t))||2) + K2η2σ2� ++ nσ2 +K +Periodical GT uses ∇F(X(t)+K−1; ξ(t)+K−1) to replace 1 +K +�K−1 +k=0 ∇F(X(t)+k; ξ(t)+k) in correction update. With +almost identical analysis, we could get a very similar equality. In addition, if we replace stochastic gradient with full +gradient, i.e, ∇f(X(t)+K−1), in periodical GT, which will improve the noise term. +Further, if η = ηsηc ≤ +√p +√ +6KL, then 6K2η2L2 +p +≤ 1 which completes the proof. +Remark 3. Shown from results above, the quantizations of γt for periodical GT and K-GT only differ in the coefficient +of stochastic noise. And using full-batch gradient can improve Periodical GT in stochastic noise to the same level as +that of K-GT. +Progress between communications +We study how the progress between communication rounds could be bounded. +Lemma C.9. We could bound the averaged progress between communication in any round t ≥ 0, and any η = ηsηc ≥ 0 +as follows, +E||¯x(t+1) − ¯x(t)||2 ≤ 2Kη2L2Et + 2K2η2E||∇f(¯x(t))||2 + (Kη)2σ2 +nK +. +Proof. From previous analysis, we guarantee 1 +n +� +i c(t) +i += 0. Then the averaged progress between communication +could be rewritten as +E||¯x(t+1) − ¯x(t)||2 = η2E|| 1 +n +� +i,k +∇Fi(x(t)+k +i +; ξ(t)+k) + K +n +� +i +c(t) +i ||2 +≤ Kη2 +n +� +i,k +2E||∇fi(x(t)+k +i +) − ∇fi(¯x(t))||2 + 2K2η2E||∇f(¯x(t))||2 + Kη2σ2 +n +≤ 2Kη2L2 +n +� +i,k +E||x(t)+k +i +− ¯x(t)||2 + 2K2η2E||∇f(¯x(t))||2 + Kη2σ2 +n +In the first inequality, note that the K random variable {ξ(t)+k}K−1 +k=0 when conditioned on communication (t) may not +be independent of each other but each has variance smaller than σ2 due to Assumption 2, and we can apply Lemma C.4. +Then the following inequalities are from the repeated application of triangle inequality. +6 + +Descent lemma for non-convex case +Lemma C.10. When function f is L-smooth, the averages ¯x(t) of the iterates of Algorithm 1 and Algorithm 2 with the +constant stepsize ηc < +1 +4ηsKL, satisfy +Ef(¯x(t+1)) − Ef(¯x(t)) ≤ −Kη +4 E||∇f(¯x(t))||2 + ηL2Et + Kη2L +2n +σ2 +Proof. Because the local functions {fi(x)} are L-smooth according to Assumption 1, it’s trivial to conclude that the +global function f(x) is also L-smooth. +Ef(¯x(t+1)) = Ef +� +¯x(t) − η +n +� +i,k +(∇Fi(x(t)+k +i +; ξ(t)+k +i +) + c(t) +i ) +� +≤ Ef(¯x(t)) + E +� +∇f(¯x(t+1)), − η +n +� +i,k +(∇Fi(x(t)+k +i +; ξ(t)+k +i +) + c(t) +i ) +� +� +�� +� +:=U ++L +2 E||¯x(t+1) − ¯x(t)||2 +From our previous analysis, we know 1 +n +� +i c(t) +i += 0, ∀t ≥ 0 forK-GT and Periodical GT (if with initialization +indicated in purposed algorithm) +U : = E +� +∇f(¯x(t+1)), − η +n +� +i,k +(∇Fi(x(t)+k +i +; ξ(t)+k +i +) + c(t) +i ) +� += E +� +∇f(¯x(t)), − η +n +� +i,k +Eξ(t)+k +i +∇Fi(x(t)+k +i +; ξ(t)+k +i +) +� += −KηE +� +∇f(¯x(t)), 1 +nK +� +i,k +∇fi(x(t)+k +i +) − ∇f(¯x(t)) + f(¯x(t)) +� += −KηE||∇f(¯x(t))||2 + +1 +nK +� +i,k +KηE +� +∇f(¯x(t)), +� +∇fi(x(t)+k +i +) − ∇fi(¯x(t)) +�� +≤ −Kη +2 E||∇f(¯x(t))||2 + Kη +2nK +� +i,k +E||∇fi(x(t)+k +i +) − ∇fi(¯x(t))||2 +≤ −Kη +2 E||∇f(¯x(t))||2 + KL2η +2nK +� +i,k +E||x(t)+k +i +− ¯x(t)||2 +Then also plug in the Lemma C.9 for E||¯x(t+1) − ¯x(t)||2, we have +Ef(¯x(t+1)) ≤ Ef(¯x(t)) + (−Kη +2 ++ K2η2L)E||∇f(¯x(t))||2 +F + (ηL2 +2 ++ Kη2L3)Et + Kη2L +2n +σ2 +Then the choice η ≤ +1 +4KL completes the proof. +Main recursion +We first construct a potential function Ht = Ef(¯x(t)) − Ef(x(⋆)) + A (Kηc)3L4 +pη2s +γt + +B +6v2 +KηcL2 +p +Ξt where constants A, +B and v can be obtained through the following lemma. +Lemma C.11 (Recursion for K-GT). For any effective stepsize of Algorithm 1 satisfying ηs = ˜O( p +KL) and ηc = ˜O(p), +there exists constants A, B, v satisfying D > 0 and D5 ≥ 0. Then we have the recursion +Ht+1 − Ht ≤ −DKηE||∇f(¯x(t))||2 + D5L2 +pK (Kη)3σ2 + +L +2nK (Kη)2σ2. +Proof. First, from previous bound on those error term γt, Ξt and Et, we could bound the difference between Ht+1 and +Ht for K-GT, while we also plug in with C > 0 the +0 ≤ −CηcL2Et + 3C(KηcL2Ξt) + 12C(Kηc)3L4γt + C 6(Kηc)2L2 +ηs +KηE||∇f(¯x(t))||2 + 3C(Kηc)3 L2σ2 +K +. +7 + +Then we have the inequality recursion for K-GT as follows, +Ht+1 − Ht ≤ +� +− A +2 + B η2 +s +v2p2 + 12C +� +� +�� +� +≤D1 +(Kηc)3L4γt ++ +� +− +B +12v2 + 3C +� +� +�� +� +≤D2 +KηcL2Ξt ++ +� +A30(KηcL)2 +p2K ++ B K2η2L2 +v2p2 +− C + ηs +� +� +�� +� +≤D3 +ηcL2Et ++ +� +− 1 +4 + A12ηs(Kηc)4L4 +p ++ C 6K2ηc2L2 +ηs +� +� +�� +� +≤D4 +KηE||∇f(¯x(t))||2 ++ +� +A2L2 +pK + B η2 +sL2 +6v2pK + C 3L2 +K +� +� +�� +� +≤ D5L2 +pK +(Kηc)3σ2 + +L +2nK (Kη)2σ2 +As long as ηc ≤ +p +96vKL, ηs = v · p → ηcηs ≤ η = +p2 +96KL and A = 72v3p + 48vp, B = 36v3p, C = vp there exists +constant v > 1 that makes D1, D2, D3 ≤ 0, D4 < 0. And D4 ≤ −D < 0, and D5 ≥ 0, which completes the +proof. +Lemma C.12 (Recursion for Periodical GT). For any effective stepsize of Algorithm 1 satisfying ηs = ˜O( p +KL) and +ηc = ˜O(p), there exists constants A, B, v satisfying D > 0 and D5 ≥ 0. Then we have the recursion +Ht+1 − Ht ≤ −DKηE||∇f(¯x(t))||2 + D5L2 +p +(Kη)3σ2 + +L +2nK (Kη)2σ2. +Proof. The sets of inequality for Periodical GT only differs in stochastic noise compared to K-GT. Then applied with +the same principle as that for K-GT, we get its recursion of potential function as follows +Ht+1 − Ht ≤ D1(Kηc)3L4γt + D2KηcL2Ξt + D3ηcL2Et + D4KηE||∇f(¯x(t))||2 ++ +� +A2L2 +p ++ B η2 +sL2 +6v2pK + C 3L2 +K +� +� +�� +� +≤ D5L2 +p +(Kηc)3σ2 + +L +2nK (Kη)2σ2 − C +2 KeK−1,t +The rest of the analysis could refer to Lemma C.11. +Remark 4. Using full gradient to improve Periodical GT has the same recursion as K-GT. +Solve the main recursion +Take K-GT as an example. Consider the telescope sum of the potential function, we can derive +1 +T + 1 +T +� +t=0 +� +Ht+1 − Ht +� += +1 +T + 1 +� +HT +1 − H0 +� +≤ +η=ηsηc +−DKη +1 +T + 1 +T +� +t=0 +E||∇f(¯x(t))||2 + D5L2 +pKη3s +(Kη)3σ2 + +L +2nK (Kη)σ2 +⇒ +ηs=v·p +1 +T + 1 +T +� +t=0 +E||∇f(¯x(t))||2 ≤ H0 − HT +1 +(T + 1)D +1 +Kη + +Lσ2 +2nKD(Kη) + D5L2σ2 +v3p4KD(Kη)2 +W.l.o.g we consider that f(x) is non-negative. Then we could neglect the effect of −HT +1. +8 + +Lemma C.13. There exists constant stepsize such that +1 +T + 1 +T +� +t=0 +E||∇f(¯x(t))||2 = O +�� +σ2LH0 +nKT ++ (σLH0 +p2KT ) +2 +3 + LH0 +p2T +� +. +Proof. The non-negative sequences {Ht}T +1 +t=0 and {E||∇f(¯xt)||}T +t=0 with positive coefficients before both Kη and +(Kη)2 satisfy the condition in Lemma C.5. Then we tune the stepsize using Lemma C.5. Then the average of +accumulation of gradient could be upper-bounded by +⇒ +1 +T + 1 +T +� +t=0 +E||∇f(¯x(t))||2 ≤ O +� +2( +Lσ2 +2nK H0 +T + 1 ) +1 +2 + 2(L2σ2 +p4K ) +1 +3 (H0(x) +T + 1 ) +2 +3 + +H0(x) +Kηmax(T + 1) +� += O +�� +σ2LH0 +nKT ++ (σLH0 +p2KT ) +2 +3 + LH0 +p2T +� +Then the convergence rate depends on the initial values of potential function H0. By the definition of potential function in +Lemma C.11, H0 is the combination of initial value for f(x0), E||X(0)− ¯X(0)||2 +F and E||C(0)+∇f( ¯X(0))(−J+I)||2 +F . +We assume that every node is guaranteed to be initialized with the same model x(0) = x(0) +i , ∀i ∈ [n]. Then we could +easily get E||X(0) − ¯X(0)||2 +F = 0. And if we initial the correction term with c(0) +i += −∇fi(x(0)) + 1 +n +� +i ∇fi(x(0)), +then E||C(0) + ∇f(X(0))(−J + I)||2 +F = 0. +H0 = f(x0) − f(x⋆) + A(Kηc)3L4 +pη2s +γ0 + B +6v2 +KηcL2 +p +Ξ0 += f(x0) − f(x⋆) := F0 +(10) +And then for arbitrary accuracy error ϵ > 0, the communication rounds needed to reach the target accuracy is +upperbounded by +T ≤ O +� σ2 +nK +1 +ϵ2 + +σ +p2√ +K +1 +ϵ +3 +2 + 1 +p2 +1 +ϵ +� +· LF0, +which concludes the proof of Theorem 3.2 for K-GT. The proof of Theorem 4.1 for Periodical GT can also be easily +derived with the same principle. +9 + +D +Experimental details +D.1 +Visualization of benchmark datasets +(a) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +class label +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +node id +random +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +class label +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +sorted +(b) +Figure 4: Data visualization. (a) Example from MNIST dataset. (b) Data partition on each node in the random and the sorted case +when there are n = 10 distributed nodes. The dot size indicates the number of samples per class allocated to each node. +We show an image example from MNIST datasets and how data of different labels is partitioned in random and sorted +case. It obviously presents in the random case, data of different labels are randomly and evenly partitioned among +nodes, but in the sorted case, each node only contains images of 1 label and the labels obtained by each node is +non-overlapping. +D.2 +Model structure +For our non-convex experiment, we use a 4-layer Convolutional Neural Network (CNN) and its details are listed in +Table 3. +Table 3: Model architecture of the benchmark experiment. For convolutional layer (Conv2D), we list parameters with sequence of +input and output dimension, kernal size, stride. For max pooling layer (MaxPool2D), we list kernal and stride. For fully connected +layer (FC), we list input and output dimension. For drop out (Dropout), we list the parameter of probability. +layer +details +1 +Conv2D(1, 10, 5, 1), MaxPool2D(2), ReLU +2 +Conv2D(10, 10, 5, 1), Dropout2D(0.5), MaxPool2D(2), ReLU +3 +FC(320, 50), ReLU +4 +FC(50, 10) +10 + diff --git a/RNAzT4oBgHgl3EQfXPwH/content/tmp_files/load_file.txt b/RNAzT4oBgHgl3EQfXPwH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..66acb2e070667fb7d2225094567f966be2b7a33e --- /dev/null +++ b/RNAzT4oBgHgl3EQfXPwH/content/tmp_files/load_file.txt @@ -0,0 +1,977 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf,len=976 +page_content='DECENTRALIZED GRADIENT TRACKING WITH LOCAL STEPS Yue Liu University of Toronto Canada Tao Lin Westlake University China Anastasia Koloskova EPFL Switzerland Sebastian U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Stich CISPA Helmholtz Center for Information Security Germany ABSTRACT Gradient tracking (GT) is an algorithm designed for solving decentralized optimization problems over a network (such as training a machine learning model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' A key feature of GT is a tracking mechanism that allows to overcome data heterogeneity between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We develop a novel decentralized tracking mechanism, K-GT, that enables communication-efficient local updates in GT while inheriting the data-independence property of GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We prove a convergence rate for K-GT on smooth non-convex functions and prove that it reduces the communication overhead asymptotically by a linear factor K, where K denotes the number of local steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We illustrate the robustness and effectiveness of this heterogeneity correction on convex and non-convex benchmark problems and on a non-convex neural network training task with the MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 1 Introduction We consider distributed optimization problems, where the objective function f(x) on model x ∈ Rd is defined as the average of n different components {f1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', fn(x)}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', f(x) = 1 n n � i=1 fi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In distributed applications, different contributors (or ‘clients’) take part in the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Such clients can be, for example, mobile edge devices, or computing nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Typically, each component fi(x) is only available to a single client (for instance, when fi(x) is defined over the training data available only locally on the client).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' This makes distributed optimization problems more difficult to solve than centralized problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Besides convergence rate in terms of iterations, communication efficiency is one of the most important metrics in distributed algorithm design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For illustration we consider the calculation of a gradient of the global function, ∇f(x) = 1 n � i ∇fi(x), that forms be basis for general first-order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Since each client only has the ability to evaluate the local gradient ∇fi(x), it is further necessary to calculate the average of these local gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Centralized algorithms [14, 23] realize such global aggregation by a central controller, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', with a parameter sever [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' However, this approach requires all clients to communicate with the central server simultaneously, resulting in a communication bottleneck at this hot point and a slowdown in clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Instead of the exact averaging, decentralized algorithms [10, 18, 31] require only partial communication through gossip averaging and reduce communication overhead by allowing a node to communicate with fewer nodes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', only its neighbors, thus avoiding having the busiest point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' How nodes are connected between each other makes up the network topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' One of the most challenging aspects in decentralized optimization is data-heterogeneity, that is when the training data is not identically and independently (non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=') distributed across the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Such non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' distributions often arise in practical applications, since, for example, training data originating from cell phones, sensors, or hospitals can have regional differences [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In this case, the local empirical losses on each client are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' This can slow down the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='01313v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='OC] 3 Jan 2023 convergence [12, 25] or even yield local overfitting (often termed client-drift) as the clients may drift away from the global optimum in the course of the optimization process [6, 9, 12, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' There are several decentralized algorithms that have been shown to mitigate heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' However, most of them are proven to converge for only strongly convex functions [1, 15, 24, 28, 32], or are proven for smooth non-convex functions but have a strict constraint on network topologies [such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Stochastic gradient tracking (GT) [11, 22, 24, 27, 35] algorithms have been proposed to address data-heterogeneity for arbitrary networks for smooth non-convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Its convergence rate only depends on the data heterogeneity at the initial point, which can be completely removed with proper initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' However, the clients are required to communicate with all their neighbors in the network after every single model update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' These methods are still therefore associated with high communication overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In order to further reduce communication overhead within distributed training, various engineering techniques have been proposed, such as using large batch [4, 20, 33], model/gradient compression [13] or asynchronized communication [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In this work, we focus on local updates to reduce communication frequency, which is often efficient in practice but remains challenging in the theoretical analysis [5, 14, 23, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' However, performing a large number of local steps can exacerbate the client-drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The resulting optimization difficulties can negate the communication savings [9, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The analysis of incorporating local steps while heterogeneity independence in the decentralized optimization is still seldom investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Integrating local updates into GT is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For instance, simply skipping communication rounds in GT (and thereby performing a number of local updates in-between) does not work well in practice1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' A concurrent work LU-GT[26] analyzed the performance of GT periodically skipping the communication but only in the deterministic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2 As a solution, we carefully design a novel tracking mechanism that enables to combine GT with local steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='3 The resulting algorithm—K-GT, where K denotes the number of local steps—is a novel decentralized method that provides communication-efficient tracking with local updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We prove that the convergence of K-GT depends only on the data heterogeneity at the starting point and that this weak data dependence can be completely circumvented with an additional round of global communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' As long as K-GT uses the same initialization as GT, K-GT inherits the heterogeneity independence property of GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We prove that K-GT (Algorithm 1) achieves asymptotically linear speed-up in terms of communication round w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' local steps K and number of clients n, and that it converges in O � σ2 nKϵ2 � rounds to an ϵ-approximate stationary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The number of communication rounds is asymptotically reduced by a factor of K compared to GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We further show that the convergence rate (including higher order terms) does not depend on the data-heterogenity if with proper initialization, opposed as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' for decentralized stochastic gradient descent (D-SGD) without tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The outline of this paper is as follows: In Section 2, we give the precise formulation of the distributed optimization problem setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In Section 3, we introduce the algorithm design of K-GT and demonstrate how it helps to correct for heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Here, our main result state its convergence rate, see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In Section 4, we generalize the gradient tracking framework and discuss about the drawbacks of other GT alternatives that could also be stemmed from the same framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We in addition contribute their convergence results and give a comparison to show that K-GT is the most communication efficiency theorectically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In Section 5, we compare the GT-variants with baseline D-SGD with numerical examples in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We summarize our main results below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We develop a novel gradient tracking algorithm for distributed optimization and analyze its convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We prove that K-GT enjoys heterogeneity-independent complexity estimates (with proper initialization) and prove that it converges asymptotically in O � σ2 nKϵ2 � rounds, where n denotes client number, K the number of local steps, σ2 the stochastic noise level and ϵ the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' This improves by a factor of K over the GT baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We provide additional theoretical insights, by studying (i) the convergence of the naïve local extension of GT, periodic GT, explaining that it performs worse than K-GT when the stochastic noise is large, and (ii) a computationally inefficient variant, large-batch GT that matches the iteration, but not the computation complexity of K-GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We empirically verify the theoretical results on strongly convex and non-convex functions and explain the impact of noise, local steps and data-heterogeneity on the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' K-GT is robust against the data-heterogeneity while improving the communication efficiency and improves generalization performance over baseline algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 1We evaluate this variant (termed periodical GT) below in Section 5, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2This concurrent work was independently developed while we were finalizing this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We will add a more detailed comparison to the next version of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 3Partial results of this paper were previously presented in YL’s master thesis [21] 2 Table 1: A comparison under different working conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ∆ ≤ n denotes the maximum degree of the communication graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' K-GT is the first fully-decentralized tracking algorithm with local steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Algorithm Settings Communication cost at the busiest point Local steps heterogeneity-robustnessa SCAFFOLD [9] O(n) \x13 \x13 GOSSIP-PGA [2] O(∆) \x13 \x17 D-SGD [12] \x13 \x17 GT [24] \x17 \x13 D2 [30] \x17 \x13 K-GT [ours] \x13 \x13 aThe data heterogeneity does not impact the worst-case convergence rate (but might require special initialization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2 Problem setting We introduce the notation and setup in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1 Decentralized Optimization Problem We consider the optimization problems as the summation from n-client loss functions, min x∈Rdf(x) := 1 n n � i=1 [fi(x) := Eξi∼DiFi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξi)] , (1) where n denotes the number of clients within the system, ξi is a random sample from Di and Di denotes the local distribution only available on node i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Di could be arbitrary and different among clients considering the applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' This setup models both empirical risk minimization and the online optimization setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In this work, we consider general smooth non-convex functions and bounded stochastic noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Assumption 1 (Smoothness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Each function fi(x) : Rd → R, ∀i ∈ [n] is differentiable and there exists a constant L > 0 such that for each x, y ∈ Rd, fi(y) ≤ fi(x) + ∇fi(x)T (y − x) + L 2 ||x − y||2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Assumption 2 (Bounded variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Each client variance is uniformly bounded, ∀i ∈ [n], ∀x ∈ Rd, Eξ∼Di||∇Fi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ) − ∇fi(x)||2 2 ≤ σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2 Communication graph The training is implemented over a decentralized network, and its topology is modelled as an undirected graph: (V, E), where V := {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' , n} is the node set and E ⊆ V × V is the edge set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Node (or client) represents a computing node, and clients communicate only along the edges e ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We denote the adjacency matrix W ∈ Rn×n, where wij = 0 means node i and j are not connected, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', eij = (i, j) /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Assumption 3 (Mixing rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Given the symmetric and doubly stochastic mixing matrix W ∈ Rn×n of nonnegative real numbers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', ∀i, j ∈ [n], wij ≥ 0, �n i=1 wij = �n j=1 wij = 1, the consensus distance decreases linearly after averaging step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' there exists a 1 ≥ p > 0 such that ||XW − ¯X||2 F ≤ (1 − p)||X − ¯X||2 F , ∀X ∈ Rd×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Note that if the commonly used network parameter ρ := ||W − 1n1T n n || [2] is strictly less than 1, then 1 ≥ p > 0 [see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The mixing rate describes the connectivity of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The larger value of p means the communication graph is better connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' p = 1 for a complete graph W = 1 n11T , and p = 0 for a disconnected graph W = In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='3 Data heterogeneity and correction When the local distributions {Di} are identical on each client, the local functions {fi(x)} are identical to each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', fi(x) ≡ f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Otherwise, heterogeneous local distributions result in heterogeneous local functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' And heterogeneity is usually measured by the discrepancy between local gradients {∇fi(x)} and global gradient ∇f(x) [9, 12] as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 3 Assumption 4 (Data-heterogeneity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' There exists constants ζ2 > 0 and B ≥ 1 such that ∀x ∈ Rd, 1 n n � i=1 ||∇fi(x)||2 ≤ ¯ζ2 + B2||∇f(x)||2 , where both ¯ζ2 and B2 represent the degree of heterogeneity within the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The baseline Decentralized SGD (D-SGD) uses naïve gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='t local model, the convergence of which inevitably are influenced by both ¯ζ2 and B [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1 Notations Gradient tracking algorithm mainly manipulates between two variables, model iterate x ∈ Rd and tracking variable z ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' More precisely, we denote vector y ∈ {x, z} as y(t)+k i on node i in local step k at communication round t, and denote its average by ¯y = 1 n � i yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The collection of vectors yi for all i ∈ [n] in matrix form is denoted by a capital letter with columns yi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', Y = [y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' , yn] ∈ Rd×n, ¯Y = [¯y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' , ¯y] = 1 nY1n1T n ∈ Rd×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Also, we extend this matrix definition to both gradient and stochastic gradient of (1) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='t model X on sample ξ = [ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' , ξn], where ξi ∼ Di, ∇F(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ) = [∇F1(x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' , ∇Fn(xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξn)] ∈ Rd×n, ∇f(X) = E(ξ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=',ξn)∇F(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ) = [∇f1(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' , ∇fn(xn)] ∈ Rd×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2 Gradient tracking Gradient tracking algorithm (GT) [27] is defined by the following update equations: X(t+1) = (X(t) − ηZ(t))W Z(t+1) = Z(t)W + G(t+1) − G(t), (2) in matrix format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Here G(t) = ∇F(X(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)) and η > 0 denotes the stepsize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' When data is heterogeneous among different nodes, {∇Fi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξi), ∀i} are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' But GT uses bias-correction to compensate heterogeneous gradient at each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' This correction is governed by the tracking variable Z that replaces the naïve gradient: Z(t+1) = ∇F(X(t+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t+1)) + Z(t)W − G(t) � �� � correction (3) Since the update (2) simultaneously updates both the model X and the tracking variable Z, there is no need to take extra consideration on the heterogeneous local gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' GT is proven to converge regardless of data heterogeneity [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 3 K-GT: Gradient Sum Tracking algorithm In this section, we present our new decentralized stochastic algorithm K-GT with its convergence analysis for general non-convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1 Algorithm In the K-GT algorithm we allow each client to perform K ≥ 1 local steps between each communication round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' To compensate to the data-heterogeneity, we use a similar correction as in (3) on top of the stochastic gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We denote the correction as ci on node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then each node repeats the following updating rule, i ∈ [n]: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Compute a local stochastic gradient ∇Fi(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξi) by sampling ξi from distribution Di;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Update the local model x(t)+k+1 i = x(t)+k i − ηc � ∇Fi(x(t)+k i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k i ) + c(t) i � using the stochastic gradients at (t) + k-th iteration and correction c(t) i in t-th communication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Repeat step (1)-(2) K times, then obtain the tracking throughout local steps, z(t) i = 1 Kηc � x(t) i − x(t)+K i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Exchange {xi, ci} with neighbors: (in matrix format): X(t+1) = � X(t) − ηs(X(t) − X(t)+K) � W , C(t+1) = C(t) + Z(t)(W − I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' (4) The complete algorithm is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1 (Gradient Sum Tracking).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Define Z(t) = 1 Kηc � X(t) − X(t)+K� as the tracking variable during communication round t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The update rule for both models X(t) and tracking variables Z(t) at communication in K-GT can be rewritten as (η = ηsηc): X(t+1) = � X(t) − KηZ(t)� W , Z(t+1) = Z(t+1)W + G(t+1) − G(t) , (5) where G(t) = 1 K � k ∇F(X(t)+k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k) denotes the mean update over the local steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The detailed proof is included in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' If K = 1 in (5), K-GT is equivalent to Gradient Tracking [27] with η = ηsηc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' K-GT essentially runs SGD if communication is the most sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' To understand the intuition behind K-GT, let us consider the global average ¯X at each iterate, which gets updated just like the standard stochastic gradient descent: ¯X(t)+k+1 = � X(t)+k − ηc(∇F(X(t)+k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k) + C(t)) � 11T n = ¯X(t) − ηc � ∇F(X(t)+k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k) + ¯C(t)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' If initialized to be C(0) = ∇F(X(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(0)) � 11T n − I � , the average of correction satisfies ¯C(t+1) = ¯C(t) + Z(t)(W − I) 11T n = ¯C(t) , ¯C(0) = ∇F(X(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(0)) � 11T n − I � 11T n ≡ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then the average of model iterate satisfies ¯X(t)+k+1 = ¯X(t) − ηc∇F(X(t)+k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k), which updates model with averaged stochastic gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' How does this correction improves D-SGD?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We consider applying the similar analysis from [30] to illustrate the effectiveness of K-GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Assume that X(t) has achieved an optimum X⋆ := x⋆1T with all local models equal to the optimum x⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Based on our analysis in appendix (Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='8), the correction will be equal to c⋆ i := −∇Fi(x⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ) + 1 n � j ∇Fj(x⋆ j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then the next local update for K-GT would be X(t)+1 = X⋆ − ηc(∇F(X⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ) + C⋆) = X⋆ − ηc∇F(X⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ) 11T n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' This illustration shows that for K-GT, the convergence when we approach a solution with only local update relies on the magnitude of E||∇F(X⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ) 11T n ||2 F , which is bounded by O(σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' However, consider the same situation for D-SGD, X(t)+1 = X⋆ − η∇F(X⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' On different nodes, ∇Fi(X⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ) deviates from each other due to data heterogeneity, and the deviation can only be characterized by ζ2 as suggested in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then the upper bound for D-SGD of the same magnitude of convergence when in the neighborhood of solution is O(σ2 + ¯ζ2) [30], which is obviously worse than that for K-GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The additional O(¯ζ2) in D-SGD from the data heterogeneity can never be improved if always using the sole stochastic gradient [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 5 Algorithm 1 K-GT: Gradient Sum Tracking 1: parameters: T: number of communication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' K: number of local steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ηc, ηs: local, communication stepsize;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' W: given topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2: Initialize: ∀i, j ∈ [n], x(0) i = x(0) j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' c(0) i = −∇Fi(x(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξi) + 1 n � j ∇Fj(x(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξj)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 3: for client i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' , n} parallel do 4: for communication: t ← 0 to T − 1 do 5: for local step: k ← 0 to K − 1 do 6: x(t)+k+1 i = x(t)+k i − ηc(∇Fi(x(t)+k i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k i ) + c(t) i ) 7: end for 8: z(t) i = 1 Kηc (x(t) i − x(t)+K i ) 9: c(t+1) i = c(t) i − z(t) i + � j wijz(t) j ▷ update tracking variable 10: x(t+1) i = � j wij(x(t) j − Kηsηcz(t) j ) ▷ update model parameters 11: end for 12: end for aThis initialization in correction c(0) i is required for heterogeneity-independent analysis in theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We demonstrate later with experiment that simply choosing c(0) i = 0 works well in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2 Main theorem: data-independent convergence on non-convex functions In this section, we present the convergence rate of K-GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Note that p is the network parameter defined in Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2 (K-GT convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For schemes as in Algorithm 1 with mixing matrices such as in Assumption 3 and arbitrary error ϵ > 0, there exists a constant stepsize ηc = O( p KL) and ηs = O(p) such that under Assumption 1 and 2 for L-smooth, (possibly non-convex) functions, it holds 1 T +1 � t E||∇f(¯x(t))||2 ≤ ϵ after O � σ2 Knϵ2 + σ p2√ Kϵ 3 2 + 1 p2ϵ � L communication rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 4 Discussion In this section, we are going to introduce and compare with other possible ways of introducing local steps to GT that has the similar communication pattern as K-GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1 Other GT alternatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1 Gradient Tracking with Periodical Communication (Periodical GT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' There is another way to incorporate local steps into above framework (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Instead of communication via fixed topology W, the communication graph changes along with time denoted by W(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Note that W(t) = I, which means there is actually no communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' If W(t) periodically alternates between {W, I}, it also reduces communication frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The full detail is concluded in Algorithm 2 (Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' K-GT suffers from less noise than Periodical GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' It is possible to reformulate local steps of Periodical GT as corrected SGD, same as that for K-GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' But Periodical GT has different update for correction at communication with C(t+1) = C(t)W + ∇F(X(t)+K−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+K−1)(W − I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' (6) The equivalence of reformulation is proven in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' However, if we simply reformulate equation (4), we obtain that K-GT uses the average of K stochastic gradient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', C(t+1) = C(t)W + 1 K � k ∇F(X(t)+k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k)(W − I) , which can reduce stochastic noise by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Periodical GT uses only one stochastic gradient, thus would suffer more from stochastic noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Using more random samples on stochastic gradient can reduce noise in Periodical GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' It is trivial to reduce the stochastic noise in (6) if using more random samples {ξ(t),s|s = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' , K − 1} to replace ∇F(X(t)+K−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+K−1) 6 Table 2: The comparison of communication rounds needed to reach target accuracy ϵ on non-convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Our results on both K-GT, Periodical GT improve the rate of D-SGD in terms of heterogeneity parameter ¯ζ2(defined in Ass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 4) when using local steps, and accelerates the rate of GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Local steps Algorithm Communication rounds K = 1 GT [11] O � σ2 nϵ2 + σ p 3 2 ϵ 3 2 + 1 p2ϵ � K > 1 D-SGD [12] O � σ2 Knϵ2 + ( ¯ζ p + σ √pK ) 1 ϵ 3 2 + 1 pϵ � K-GT [ours] O � σ2 Knϵ2 + σ p2√ Kϵ 3 2 + 1 p2ϵ � Periodical GT [ours] O � σ2 Knϵ2 + σ p2ϵ 3 2 + 1 p2ϵ � Periodical GT w/ full gradient [ours] O � σ2 Knϵ2 + σ p2√ Kϵ 3 2 + 1 p2ϵ � with ∇F(X(t)+K−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+K−1) = 1 K � s ∇F(X(t)+K−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+K−1,s) , then the correction C(t) has the same level of stochastic noise as K-GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' However, using more sample to calculate SGD requires a lot more extra computation than K-GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1 (Periodical GT convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For schemes as in Algorithm 2 (Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1) with mixing matrices such as in Assumption 3 and arbitrary error ϵ > 0, there exists a constant stepsize η = O( p2 KL) such that under Assumption 1 and 2 for L-smooth, (possibly non-convex) functions, it holds 1 T +1 � t E||∇f(¯x(t))||2 ≤ ϵ after O � σ2 Knϵ2 + σ p2ϵ 3 2 + 1 p2ϵ � L communication rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Conversely, if we consider using the full-batch tracking Algorithm 3 (Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1), then the convergence rate can be improved to O � σ2 Knϵ2 + σ p2√ Kϵ 3 2 + 1 p2ϵ � L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Note that the latter result refers to full batch tracking which comes at additional computation cost each communication round (in contrast to K-GT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2 Gradient Tracking with Large Batch (Large-batch GT) Apart from local training, large-batch training is also popular to achieve acceleration in distributed setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Similar to Large-batch SGD, we calculate G(t) = � k ∇F(X(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t),k) in (2) with K i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' random samples, {ξ(t),k|k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' , K − 1} and make W(t) = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' It is theoretically workable to improve the asymptotical communication rounds needed to reach the desired accuracy ϵ from O � σ2 nϵ2 � [11] to O � σ2 nKϵ2 � , while remains heterogeneity-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We empirically show that Large-batch GT remains heterogeneity-independent and has the same communication performance as K-GT (Figure 1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2 Convergence comparison We summarized the convergence rate for the related decentralized algorithms in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In order to analyze the convergence for other methods that depend on data heterogeneity, there is an addition assumption to measure data heterogeneity [9, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' K-GT achieves acceleration by local steps in high-noise regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' When ϵ is sufficiently small, the noise dominates the convergence rate (σ > 0) and it is not affected by graph parameter p for GT, Periodical GT and K-GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then after enough transient time, Periodical GT and K-GT with O( σ2 nKϵ2 ) achieves linear speedup by K compared to GT with rate O( σ2 nϵ2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In addition, the transient time for K-GT also decreases with O( 1 √ K ) comparing to GT baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' GT methods are in general more sensitive to the network parameter than diffusion methods [34], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', D-SGD, in the non-asymptotical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In our analysis of K-GT, the dependency on the network parameter p is worse than for vanilla 7 GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Combining our analysis with the tighter analysis of GT presented in concurrent work [11] would be an interesting future direction—however in this work we focused on the aspect of equipping GT with local steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The impact of data heterogeneity is removable for K-GT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' K-GT does not completely solve data heterogeneity in general, and depends on the data heterogeneity at the initial point, which is the same case in GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' It has been proven for GT that in the non-asymptotic regime a weaker dependence on the data heterogeneity at the initial point actually remains [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' However, with a single round of global communication for the initial iterates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 1), we can remove the heterogeneity from the complexity estimates for GT, K-GT and Periodical GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Table 2 removes the initialization terms from the rate to simplify the presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' On the contrary, heterogeneity ¯ζ2 under no circumstance can be eliminated for D-SGD and slows down its convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Periodical GT suffers more from noise comparing to K-GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In asymptotical regime, the transient time for K-GT O( σ √ K ) decrease with local steps while Periodical GT O(σ) does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' But this noise term can be improved as discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' If we consider a full gradient in equation (6), Periodical GT performs similar to K-GT at the expense of extra computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 5 Experimental results We evaluate the effectiveness of K-GT by comparing it with D-SGD and periodical GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1 Setting We conduct experiments in two settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' SYNTHETIC DATASETS: We first construct the distributed least squares objective with fi(x) = 1 2||Aix−bi||2 with fixed Hessian A2 i = i2 n · Id, and sample each bi ∼ N(0, ¯ζ2 i2 · Id) for each client i ∈ [n], where ¯ζ2 can control the deviation between local objectives [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Stochastic noise is controlled by adding Gaussian noise with σ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' REAL-WORLD DATASET, MNIST [3]: We test the case that all clients collaboratively train a convolutional neural network (CNN)4 on real-world dataset, MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In total this dataset contains 60,000 images of size 28×28 and 10 labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We use a ring topology for both sets of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For simplicity, instead of using the initialization in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 1, we initialize ci = 0 for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='For data partition on MNIST, we consider both homogeneous and heterogeneous cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The homogeneous dataset is first shuffled and then uniformly partitioned among all the clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We call this the ‘random’ setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The heterogeneous datasets is created when each client only has exclusive access to subset of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We call this the ‘sorted’ case, and the data variation across clients is maximized at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We use n = 5 and n = 10 clients and each client has access to one and two classes case accordingly, and the case n = 10 has more severe heterogeneity condition than the case n = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Parameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For SYNTHETIC DATASETS, we use the same learning rate ηs=1 and η=1e-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For MNIST, we use the best constant learning rate tuned from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='001} for algorithms and batch size 128 on each client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Note that even though our algorithm is purposed with constant learning rate, using more sophisticated and time-varying learning rate scheduler would definitely bring much better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We mainly illustrate the acceleration and robustness in convergence rate of the K-GT compared to baseline D-SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We also consider the performance of several GT-variants that supports local steps discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2 Numerical results K-GT is the most robust against heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In the convex case, client drift only happens for D-SGD suggested by Figure 1 in which the larger value ¯ζ2 ̸= 0 gets, the poorer model quality D-SGD ends up with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' However, K-GT, Periodical GT (w/ and w/o full grad) and Large-batch GT do not suffer from ’client-drift’ and ultimately reach the consistent level of model quality regardless of increasing of ¯ζ and K (number of either local steps or random samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In the non-convex case, since it’s known the optimality condition and optimization trajectory is more complex than the convex case, generalization performance of all methods in Figure 2 cannot fully recover the baseline performance when data partition is random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' However, K-GT could always outperform when data partition is non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' and the improvement is more significant when the degree of heterogeneity is increasing from Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 4Here we only consider a very simple network without Batch Norm layers [7] for simplicity, since it inherently assumes that the data distribution is uniform across different batches, which is not the case that we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The detailed network structure is listed in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 8 0 2000 4000 10 5 10 3 10 1 101 103 || f(x)||2 K = 1 2 = 0 0 100 200 communication rounds 10 5 10 3 10 1 101 103 || f(x)||2 K = 20 0 2000 4000 10 5 10 3 10 1 101 103 2 = 1 0 100 200 communication rounds 10 5 10 3 10 1 101 103 0 2000 4000 10 5 10 3 10 1 101 103 2 = 10 0 100 200 communication rounds 10 5 10 3 10 1 101 103 D-SGD K-GT Periodical GT Periodical GT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' w/ full grad Large-batch GT Large-batch SGD Figure 1: Training SYNTHETIC convex functions over ring by 10 clients with noise σ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In total 5000 communication rounds for K = 1 (top row) while only 250 rounds for K = 20 (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' All uses the same learning rate and are averaged by three repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The client-drift for D-SGD is even more severe with increasing heterogeneity (larger ¯ζ) as well as K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' K-GT, GT, and GT w/ full grad are consistent for different ¯ζ2 while achieving communication reduction when K > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 0 40 80 120 160 200 60 70 80 90 100 top-1 test acc(%) n = 5 0 40 80 120 160 200 computation epochs 60 70 80 90 100 top-1 test acc(%) n = 10 DSGD, random DSGD, sorted K-GT, random K-GT, sorted (a) K = 1, and data is partitioned either random or sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 0 20 40 60 80 100 60 70 80 90 100 top-1 test acc(%) n = 5 0 40 80 120 160 200 communication rounds 60 70 80 90 100 top-1 test acc(%) n = 10 DSGD K-GT Periodical GT Periodical GT w/ full grad (b) K = 1 n epoch, and data is partitioned sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Figure 2: Generalization performance on MNIST among D-SGD (blue), K-GT(red), GT w/o (orange) and w/ (green) full gradient for n = 5 (top row) and n = 10 (bottom row) clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The x-axis corresponds to (a) the number of pass over overall dataset(epoch), and (b) the number of communication rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In (a), when K = 1, K-GT and Periodical GT are identical to GT baseline (remark 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Learning rates are tuned to be the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Note that 1 epoch of passing over global dataset is equivalent to 470 times computation on SGD when mini-batch sized 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' And for (b), since nK = 470 is fixed for both n = 5 (top right) and n = 10 (top left), the number of local steps between communication rounds is Kn=5 = 2Kn=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 9 0 20 40 60 80 computation epochs 0 20 40 60 80 100 top-1 test acc(%) K = 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Bloc = 128 D-SGD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' B = Bloc Large-batch SGD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' B = KBloc K-GT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' B = Bloc Large-batch GT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' B = KBloc Figure 3: Generalization performance comparison on MNIST between large-batch training and training with local steps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' where large-batch training uses KBloc local batch size and communicates every update,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' while training with local steps uses Bloc local batch size and communicates periodically every K local update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Local step reduces communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' From K = 1 to K = 20 in Figure 1, K-GT and other GT alternatives reach the same target after 2000 rounds to only 100 rounds, achieving linear reduction in communication with the help of local steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' However, more local steps makes D-SGD suffer even more in model quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' At the same time, introducing local steps into the training of non-convex functions would still achieve communication reduction but not by a linear factor of K as in the convex case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Within Figure 2b, we fixe nK = 1 epoch over the data such that for no matter which n = 5 or n = 10 client communicates once after 1 epoch of computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Compared to K = 1 in Figure 1, the acceleration when K > 1 is still by a huge amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' However, note that introducing more local steps when data partition is heterogeneous would result in more severe quality loss, but K-GT still outperforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Large-batch GT has similar performance to tracking with local steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' From both convex (Figure 1) and non- convex (Figure 3) functions, either K-GT or Large-batch GT, after the same number of communication rounds while the simultaneously the same number of computation epochs, reaches the similar level of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' But for D-SGD, training with local steps could be even more stable and generalizes better than the Large-batch, which has been empirically investigated in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 6 Conclusion Decentralized learning is a promising building block for the democratization of Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Especially in Edge AI applications, users’ data does not follow a uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' This requires robustness of decentralized learning algorithms to data heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We propose a novel decentralized optimization algorithm (K-GT) supports communication efficient local update steps and overcomes data dissimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The tracking mechanism uses the accumulated gradient sum, akin to momentum, thereby reducing variance across local updates without the need of large batch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We demonstrated the superiority of K-GT with both convergence guarantees and empirical evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 10 References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Alghunaim and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Sayed, Linear convergence of primal–dual gradient methods and their performance in distributed optimization, Automatica 117 (2020), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 109003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [2] Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', Accelerating gossip SGD with periodic global averaging, in ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Deng, The mnist database of handwritten digit images for machine learning research, IEEE Signal Processing Magazine 29 (2012), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 141–142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [4] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Goyal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Dollár, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Girshick, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Noordhuis, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Wesolowski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Kyrola, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Tulloch, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Jia, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' He, Accurate, large minibatch SGD: training ImageNet in 1 hour, arXiv preprint arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='02677 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Haddadpour, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Kamani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Mahdavi, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Cadambe, Local SGD with Periodic Averaging: Tighter Analysis and Adaptive Synchronization, in NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [6] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Hsieh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Phanishayee, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Mutlu, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Gibbons, The Non-IID Data Quagmire of Decentralized Machine Learning, in ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift, in ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [8] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Kairouz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' McMahan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Avent, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Bellet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Bennis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Bhagoji, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Bonawitz, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Charles, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Cormode, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Cummings, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' D’Oliveira, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Eichner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Rouayheb, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Evans, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Gardner, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Garrett, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Gascón, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Ghazi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Gibbons, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Gruteser, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Harchaoui, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' He, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' He, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Huo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Hutchinson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Hsu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Jaggi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Javidi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Joshi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Khodak, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Koneˇcný, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Korolova, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Koushanfar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Koyejo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lepoint, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Liu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Mittal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Mohri, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Nock, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Özgür, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Pagh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Raykova, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Qi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Ramage, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Raskar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Song, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Song, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Stich, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Sun, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Suresh, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Tramèr, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Vepakomma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Xiong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Xu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Yang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Yu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Yu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Zhao, Advances and open problems in federated learning, Foundations and Trends® in Machine Learning 14 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 1–210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Karimireddy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Kale, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Mohri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Reddi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Stich, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Suresh, SCAFFOLD: Stochastic Controlled Averaging for Federated Learning, in ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Koloskova, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Stich, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Jaggi, Decentralized deep learning with arbitrary communication compression, ICLR (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Koloskova, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lin, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Stich, An improved analysis of gradient tracking for decentralized machine learning, NeurIPS (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Koloskova, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Loizou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Boreiri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Jaggi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Stich, A Unified Theory of Decentralized SGD with Changing Topology and Local Updates, in ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Koloskova, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Stich, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Jaggi, Decentralized Stochastic Optimization and Gossip Algorithms with Compressed Communication, in ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Koneˇcn`y, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' McMahan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Yu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Richtárik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Suresh, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Bacon, Federated learning: Strategies for improving communication efficiency, arXiv preprint arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='05492 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [15] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Cen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Chen, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Chi, Communication-Efficient Distributed Optimization in Networks with Gradient Tracking and Variance Reduction, in AISTATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Andersen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Smola, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Yu, Communication Efficient Distributed Machine Learning with the Parameter Server, in NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [17] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Li, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Sahu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Zaheer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Sanjabi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Talwalkar, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Smith, Federated Optimization in Heterogeneous Networks, in MLSys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [18] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lian, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Hsieh, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Zhang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Liu, Can Decentralized Algorithms Outperform Centralized Algorithms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' A Case Study for Decentralized Parallel Stochastic Gradient Descent, in NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [19] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lian, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Zhang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Liu, Asynchronous Decentralized Parallel Stochastic Gradient Descent, in ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [20] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Stich, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Patel, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Jaggi, Don’t Use Large Mini-batches, Use Local SGD, in ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [21] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Liu, Variance reduction on decentralized training over heterogeneous data, Master’s thesis, ETH Zürich, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Available at https://pub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='tik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='ch/students/2020-HS/MA-2020-32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 11 [22] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lorenzo and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Scutari, NEXT: In-network nonconvex optimization, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Signal and Information Processing over Networks 2 (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 120–136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [23] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' McMahan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Moore, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Ramage, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Hampson, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' y Arcas, Communication-efficient learning of deep networks from decentralized data, in AISTATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' PMLR, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 1273–1282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [24] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Nedi´c, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Olshevsky, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Shi, Achieving geometric convergence for distributed optimization over time- varying graphs, SIAM Journal on Optimization 27 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2597–2633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Nemirovski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Juditsky, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lan, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Shapiro, Robust stochastic approximation approach to stochastic programming, SIAM Journal on Optimization 19 (2009), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 1574–1609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [26] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Nguyen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Alghunaim, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Yuan, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Uribe, On the performance of gradient tracking with local updates (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Available at https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='org/abs/2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='04757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Pu and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Nedi´c, A Distributed Stochastic Gradient Tracking Method, in ICDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [28] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Shi, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Ling, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Wu, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Yin, EXTRA: An exact first-order algorithm for decentralized consensus optimization, SIAM Journal on Optimization 25 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 944–966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [29] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Stich, Local SGD Converges Fast and Communicates Little, in ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [30] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Tang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Yan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Zhang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Liu, D2: Decentralized Training over Decentralized Data, in ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Tsitsiklis, Problems in decentralized decision making and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', Massachusetts Inst of Tech Cambridge Lab for Information and Decision Systems, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [32] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Xin, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Khan, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Kar, Variance-reduced decentralized stochastic optimization with accelerated conver- gence, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Signal Process 68 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 6255–6271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [33] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' You, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Hsieh, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Demmel, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Keutzer, ImageNet Training in Minutes, in ICPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [34] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Yuan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Xu, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Ling, Can primal methods outperform primal-dual methods in decentralized dynamic optimization?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Signal Process 68 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 4466–4480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' [35] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Zhang and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' You, Decentralized stochastic gradient tracking for non-convex empirical risk minimization, arXiv preprint arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='02712 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 12 A Algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1 Periodical Algorithm Algorithm 2 Periodical GT: GT with periodical communication 1: parameters: 2: T: number of communication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' K: number of local steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ηs, ηc: communication, local stepsize;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' W: given topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 3: Initialize: x0 i = x0 j, z0 i = 1 n � i Fi(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξi) = z0 j, ∀ i, j ∈ [n]a 4: for node i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', n} parallel do 5: for communication: t ← 0 to T − 1 do 6: for local steps: k ← 0 to K − 2 do 7: x(t)+k+1 i = x(t)+k i − ηcz(t)+k i 8: z(t)+k+1 i = z(t)+k i + ∇Fi(x(t)+k+1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k+1 i ) − ∇Fi(x(t)+k i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k i ) 9: end for 10: x(t)+K j = x(t)+K−1 i − ηcz(t)+K−1 i 11: x(t+1) i = � j wij � x(t) j − ηs(x(t) j − x(t)+K j ) � 12: z(t+1) i = � j wijz(t)+K−1 j + ∇Fi(x(t+1) i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t+1) i ) − ∇Fi(x(t)+K−1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+K−1 i ) 13: end for 14: end for aThis initialization in tracking variable z(0) i is required for heterogeneity-independent analysis in theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In fact, we show later with experiment that z(0) i = ∇Fi(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ) works well in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Algorithm 3 Periodical GT with full-batch gradient 1: parameters: 2: T: number of communication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' K: number of local steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ηs, ηc: communication, local stepsize;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' W: given topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 3: Initialize: x0 i = x0 j, c0 i = −∇Fi(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξi) + 1 n � j ∇Fj(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξj)a 4: for node i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', n} parallel do 5: for communication: t ← 0 to T − 1 do 6: for local steps: k ← 0 to K − 2 do 7: z(t)+k i = ∇Fi(x(t)+k i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k i ) + c(t) i 8: x(t)+k+1 i = x(t)+k i − ηcz(t)+k i 9: end for 10: Compute full gradient on x(t)+K−1 i , gi = ∇fi(x(t)+K−1 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 11: x(t)+K j = x(t)+K−1 i − ηc(gi + c(t) i ) 12: x(t+1) i = � j wij � x(t)+K−1 j − ηs(x(t) j − x(t)+K j ) � 13: c(t+1) i = � j wijc(t) j + � j wijgj − gi 14: end for 15: end for aThis initialization in correction c(0) i is required for heterogeneity-independent analysis in theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In fact, we show later with experiment that c(0) i = 0 works well in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' B Proof of proposition In this section, we will prove the propositions previously discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1 Tracking Property of K-GT Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1 (Gradient Sum Tracking).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Define Z(t) = 1 Kηc � X(t) − X(t)+K� as the tracking variable during communication round t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The update rule for both models X(t) and tracking variables Z(t) at communication in K-GT 1 can be rewritten as (η = ηsηc): X(t+1) = � X(t) − KηZ(t)� W , Z(t+1) = Z(t+1)W + G(t+1) − G(t) , (5) where G(t) = 1 K � k ∇F(X(t)+k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k) denotes the mean update over the local steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The updating schemes of the model for K-GT are shown in equation (4), then if we define Z(t) = 1 Kηc � X(t) − X(t)+K� , and η = ηsηc, then with simply reformulating we could derive the set of equations shown above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2 Periodical Gradient Tracking reformulation The periodical GT is actually time-varying GT with skipping communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' That is W(t) = W in equation (2) when mod(t, K)=0, otherwise W(t) = I no communication and local step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then we adopt the notation for K-GT that we denote the model at k-th local step after t-th communication round as X(t)+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' And the same principle is applied to tracking variable Z(t)+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In the following sections, we will first show that Periodical GT can be equivalently reformulated and corrected SGD with constant correction throughout local steps, and then provide the update scheme for both correction and model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1 Corrected SGD Claim B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The local tracking variable during local steps can be equivalently rewritten as corrected SGD with correction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', Z(t)+k+1 = ∇F(X(t)+k+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k+1) + Z(t)+k − ∇F(X(t)+k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k) � �� � C(t)+k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' And the correction C remains unchanged throughout local steps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', C(t)+k+1 = C(t)+k, ∀k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', K − 1}, and is updated only at each time of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We know that local model is updated with Z instead of ∇F(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We define the deviation of Z from the SGD as C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' By contradiction we assume that deviation is different for each local iterate (t) + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' That’s C(t)+k+1 ̸= C(t)+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then for each local update, we have Z(t)+k+1 = Z(t)+k + ∇F(X(t)+k+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k+1) − ∇F(X(t)+k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k) Z(t)+k+1 − ∇F(X(t)+k+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k+1) = Z(t)+k − ∇F(X(t)+k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k) C(t)+k+1 = C(t)+k, ∀k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', K − 1} which contradicts the assumed fact that C(t)+k+1 ̸= C(t)+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2 Updating scheme reformulation Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' If we additionally consider separate step sizes for local steps and communication, we can equivalently rewrite Periodical GT as follows, Local steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We consider local steps as corrected SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The correction C ∈ Rd×n captures the difference between local update and communication update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For local steps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='e, ∀k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', K − 1}, X(t)+0 ≡ X(t), X(t)+k+1 = X(t)+k − ηc(∇F(X(t)+k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k) + C(t)), (7) where C(t) is constant for all local steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then it synchronizes both X and C, X(t+1) = � X(t) − ηs(X(t) − X(t)+K) � W C(t+1) = C(t)W + ∇F(X(t)+K−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+K−1)(W − I) (8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' By Claim B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2, the local update is equivalent to corrected SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Note that different stepsizes ηc and ηs are used for model update of local step and communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 2 The correction C is constant during local steps by Claim B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2, then consider its update during communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Z(t+1) = Z(t)+K−1W + ∇F(X(t+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t+1)) − ∇F(X(t)+K−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+K−1) ⇔ (∇F(X(t+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t+1)) + C(t+1)) = � ∇F(X(t)+K−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+K−1) + C(t)� W + ∇F(X(t+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t+1)) − ∇F(X(t)+K−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+K−1) ⇔ C(t+1) = C(t)W + ∇F(X(t)+K−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+K−1)(W − I) C Proof of theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1 Technical tools In this section, we mainly introduce some analytical tools that help in convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1 (Implications of the smoothness Assumption 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Assumption 1 implies ∀i and ∀x, y ∈ Rd, ||∇fi(x) − ∇fi(y)|| ≤ L||x − y||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For arbitrary set of n vectors {ai}n i=1, ai ∈ Rd, || 1 n �n i ai||2 ≤ 1 n �n i ||ai||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For given two vectors a, b ∈ Rd, 2⟨a, b⟩ ≤ α||a||2 + 1 α||b||2, α > 0, which is equivalent to ||a + b||2 ≤ (1 + α)||a||2 + (1 + 1 α)||b||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Above inequality also holds for matrix in Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For A, B ∈ Rd×n, ||AB||F ≤ ||A||F ||B||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='4 (Variance upperbound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' If there exist n zero-mean random variables {ξi}n i=1 that may not be independent of each other, but all have variance smaller than σ2, then the variance of sum is upperbounded by E|| � i ξi||2 ≤ nσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' E|| � i ξi||2 ≤ E � n � i ||ξi||2� ≤ nσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='5 (Unrolling recursion [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For any parameters r0 ≥ 0, b ≥ 0, e ≥ 0, u ≥ 0 there exists constant stepsize η ≤ 1 u such that ΨT := r0 T + 1 1 η + bη + eη2 ≤ 2( br0 T + 1) 1 2 + 2e 1 3 ( r0 T + 1) 2 3 + ur0 T + 1 Additional definitions Before proceeding with the proof of the convergence theorem, we need some addition set of definitions of the various errors we track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For simplicity, we define the special matrix J = 1n1T n n as it could be used to calculate the averaged matrix, XJ = ¯X = [¯x ¯x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ¯x] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We define the client variance (or consensus distance) to be how much each node deviates from their averaged model: Ξt = 1 n �n i E||x(t) i − ¯x(t)||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Since we are doing local steps between communication, we define the local progress to be how much each node moves from the globally averaged starting point as client-drift: at k-th local step: ek,t := 1 n �n i E||x(t)+k i − ¯x(t)||2 accumulation of local steps: Et := �K−1 k=0 ek,t = �K−1 k=0 1 n �n i E||x(t)+k i − ¯x(t)||2 Because we update model with correction, the corrected gradient will be aligned with the direction of the global update instead of the local update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The correction is updated every communication, and remains constant during local steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We define the quality of this correction to be how much it approximates the true deviation between global update and local update, γt = 1 nL2 E||C(t) + ∇f( ¯X(t)) − ∇f( ¯X(t))J||2 F , where J = 1 n11T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2 Convergence analysis This section we will show the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Since from the previous analysis that K-GT and periodical GT are equivalent to corrected SGD for local step, and have similar pattern during communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We can analyze them within the same prove framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In order to prove the theorems, we first provide the recursion for client-drift, consensus distance and qualify of correction in following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 3 Bounding the client drift We will next consider the progress made within local steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' That’s the accumulated model update before next communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Suppose the local step-size for node ηc ≤ 1 8KL, and for arbitrary communication step size ηs ≥ 0, we could bound the drift as Et ≤ 3(KΞt) + 12K2η2 cL2(Kγt) + 6K2ηc 2(KE||∇f(¯x(t))||2) + 3K2ηc 2σ2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' First, observe that K = 1, Et = 1 nE||X(t) − ¯X(t)||2 F , 0 ≤ 2 nE||X(t) − ¯X(t)||2 F + 6ηc 2E||∇f(¯x(t))||2 + 3ηc 2σ2, the inequality will always hold since RHS is always positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then the lemma is trivially proven for K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then we consider the case for K ≥ 2, and nek,t := E||X(t)+k − ¯X(t)||2 F = E||X(t)+k−1 − ηc � ∇F(X(t)+k−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k) + C(t)� − ¯X(t)||2 F ≤ (1 + 1 K − 1)E||X(t)+k−1 − ¯X(t)||2 F + nηc 2σ2 + Kηc 2E||∇f(X(t)+k−1) − ∇f( ¯X(t)) + C(t) + ∇f( ¯X(t))(I − J) + ∇f( ¯X(t))J||2 F ≤ (1 + 1 K − 1 + 4Kηc 2L2) � �� � :=C E||X(t)+k−1 − ¯X(t)||2 F + 4Kηc 2L2nγt + 2Kηc 2nE||∇f(¯x(t))||2 + nηc 2σ2 ≤ CkE||X(t) − ¯X(t)||2 F + k−1 � r=0 Cr� 4Kηc 2L2nγt + 2Kηc 2nE||∇f(¯x(t))||2 + nηc 2σ2� If ηc ≤ 1 8KL, then 4K(ηcL)2 ≤ 1 16K < 1 16(K−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Since C > 1, then Ck ≤ CK ≤ (1+ 1 K−1+ 1 16(K−1))K ≤ e1+ 1 16 ≤ 3, and �k−1 r Cr ≤ KCK ≤ 3K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We could rewrite the bound on client drift at kth local step, nek,t ≤ 3Ξt + 3K � 4Kηc 2L2nγt + 2Kηc 2nE||∇f(¯x(t))||2 + nηc 2σ2� (9) Clearly, in inequality (9), the RHS is independent of time step k ∈ [0, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then the accumulated progress within local steps Et could be formulated by Et := K−1 � k=0 ek,t ≤ 3(KΞt) + 12K2ηc 2L2(Kγt) + 6K2ηc 2(KE||∇f(¯x(t))||2) + 3K2ηc 2σ2 Consensus distance We then consider how the consensus distance for communicated model is developed between communications after local training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For any effective step-size η = ηsηc, we have the descent lemma for Ξt as Ξt+1 ≤ (1 − p 2)Ξt + 6Kη2L2 p Et + 6K2η2L2 p γt + Kη2σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We know that the update between two communication round is as follows, X(t+1) = � X(t) − KηZ(t)� W, 4 where Z(t) = 1 K � k � ∇F(X(t)+k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k) + C(t)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then consensus distance at time (t + 1) can be measured by nΞt+1 = E||X(t+1) − ¯X(t+1)||2 F = E|| � X(t) − η K−1 � k=0 (∇F(X(t)+k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k) + C(t)) � (W − J)||2 F ≤ (1 − p)E|| � X(t) − η K−1 � k=0 (∇f(X(t)+k) + C(t)) � (I − J)||2 F + nKη2σ2 ≤ nKη2σ2 + (1 + α)(1 − p)E||X(t)(I − J)||2 F + (1 + 1 α)η2E|| K−1 � k=0 ∇f(X(t)+k)(I − J) ± K∇f( ¯X(t))(I − J) + KC(t)||2 F ≤ α= p 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 1 p ≤1 nKη2σ2 + (1 − p 2)E||X(t) − ¯X(t)||2 F + 6 p � Kη2L2||I − J||2 K−1 � k=0 E||X(t)+k − ¯X(t)||2 F + K2η2 L2 L2 E||f( ¯X(t))(I − J) + C(t)||2 F � ≤ (1 − p 2)nΞt + 6Kη2L2 p nEt + 6K2η2L2 p nγt + nKη2σ2 Quality measure of correction We now bound the quality measure of correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The correction is thought to depict the deviation of local and global gradient of the ideally averaged model ¯X at the time of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' That is, quality measure of correction is defined to be γt = 1 nL2 E||C(t) + ∇f( ¯X(t)) − ∇f( ¯X(t))J||2 F , where J = 1 n11T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' How to estimate correction, K-GT and periodical GT have different options, which is carefully discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For any effective step-size η = ηsηc ≤ √p √ 6KL, we have the descent lemma for γ in periodical GT as follow, Kγt+1 ≤ (1 − p 2)Kγt + 24 p (KeK−1,t) + 2 pEt + 12K2η p KηE||∇f(¯x(t))||2 + 2Kσ2 L2 , and if we instead of using the average of local steps for K-GT in correction, we have the descent lemma for γ as follow, Kγt+1 ≤ (1 − p 2)Kγt + 30 p Et + 12K2η2 p KE||∇f(¯x(t))||2 + 2σ2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The averaged correction between two consecutive communication round satisfies C(t+1)J = C(t)J + 1 Kηc (X(t) − X(t)+K)(W − I)J = C(t)J for K-GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We assume that the correction is initialized with arbitrary value as long as its globally average always equals to zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=', C(t)J = C(0)J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Recall the definition of γt, note that (C(t) + ∇f( ¯X(t)) − ∇f( ¯X(t)J)J = C(t)J + ∇f( ¯X(t))(J − J) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' It’s easy to check that Periodical GT has the same property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 5 Then for K-GT, we have the recursion of quality measure can be formulated as follows, nL2γt+1 := E||C(t+1) + ∇f( ¯X(t+1))(I − J)||2 F = E||C(t)W + 1 K K−1 � k=0 ∇F(X(t)+k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k)(W − I) + ∇f( ¯X(t+1))(I − J)||2 F = E|| � C(t) + ∇f( ¯X(t))(I − J) � W + � 1 K K−1 � k=0 ∇f(X(t)+k) − ∇f( ¯X(t)) � (W − I) + � ∇f( ¯X(t+1)) − ∇f( ¯X(t)) � (I − J)||2 F + nσ2 K ≤ (1 + α)(1 − p)nL2γt + 2(1 + 1 α) � ||W − I||2 1 K K−1 � k=0 E||∇f(X(t)+k) − ∇f( ¯X(t))||2 F + ||I − J||2E||∇f( ¯X(t+1)) − ∇f( ¯X(t))||2 F � + nσ2 K (due to ||W − I|| ≤ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ||I − J|| ≤ 1) ≤ α= p 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 1 p ≤1 (1 − p 2)nL2γt + 6 p � 4 K K−1 � k=0 L2E||X(t)+k − ¯X(t)||2 F + nL2E||¯x(t+1) − ¯x(t)||2� + nσ2 K ≤ (1 − p 2)nγt + 6L2 pK � 4nEt + 2K2η2L2nEt + 2K2η2n(KE||∇f(¯x(t))||2) + K2η2σ2� + nσ2 K Periodical GT uses ∇F(X(t)+K−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+K−1) to replace 1 K �K−1 k=0 ∇F(X(t)+k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k) in correction update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' With almost identical analysis, we could get a very similar equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' In addition, if we replace stochastic gradient with full gradient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='e, ∇f(X(t)+K−1), in periodical GT, which will improve the noise term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Further, if η = ηsηc ≤ √p √ 6KL, then 6K2η2L2 p ≤ 1 which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Shown from results above, the quantizations of γt for periodical GT and K-GT only differ in the coefficient of stochastic noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' And using full-batch gradient can improve Periodical GT in stochastic noise to the same level as that of K-GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Progress between communications We study how the progress between communication rounds could be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We could bound the averaged progress between communication in any round t ≥ 0, and any η = ηsηc ≥ 0 as follows, E||¯x(t+1) − ¯x(t)||2 ≤ 2Kη2L2Et + 2K2η2E||∇f(¯x(t))||2 + (Kη)2σ2 nK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' From previous analysis, we guarantee 1 n � i c(t) i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then the averaged progress between communication could be rewritten as E||¯x(t+1) − ¯x(t)||2 = η2E|| 1 n � i,k ∇Fi(x(t)+k i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k) + K n � i c(t) i ||2 ≤ Kη2 n � i,k 2E||∇fi(x(t)+k i ) − ∇fi(¯x(t))||2 + 2K2η2E||∇f(¯x(t))||2 + Kη2σ2 n ≤ 2Kη2L2 n � i,k E||x(t)+k i − ¯x(t)||2 + 2K2η2E||∇f(¯x(t))||2 + Kη2σ2 n In the first inequality, note that the K random variable {ξ(t)+k}K−1 k=0 when conditioned on communication (t) may not be independent of each other but each has variance smaller than σ2 due to Assumption 2, and we can apply Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then the following inequalities are from the repeated application of triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 6 Descent lemma for non-convex case Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' When function f is L-smooth, the averages ¯x(t) of the iterates of Algorithm 1 and Algorithm 2 with the constant stepsize ηc < 1 4ηsKL, satisfy Ef(¯x(t+1)) − Ef(¯x(t)) ≤ −Kη 4 E||∇f(¯x(t))||2 + ηL2Et + Kη2L 2n σ2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Because the local functions {fi(x)} are L-smooth according to Assumption 1, it’s trivial to conclude that the global function f(x) is also L-smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Ef(¯x(t+1)) = Ef � ¯x(t) − η n � i,k (∇Fi(x(t)+k i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k i ) + c(t) i ) � ≤ Ef(¯x(t)) + E � ∇f(¯x(t+1)), − η n � i,k (∇Fi(x(t)+k i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k i ) + c(t) i ) � � �� � :=U +L 2 E||¯x(t+1) − ¯x(t)||2 From our previous analysis, we know 1 n � i c(t) i = 0, ∀t ≥ 0 forK-GT and Periodical GT (if with initialization indicated in purposed algorithm) U : = E � ∇f(¯x(t+1)), − η n � i,k (∇Fi(x(t)+k i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k i ) + c(t) i ) � = E � ∇f(¯x(t)), − η n � i,k Eξ(t)+k i ∇Fi(x(t)+k i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ξ(t)+k i ) � = −KηE � ∇f(¯x(t)), 1 nK � i,k ∇fi(x(t)+k i ) − ∇f(¯x(t)) + f(¯x(t)) � = −KηE||∇f(¯x(t))||2 + 1 nK � i,k KηE � ∇f(¯x(t)), � ∇fi(x(t)+k i ) − ∇fi(¯x(t)) �� ≤ −Kη 2 E||∇f(¯x(t))||2 + Kη 2nK � i,k E||∇fi(x(t)+k i ) − ∇fi(¯x(t))||2 ≤ −Kη 2 E||∇f(¯x(t))||2 + KL2η 2nK � i,k E||x(t)+k i − ¯x(t)||2 Then also plug in the Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='9 for E||¯x(t+1) − ¯x(t)||2, we have Ef(¯x(t+1)) ≤ Ef(¯x(t)) + (−Kη 2 + K2η2L)E||∇f(¯x(t))||2 F + (ηL2 2 + Kη2L3)Et + Kη2L 2n σ2 Then the choice η ≤ 1 4KL completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Main recursion We first construct a potential function Ht = Ef(¯x(t)) − Ef(x(⋆)) + A (Kηc)3L4 pη2s γt + B 6v2 KηcL2 p Ξt where constants A, B and v can be obtained through the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='11 (Recursion for K-GT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For any effective stepsize of Algorithm 1 satisfying ηs = ˜O( p KL) and ηc = ˜O(p), there exists constants A, B, v satisfying D > 0 and D5 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then we have the recursion Ht+1 − Ht ≤ −DKηE||∇f(¯x(t))||2 + D5L2 pK (Kη)3σ2 + L 2nK (Kη)2σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' First, from previous bound on those error term γt, Ξt and Et, we could bound the difference between Ht+1 and Ht for K-GT, while we also plug in with C > 0 the 0 ≤ −CηcL2Et + 3C(KηcL2Ξt) + 12C(Kηc)3L4γt + C 6(Kηc)2L2 ηs KηE||∇f(¯x(t))||2 + 3C(Kηc)3 L2σ2 K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 7 Then we have the inequality recursion for K-GT as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Ht+1 − Ht ≤ � − A 2 + B η2 s v2p2 + 12C � � �� � ≤D1 (Kηc)3L4γt + � − B 12v2 + 3C � � �� � ≤D2 KηcL2Ξt + � A30(KηcL)2 p2K + B K2η2L2 v2p2 − C + ηs � � �� � ≤D3 ηcL2Et + � − 1 4 + A12ηs(Kηc)4L4 p + C 6K2ηc2L2 ηs � � �� � ≤D4 KηE||∇f(¯x(t))||2 + � A2L2 pK + B η2 sL2 6v2pK + C 3L2 K � � �� � ≤ D5L2 pK (Kηc)3σ2 + L 2nK (Kη)2σ2 As long as ηc ≤ p 96vKL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' ηs = v · p → ηcηs ≤ η = p2 96KL and A = 72v3p + 48vp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' B = 36v3p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' C = vp there exists constant v > 1 that makes D1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' D3 ≤ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' D4 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' And D4 ≤ −D < 0, and D5 ≥ 0, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='12 (Recursion for Periodical GT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For any effective stepsize of Algorithm 1 satisfying ηs = ˜O( p KL) and ηc = ˜O(p), there exists constants A, B, v satisfying D > 0 and D5 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then we have the recursion Ht+1 − Ht ≤ −DKηE||∇f(¯x(t))||2 + D5L2 p (Kη)3σ2 + L 2nK (Kη)2σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The sets of inequality for Periodical GT only differs in stochastic noise compared to K-GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then applied with the same principle as that for K-GT, we get its recursion of potential function as follows Ht+1 − Ht ≤ D1(Kηc)3L4γt + D2KηcL2Ξt + D3ηcL2Et + D4KηE||∇f(¯x(t))||2 + � A2L2 p + B η2 sL2 6v2pK + C 3L2 K � � �� � ≤ D5L2 p (Kηc)3σ2 + L 2nK (Kη)2σ2 − C 2 KeK−1,t The rest of the analysis could refer to Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Using full gradient to improve Periodical GT has the same recursion as K-GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Solve the main recursion Take K-GT as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Consider the telescope sum of the potential function, we can derive 1 T + 1 T � t=0 � Ht+1 − Ht � = 1 T + 1 � HT +1 − H0 � ≤ η=ηsηc −DKη 1 T + 1 T � t=0 E||∇f(¯x(t))||2 + D5L2 pKη3s (Kη)3σ2 + L 2nK (Kη)σ2 ⇒ ηs=v·p 1 T + 1 T � t=0 E||∇f(¯x(t))||2 ≤ H0 − HT +1 (T + 1)D 1 Kη + Lσ2 2nKD(Kη) + D5L2σ2 v3p4KD(Kη)2 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='g we consider that f(x) is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then we could neglect the effect of −HT +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 8 Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' There exists constant stepsize such that 1 T + 1 T � t=0 E||∇f(¯x(t))||2 = O �� σ2LH0 nKT + (σLH0 p2KT ) 2 3 + LH0 p2T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The non-negative sequences {Ht}T +1 t=0 and {E||∇f(¯xt)||}T t=0 with positive coefficients before both Kη and (Kη)2 satisfy the condition in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then we tune the stepsize using Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then the average of accumulation of gradient could be upper-bounded by ⇒ 1 T + 1 T � t=0 E||∇f(¯x(t))||2 ≤ O � 2( Lσ2 2nK H0 T + 1 ) 1 2 + 2(L2σ2 p4K ) 1 3 (H0(x) T + 1 ) 2 3 + H0(x) Kηmax(T + 1) � = O �� σ2LH0 nKT + (σLH0 p2KT ) 2 3 + LH0 p2T � Then the convergence rate depends on the initial values of potential function H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' By the definition of potential function in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='11, H0 is the combination of initial value for f(x0), E||X(0)− ¯X(0)||2 F and E||C(0)+∇f( ¯X(0))(−J+I)||2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We assume that every node is guaranteed to be initialized with the same model x(0) = x(0) i , ∀i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Then we could easily get E||X(0) − ¯X(0)||2 F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' And if we initial the correction term with c(0) i = −∇fi(x(0)) + 1 n � i ∇fi(x(0)), then E||C(0) + ∇f(X(0))(−J + I)||2 F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' H0 = f(x0) − f(x⋆) + A(Kηc)3L4 pη2s γ0 + B 6v2 KηcL2 p Ξ0 = f(x0) − f(x⋆) := F0 (10) And then for arbitrary accuracy error ϵ > 0, the communication rounds needed to reach the target accuracy is upperbounded by T ≤ O � σ2 nK 1 ϵ2 + σ p2√ K 1 ϵ 3 2 + 1 p2 1 ϵ � LF0, which concludes the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2 for K-GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1 for Periodical GT can also be easily derived with the same principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' 9 D Experimental details D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='1 Visualization of benchmark datasets (a) 0 1 2 3 4 5 6 7 8 9 class label 1 2 3 4 5 6 7 8 9 10 node id random 0 1 2 3 4 5 6 7 8 9 class label 1 2 3 4 5 6 7 8 9 10 sorted (b) Figure 4: Data visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' (a) Example from MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' (b) Data partition on each node in the random and the sorted case when there are n = 10 distributed nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' The dot size indicates the number of samples per class allocated to each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' We show an image example from MNIST datasets and how data of different labels is partitioned in random and sorted case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' It obviously presents in the random case, data of different labels are randomly and evenly partitioned among nodes, but in the sorted case, each node only contains images of 1 label and the labels obtained by each node is non-overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='2 Model structure For our non-convex experiment, we use a 4-layer Convolutional Neural Network (CNN) and its details are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' Table 3: Model architecture of the benchmark experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For convolutional layer (Conv2D), we list parameters with sequence of input and output dimension, kernal size, stride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For max pooling layer (MaxPool2D), we list kernal and stride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For fully connected layer (FC), we list input and output dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' For drop out (Dropout), we list the parameter of probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content=' layer details 1 Conv2D(1, 10, 5, 1), MaxPool2D(2), ReLU 2 Conv2D(10, 10, 5, 1), Dropout2D(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} +page_content='5), MaxPool2D(2), ReLU 3 FC(320, 50), ReLU 4 FC(50, 10) 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf'} diff --git a/UdE0T4oBgHgl3EQflgEz/content/2301.02486v1.pdf 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0000000000000000000000000000000000000000..dab20617e383d0c07ce2fdc8c568ac65d74db845 --- /dev/null +++ b/W9AzT4oBgHgl3EQfYfyH/content/tmp_files/2301.01336v1.pdf.txt @@ -0,0 +1,965 @@ +Optimal Decoy Resource Allocation for Proactive Defense in +Probabilistic Attack Graphs +Haoxiang Ma +University of Florida +Gainesville, United State +hma2@ufl.edu +Shuo Han +University of Illinois Chicago +Chicago, United State +hanshuo@uic.edu +Nandi Leslie +Raytheon Technologies +Arlington County, United State +nandi.o.leslie@raytheon.com +Charles Kamhoua +U.S. Army Research Laboratory +Gainesville, United State +charles.a.kamhoua.civ@mail.mil +Jie Fu +University of Florida +Gainesville, United State +fujie@ufl.edu +ABSTRACT +This paper investigates the problem of synthesizing proactive de- +fense systems in which the defender can allocate deceptive targets +and modify the cost of actions for the attacker who aims to com- +promise security assets in this system. We model the interaction +of the attacker and the system using a formal security model– a +probabilistic attack graph. By allocating fake targets/decoys, the +defender aims to distract the attacker from compromising true tar- +gets. By increasing the cost of some attack actions, the defender +aims to discourage the attacker from committing to certain poli- +cies and thereby improve the defense. To optimize the defense +given limited decoy resources and operational constraints, we for- +mulate the synthesis problem as a bi-level optimization problem, +while the defender designs the system, in anticipation of the at- +tacker’s best response given that the attacker has disinformation +about the system due to the use of deception. Though the general +formulation with bi-level optimization is NP-hard, we show that +under certain assumptions, the problem can be transformed into +a constrained optimization problem. We proposed an algorithm +to approximately solve this constrained optimization problem us- +ing a novel, incentive-design method for projected gradient ascent. +We demonstrate the effectiveness of the proposed method using +extensive numerical experiments. +KEYWORDS +Attack Graph, Deception, Markov Decision Process +ACM Reference Format: +Haoxiang Ma, Shuo Han, Nandi Leslie, Charles Kamhoua, and Jie Fu. 2023. +Optimal Decoy Resource Allocation for Proactive Defense in Probabilistic +Attack Graphs. In Proc. of the 22nd International Conference on Autonomous +Agents and Multiagent Systems (AAMAS 2023), London, United Kingdom, +May 29 – June 2, 2023, IFAAMAS, 8 pages. +1 +INTRODUCTION +Proactive defense refers to a class of defense mechanisms for the +defender to detect any ongoing attacks, distract the attacker with +deception, or use randomization to increase the difficulty of an +attack to the system. In this paper, we propose a mathematical +Proc. of the 22nd International Conference on Autonomous Agents and Multiagent Sys- +tems (AAMAS 2023), A. Ricci, W. Yeoh, N. Agmon, B. An (eds.), May 29 – June 2, 2023, +London, United Kingdom. © 2023 International Foundation for Autonomous Agents +and Multiagent Systems (www.ifaamas.org). All rights reserved. +framework and solution approach for synthesizing a proactive +defense system with deception. +We start by formulating the attack planning problem using a +probabilistic attack graph, which can be viewed as a Markov de- +cision process (MDP) with a set of attack target states. Attack +graphs(AGs)[7] can be used in modeling computer networks. They +are widely used in network security to identify the minimal subset +of vulnerability/sensors to be used in order to prevent all known +attacks[13, 16]. Probabilistic attack graphs introduce uncertain out- +comes of attack actions that account for action failures in a sto- +chastic environment. For example, in [5, 6], probabilistic transitions +in attack graphs capture uncertainties originated from network- +based randomization. Under the probabilistic attack graph modeling +framework, we investigate how to allocate decoy resources as fake +targets to distract the attacker into attacking the fake targets, and +how to modify the attack action costs to discourage the attacker +from reaching the true targets. +The joint design of decoy resource allocation and action cost +modification can be cast as a bi-level optimization problem, which +is generally NP-hard [2]. Under the assumption that potential de- +coy states are predefined and the defender only needs to allocate +resources/rewards to decoys, we prove the bi-level optimization can +be equivalently expressed as a constrained optimization problem. +To solve the constrained optimization problem using a projected +gradient ascent efficiently, we build two important relations: First, +we show that the projection step of a defender’s desired attack +policy to the set of realizable attack policy space can be performed +using Inverse Reinforcement Learning (IRL) [21]. Essentially, IRL is +to shape the attacker’s perceived reward so that the rational attacker +will mimic a strategy chosen by the defender. Second, the gradient +ascent step can be performed using policy improvement, which +is a subroutine in policy iteration with respect to maximizing the +defender’s total reward. The project gradient ascent is ensured to +converge to a (local) optimal solution to this nonconvex constrained +optimization problem. +Related work. The synthesis of proactive defense strategies stud- +ied here is closely related to the Stackelberg security game(SSG) +(surveyed in [18]) and its solution via bi-level optimization. In an +SSG, the defender is to protect a set of targets with limited re- +sources, while the attacker selects the optimal attack strategy given +the knowledge of the defender’s strategy. In [12], the authors study +arXiv:2301.01336v1 [cs.MA] 3 Jan 2023 + +security countermeasure-allocation and use attack graphs to evalu- +ate the network’s security given the allocated resources. However, +the SSG does not account for the asymmetric information intro- +duced by the use of deception. In [20], the authors introduce reward +shaping to motivate the agent to behave as the target policy. How- +ever, in our setting, the target policy may be infeasible, because the +defender aims to lure the attacker to reach a fake target, while the +attacker may not intentionally avoid true targets. +Deceptions create incorrect/incomplete information to the at- +tacker. In [19], the authors formulate a security game to allocate +limited decoy resources to mask a network configuration from the +cyber attacker. The decoy-based deception manipulates the adver- +sary’s perception of the payoff matrix. In [1], the authors study +honeypot allocation in deterministic attack graphs and determine +the optimal allocation strategy using the minimax theorem. In [10], +the authors study directed acyclic attack graphs that can be modi- +fied by the defender using deceptive and protective resources. They +propose a mixed-integer linear program (MILP)-based algorithm +to determine the allocation of deceptive and protective resources +in the graph. In [3], they harden the network by using honeypots +so that the attacker can not discriminate between a true target and +a fake target. In [11], the authors assign fake edges in the attack +graph in order to interdict the attacker and employ MILP to find +the optimal solution. +Compared to existing work, our work makes the following con- +tributions: First, we do not assume any graph structure in the attack +graph and consider probabilistic attack graphs instead of determin- +istic ones. As the attacker can take a randomized strategy in the +probabilistic attack graph, it is not possible to construct a payoff +matrix and apply the minimax theorem for decoy resource alloca- +tion. Second, we consider simultaneously allocating limited decoy +resources and modifying the cost of attack actions and analyze +the best response of the attacker given the disinformation caused +by deception. Third, we proposed an efficient incentive-design in- +spired algorithm for synthesizing the defense strategy Under the +assumption that the attacker is rational and can not distinguish +decoys from the true targets, by modifying the action reward and +allocating decoy resources properly, we show that it is possible +to shape the attacker’s behavior so that the misperceived attacker +is incentivized to commit an attack strategy that maximizes the +defender’s reward. Finally, we test the scalability of our method on +different problem sizes. +2 +PRELIMINARIES AND PROBLEM +FORMULATION +Notations. Let R denote the set of real numbers and R𝑛 the set +of real 𝑛-vectors. Let R𝑛 +>0 (resp. R𝑛 +<0) be the set of positive (resp. +negative) real 𝑛-vectors. We use 1 to represent the vector of all ones. +Given a vector 𝑧 ∈ R𝑛, let 𝑧𝑖 be the 𝑖-th component. Given a finite +set 𝑍, the set of probability distributions over 𝑍 is represented +as Dist(𝑍). Given 𝑑 ∈ Dist(𝑍), the support of 𝑑 is denoted as +Supp(𝑑) = {𝑧 ∈ 𝑍 | 𝑑(𝑧) > 0}. Let 𝐼𝐵 be the indicator function, i.e., +𝐼𝐵(𝑥) = 1 if 𝑥 ∈ 𝐵, and 𝐼𝐵(𝑥) = 0 otherwise. +We consider the adversarial interaction between a defender +(player 1, pronoun she/her) and an attacker (player 2, pronoun +he/him/his) in a system equipped with proactive defense (formally +defined later). We first introduce a formal model, called probabilistic +attack graph, to capture how the attacker plans to achieve the attack +objective. Then, we introduce proactive defense countermeasures +with deception. +Attack Planning Problem. The attack planning problem is mod- +eled as a probabilistic attack graph, +𝑀 = (𝑆,𝐴, 𝑃,𝜈,𝛾, 𝐹, 𝑅2), +where 𝑆 is a set of states (nodes in the attack graph), 𝐴 is a set of +attack actions, 𝑃 : 𝑆 × 𝐴 → Dist(𝑆) is a probabilistic transition +function such that 𝑃(𝑠′|𝑠,𝑎) is the probability of reaching state 𝑠′ +given action 𝑎 being taken at state 𝑠, 𝜈 ∈ Dist(𝑆) is the initial state +distribution, 𝛾 ∈ (0, 1] is a discount factor. The attack’s objective +is described by a set 𝐹 of target states and a target reward function +𝑅2 : 𝐹 × 𝐴 → R≥0, which assigns each state-action pair (𝑠,𝑎) +where 𝑠 ∈ 𝐹 and 𝑎 ∈ 𝐴 to a nonnegative value of reaching that +target for the attacker. The reward function can be extended to the +entire state space by defining 𝑅2(𝑠,𝑎) = 0 for any 𝑠 ∈ 𝑆 \ 𝐹,𝑎 ∈ 𝐴. +To capture the termination of attacks, we introduce a unique sink +state 𝑠sink ∈ 𝑆 \ 𝐹 such that 𝑃(𝑠sink|𝑠sink,𝑎) = 1 for all 𝑎 ∈ 𝐴 and +𝑃(𝑠sink|𝑠,𝑎) = 1 for any target 𝑠 ∈ 𝐹 and 𝑎 ∈ 𝐴. +The probabilistic attack graph characterizes goal-directed attacks +encountered in cyber security [8, 14], in which by reaching a tar- +get state, the attacker compromises certain critical network hosts. +Probabilistic attack graphs [10, 17] capture the uncertain outcomes +of the attack actions using the probabilistic transition function and +generalize deterministic attack graphs [7]. +The attacker is to maximize his discounted total reward, starting +from the initial state 𝑆0 ∼ 𝜈. A randomized, finite-memory attack +policy is a function 𝜋 : 𝑆∗ → Dist(𝐴), which maps a finite run +𝜌 ∈ 𝑆∗ into a distribution 𝜋(𝜌) over actions. A policy is called +Markovian if it only depends on the most recent state, i.e., 𝜋 : 𝑆 → +Dist(𝐴). We only consider Markovian policies because it suffices +to search within Markovian policies for an optimal attack policy. +Let (Ω, F ) be the canonical sample space for (𝑆0,𝐴0, (𝑆𝑡,𝐴𝑡)𝑡>1) +with the Borel 𝜎-algebra F = B(Ω) and Ω = 𝑆 × 𝐴 × �∞ +𝑡=1(𝑆 × 𝐴). +The probability measure Pr𝜋 on (Ω, F ) induced by a Markov policy +𝜋 satisfies: Pr𝜋 (𝑆0 = 𝑠) = 𝜇0(𝑠), Pr𝜋 (𝐴0 = 𝑎 | 𝑆0 = 𝑠) = 𝜋(𝑠,𝑎), +and Pr𝜋 (𝑆𝑡 = 𝑠 | (𝑆𝑘,𝐴𝑘)𝑘<𝑡) = 𝑃(𝑠 | 𝑆𝑘,𝐴𝑘), and Pr𝜋 (𝐴𝑡 = 𝑎 | +(𝑆𝑘,𝐴𝑘)𝑘<𝑡,𝑆𝑡) = 𝜋(𝑆𝑡,𝑎). +Given a Markovian policy 𝜋 : 𝑆 → Dist(𝐴), we define the at- +tacker’s value function 𝑉 𝜋 +2 : 𝑆 → R as +𝑉 𝜋 +2 (𝑠) = E𝜋 [ +∞ +∑︁ +𝑘=0 +𝛾𝑘𝑅2(𝑆𝑘,𝐴𝑘)|𝑆0 = 𝑠], +where E𝜋 is the expectation given the probability measure Pr𝜋. +Proactive Defense with Deception. We assume that the defender +knows the attacker’s objective given by the tuple ⟨𝐹, 𝑅2⟩, i.e., the +target states and target reward function. The defender’s proactive +defense mechanisms are the following: +• Defend by deception: The defender employs a deception +method called “revealing the fake”. Specifically, the defender +has a set 𝐷 ⊂ 𝑆 \ 𝐹 of states in the MDP 𝑀 that can be set +to be fake target states with fake target rewards �𝑦 ∈ R|𝐷 |. + +The attacker cannot distinguish the real targets 𝐹 from fake +targets 𝐷. +• Defend by state-action reward modification: The defender +has a set𝑊 ⊂ (𝑆\(𝐹∪𝐷))×𝐴 of state action pairs in the MDP +𝑀 whose reward can be modified. Once the reward of the +state action pair (𝑠,𝑎) is modified, the attacker’s perceived +reward 𝑅2(𝑠,𝑎) < 0, i.e., the cost of attack action 𝑎 at state 𝑠 +is −𝑅2(𝑠,𝑎). +The defender can determine how to allocate her decoy resource +and limited state-action reward modification ability. +Definition 1 (Decoy allocation under constraints). The defender’s +decoy allocation design is a nonnegative real-valued vector �𝑦 ∈ R|𝑆 | +≥0 +satisfying �𝑦(𝑠) = 0 for any 𝑠 ∈ 𝑆 \𝐷 and constrained by 1T�𝑦 ≤ ℎ for +some ℎ ≥ 0. Given a decoy allocation �𝑦, the attacker’s perceptual +reward function is defined by +𝑅 �𝑦 +2 (𝑠,𝑎) = +� �𝑦(𝑠) +if �𝑦(𝑠) > 0, +𝑅2(𝑠,𝑎) +if �𝑦(𝑠) = 0. +Definition 2 (Action reward modification). Given a set 𝑊 +⊂ +(𝑆 \ (𝐹 ∪ 𝐷)) × 𝐴, the defender’s action reward modification is a +nonpositive reward-valued vector �𝑥 ∈ R|𝑆×𝐴| +≤0 +satisfying �𝑥(𝑠,𝑎) = 0 +for any (𝑠,𝑎) ∉ 𝑊 . Given an action reward modification �𝑥, the +attacker’s perceptual reward function is defined by +𝑅 �𝑥 +2 (𝑠,𝑎) = +� �𝑥(𝑠,𝑎) +if �𝑥(𝑠,𝑎) < 0, +𝑅2(𝑠,𝑎) +if �𝑥(𝑠,𝑎) = 0. +Note that the defender does not consider modifying the state- +action reward for (fake or real) target states 𝐹 ∪ 𝐷 because once a +state in 𝐹 ∪ 𝐷 is reached, the attack is terminated. +Definition 3. The defender’s proactive defense strategy is a tu- +ple (�𝑥, �𝑦) including an action reward modification �𝑥 and a decoy +allocation design �𝑦. +Because the action reward modification is independent of the +decoy allocation design, the reward function given a defender’s +strategy (�𝑥, �𝑦) is the composition of 𝑅 �𝑥 +2 and 𝑅 �𝑦 +2 and thus omitted. +Assumption 1. The attack process terminates under two cases: +Either the attack succeeds, in which the attacker reaches a target +𝑠 ∈ 𝐹, or the attack is interdicted, in which the attacker reaches a +state allocated with a decoy. +Our problem can be informally stated as follows. +Problem 1. In the attack planning scenario we mentioned above, +determine the defender’s strategy to allocate decoy resources and +modify action reward so as to maximize the probability that the +attacker reaches a fake target given the best response of the attacker. +3 +MAIN RESULTS +In this section, we first define the attacker’s perceptual planning +problem for a fixed action reward modification and decoy resource +allocation (�𝑥, �𝑦). Then we show that the design of the proactive +defense can be formulated as a bi-level optimization problem. We +investigate the special property of the formulated bi-level opti- +mization problem to develop an optimization-based approach for +synthesizing the proactive defense strategy. +3.1 +A Bi-level Optimization Formulation +The defender’s strategy changes how the attacker perceives the +attack planning problem as follows: +Definition 4 (Perceptual attack planning problem with modified +reward and decoys). Let the action reward modification be �𝑥 and +decoy allocation be �𝑦, and the attacker’s original planning problem +𝑀 = (𝑆,𝐴, 𝑃,𝜈,𝛾, 𝐹, 𝑅2), the perceptual planning problem of the +attacker is defined by the following MDP with terminating states: +𝑀(�𝑥, �𝑦) = (𝑆,𝐴, 𝑃 �𝑦,𝜈,𝛾, 𝐹 ∪ 𝐷 �𝑦, 𝑅 �𝑥, �𝑦 +2 +), +where 𝑆,𝐴,𝜈,𝛾 are the same as those in 𝑀, 𝐷 �𝑦 = {𝑠 ∈ 𝐷 | �𝑦(𝑠) ≠ 0} +are decoy target states and absorbing. The transition function 𝑃 �𝑦 is +obtained from the original transition function 𝑃 by only making all +states in 𝐷 �𝑦 absorbing. The reward 𝑅 �𝑥, �𝑦 +2 +is defined based on Def. 1 +and Def. 2. +The perceptual value for the attacker is +𝑉 𝜋 +2 (𝜈; �𝑥, �𝑦) = E𝜋 +� ∞ +∑︁ +𝑘=0 +𝛾𝑘𝑅 �𝑥, �𝑦 +2 +(𝑆𝑘,𝐴𝑘) | 𝑆0 ∼ 𝜈 +� +, +where E𝜋 is the expectation given the probability measure Pr𝜋 in +duced by 𝜋 from the MDP 𝑀(�𝑥, �𝑦). +The defender’s deception objective is given by a reward function +𝑅 �𝑦 +1 : 𝑆 → R, defined by +𝑅 �𝑦 +1 (𝑠) = +� +1 +if 𝑠 ∈ 𝐷 �𝑦, +0 +otherwise. +(1) +Given the probability measure Pr𝜋, we denote the defender’s +value by +𝑉 𝜋 +1 (𝜈; �𝑦) = E𝜋 +� ∞ +∑︁ +𝑘=0 +𝛾𝑘𝑅1(𝑆𝑘) | 𝑆0 ∼ 𝜈 +� +. +With this reward definition, the value 𝑉 𝜋 +1 (𝜈; �𝑦) is the probability +of the attacker reaching a fake target in 𝐷 �𝑦. +To formalize the deception objective, we introduce the notion of +a defender’s preferred attack policy as follows. +Definition 5 (A defender’s preferred attack policy). Given the +perceptual planning problem of the attacker 𝑀(�𝑥, �𝑦) where (�𝑥, �𝑦) is +a fixed proactive defense strategy, let 𝜋 and 𝜋 ′ be two attack policies +that achieve the same value for the attacker, i.e., 𝑉 𝜋 +2 (𝜈; �𝑥, �𝑦) = +𝑉 𝜋′ +2 (𝜈; �𝑥, �𝑦). Policy 𝜋 is strictly preferred to 𝜋 ′ by the defender if +and only if +𝑉 𝜋 +1 (𝜈; �𝑦) > 𝑉 𝜋′ +1 (𝜈; �𝑦). +In words, if two policies are equally good for the attacker, the +one with a higher probability to reach a fake target is preferred by +the defender. +Then the problem of synthesizing an optimal proactive defense +strategy (�𝑥, �𝑦) can be mathematically formulated as +Problem 2. +max. +�𝑥 ∈𝑋, �𝑦∈𝑌 +𝑉 𝜋∗ +1 (𝜈; �𝑦) +s.t. +𝜋∗ ∈ argmax +𝜋 +𝑉 𝜋 +2 (𝜈; �𝑥, �𝑦). + +where 𝑋 = R|𝑊 | +≤0 and 𝑌 = {�𝑦 | ∀𝑠 ∈ 𝑆 \ 𝐷, �𝑦(𝑠) = 0 and 1T�𝑦 ≤ ℎ} +are the ranges for variables �𝑥 and �𝑦 correspondingly. +In words, the defender decides (�𝑥, �𝑦) so that the attacker’s best +response in his perceptual attack planning problem turns out to be +an attack policy most preferred by the defender, as it maximizes +the defender’s value. +3.2 +Transforming into a Constrained +Optimization Problem +The bi-level optimization problem is known to be strongly NP-hard +[4]. However, under certain conditions, the bi-level optimization +problem can be shown to be equivalent to a constrained optimiza- +tion problem. +Let Π(�𝑥, �𝑦) = {𝜋 | 𝑉 𝜋 +2 (𝜈; �𝑥, �𝑦) = max𝜋 𝑉 𝜋 +2 (𝜈; �𝑥, �𝑦)} , which is +the set of optimal policies in the attacker’s perceived planning +problem with respect to a choice of variables �𝑥 and �𝑦. The bi-level +optimization problem is then equivalently written as the following +constrained optimization problem: +max. +𝜋∗,�𝑥 ∈𝑋, �𝑦∈𝑌 +𝑉 𝜋∗ +1 (𝜈; �𝑦) +s.t. +𝜋∗ ∈ Π(�𝑥, �𝑦). +(2) +This, in turn, is equivalent to +max. +𝜋∗ +𝑉 𝜋∗ +1 (𝜈; �𝑦) +s.t. +𝜋∗ ∈ +� +�𝑥 ∈𝑋, �𝑦∈𝑌 +Π(�𝑥, �𝑦). +(3) +Here, the constraint means the attacker’s response 𝜋∗ can be se- +lected from the collection of optimal attack policies given all possi- +ble values for �𝑥, �𝑦. +By the definition of the defender’s value function, it is noted that +𝑉 𝜋 +1 (𝜈; �𝑦) does not depend on the exact value of �𝑦 but only depends +on whether �𝑦(𝑠) > 0 for each state 𝑠 ∈ 𝐷. Formally, +Lemma 1. For any �𝑦1, �𝑦2 ∈ 𝑌, if �𝑦1(𝑠) = 0 =⇒ �𝑦2(𝑠) = 0 and +vice versa, then 𝑉 𝜋 +1 (𝜈; �𝑦1) = 𝑉 𝜋 +1 (𝜈; �𝑦2). +Proof. Given two different vectors �𝑦1 and �𝑦2, we can construct +two MDPs: 𝑀1 � 𝑀(�𝑥, �𝑦1) = (𝑆,𝐴, 𝑃 �𝑦1,𝜈,𝛾, 𝐹, 𝑅1) and 𝑀2 � +𝑀(�𝑥, �𝑦2) = (𝑆,𝐴, 𝑃 �𝑦2,𝜈,𝛾, 𝐹, 𝑅1), respectively. +If �𝑦1(𝑠) = 0 if and only if �𝑦2(𝑠) = 0, then the transition functions +𝑃 �𝑦1 of 𝑀1 and 𝑃 �𝑦2 of 𝑀2 are the same (see Def. 4). +Further, the defender’s reward function 𝑅 �𝑦1 +1 also equals to 𝑅 �𝑦2 +1 +(see (1)), given both the transition dynamics and reward are the +same, we have 𝑉 𝜋 +1 (𝜈; �𝑦1) = 𝑉 𝜋 +1 (𝜈; �𝑦2). +□ +Next, to remove the dependency of 𝑉 𝜋 +1 (𝜈; �𝑦) on �𝑦, we make the +following assumption: +Assumption 2. The set 𝐷 �𝑦 = {𝑠 ∈ 𝐷 | �𝑦(𝑠) ≠ 0} of states where +decoys are allocated is given. +Under this assumption, we simply assume all states in the given +set 𝐷 have to be assigned with nonzero decoy resources. That is +𝐷 �𝑦 = 𝐷. +This assumption further reduces the defender’s synthesis prob- +lem into a constrained optimization problem. +max. +𝜋∗ +𝑉 𝜋∗ +1 (𝜈) +s.t. +𝜋∗ ∈ Π ≜ +� +�𝑦∈𝑌,�𝑥 ∈𝑋 +Π(�𝑥, �𝑦), +�𝑦(𝑠) > 0, ∀𝑠 ∈ 𝐷. +(4) +Because the above problem is a standard constrained optimiza- +tion problem, one can obtain a locally optimal solution using the +projected gradient method: +𝜋𝑘+1 = projΠ (𝜋𝑘 + 𝜂∇𝑉 𝜋𝑘 +1 +(𝜈)). +where projΠ (𝜋) denotes projecting policy 𝜋 onto the policy space +Π and 𝜂 is the step size. +3.3 +Connecting Inverse-reinforcement +Learning with Project Gradient Ascent +A key step in performing projected gradient ascent is to evaluate, for +any policy ˆ𝜋, the projection projΠ ( ˆ𝜋). However, this is nontrivial +because the set ¯Π includes a set of attack policies, each of which +corresponds to a choice of vectors (�𝑥, �𝑦). As a result, ¯Π does not +have a compact representation. Next, we propose a novel algorithm +that computes the projection. +First, by the definition of projection, it is noted that this pro- +jection step is equivalent to solving the following optimization +problem: +min. +𝜋 +D(𝜋, ˆ𝜋) +s.t. +𝜋 ∈ Π, +�𝑦(𝑠) > 0;∀𝑠 ∈ 𝐷. +(5) +where D(𝜋, ˆ𝜋) is the distance between the two policies 𝜋, ˆ𝜋. +The distance function D can be chosen to be the Kullback–Leibler +(KL)-divergence between policy-induced Markov chains, defined +as follows. +Definition 6. Given an MDP 𝑀 = (𝑆,𝐴, 𝑃,𝜈) and two Markovian +policies 𝜋1, 𝜋2. Let 𝑀𝜋1 = (𝑆, 𝑃1,𝜈) and 𝑀𝜋2 = (𝑆, 𝑃2,𝜈) be two +Markov chains induced from 𝑀 under 𝜋1 and 𝜋2, respectively. The +KL divergence DKL +�𝑀𝜋1 ∥𝑀𝜋2 +� (relative entropy from 𝑀𝜋2 to 𝑀𝜋1) +is defined by +DKL +�𝑀𝜋1 ∥𝑀𝜋2 +� = +∑︁ +𝜌 ∈𝑆∗ +Pr1(𝜌) log Pr1(𝜌) +Pr2(𝜌) , +where Pr𝑖 (𝜌) is the probability of a path 𝜌 in the Markov chain +𝑀𝜋𝑖 for 𝑖 = 1, 2. +The KL divergence in (5) can be expressed as +DKL +�𝑀𝜋 (�𝑥, �𝑦)∥𝑀�𝜋 (�𝑥, �𝑦)� = +∑︁ +𝜌 +� +Pr(𝜌) log +� +Pr(𝜌) +Pr(𝜌|�𝑥, �𝑦) += +∑︁ +𝜌 +� +Pr(𝜌) log � +Pr(𝜌) − +∑︁ +𝜌 +� +Pr(𝜌) log Pr(𝜌|�𝑥, �𝑦), +(6) +where � +Pr(𝜌) is the probability of path 𝜌 in the Markov chain +𝑀�𝜋 (�𝑥, �𝑦), and Pr(𝜌|�𝑦) is the probability of path 𝜌 in the Markov +chain 𝑀𝜋 (�𝑥, �𝑦) induced by a policy 𝜋. + +Because the first term in the sum in (6) is a constant for ˆ𝜋 is +fixed, the KL divergence minimization problem is equivalent to the +following maximization problem: +max. +�𝑥 ∈𝑋, �𝑦∈𝑌 +∑︁ +𝜌 +� +Pr(𝜌) log Pr(𝜌|�𝑥, �𝑦) +(7) +s.t. +�𝑦(𝑠) > 0;∀𝑠 ∈ 𝐷, +(8) +1T�𝑦 ≤ ℎ. +(9) +Problem (7) can be solved by an extension of the Maximum Entropy +(MAXENT) IRL algorithm [21], which was originally developed in +the absence of constraints. It is well-known that IRL is to infer, from +the expert demonstration, a reward function for which the expert +policy generating the demonstrations is optimal. The use of IRL to +perform the projection is intuitively understood as follows: The +goal is to compute a pair of vectors (�𝑥, �𝑦) that alters the attacker’s +perceived reward function so that the attacker’s optimal policy +given (�𝑥, �𝑦) is closed to the “expert policy” ˆ𝜋, under the constraints +of �𝑦. +To handle the decoy resource constraint (9), we approximate the +constraint using a logarithmic barrier function and compute the +optimal solution �𝑦∗ using gradient-based numerical optimization. +Considering the constraint 1T�𝑦 ≤ ℎ, we implement the barrier +function in order to approximate the inequality constraints and +rewrite the optimization problem as: +max +�𝑥, �𝑦 +∑︁ +𝜌 +� +Pr(𝜌) log Pr(𝜌|�𝑥, �𝑦) + 1 +𝑡 log(ℎ − 1T�𝑦) +subject to: �𝑦(𝑠) = 0, +∀𝑠 ∈ 𝑆 \ 𝐷. +where 𝑡 is the weighting parameter of the logarithmic barrier func- +tion. In our experiment, 𝑡 is fixed to be 1000. +Since constraint �𝑦(𝑠) = 0, ∀𝑠 ∈ 𝑆 \𝐷, can be incorporated into the +domain of decision variables �𝑦, we can use gradient ascent to obtain +the optimal �𝑥∗, �𝑦∗ that maximizes the objective function. Specifi- +cally, �𝑥 and �𝑦 can be updated via �𝑥𝑘+1 = projX (�𝑥𝑘 + 𝜂𝑥∇𝐿(�𝑥, �𝑦)), +�𝑦𝑘+1 = projY (�𝑦𝑘 + 𝜂𝑦∇𝐿(�𝑥, �𝑦)). +3.4 +Policy Improvement for Gradient Ascent +Step +After the projection step to obtain a policy 𝜋𝑘 and the corresponding +vector (�𝑥, �𝑦), we aim to compute a one-step gradient ascent to +improve the objective function’s value +𝑉 𝑘+1 +1 +(𝜈) = 𝑉 𝑘 +1 (𝜈) + ∇𝑉 𝑘 +1 (𝜈), +where 𝑉 𝑘 +1 (𝜈) is the defender’s value evaluated given the attack +policy 𝜋𝑘 at the 𝑘-th iteration. +For this step, we perform a policy improvement step with respect +to the defender’s reward function 𝑅 �𝑦 +1 , which now is independent +of �𝑦 because the set 𝐷 �𝑦 is fixed to be a constant set 𝐷. It is shown +in [9, 15] that policy improvement is a one-step Newton update of +optimizing the value function. +Specifically, the policy improvement is to compute +˜𝜋𝑘+1(𝑠,𝑎) = +exp ((𝑅1(𝑠,𝑎) + 𝛾𝑉 𝑘 +1 (𝑠′))/𝜏) +� +𝑎∈𝐴 exp ((𝑅1(𝑠,𝑎) + 𝛾𝑉 𝑘 +1 (𝑠′))/𝜏) +, +The policy at iteration 𝑘 + 1 is obtained by performing the pro- +jection step ((5)) in which ˆ𝜋 ≜ ˜𝜋𝑘+1. +The iteration stops when |𝑉 𝑘+1 +1 +(𝜈) − 𝑉 𝑘 +1 (𝜈)| ≤ 𝜖 where 𝜖 is +a manually defined threshold. The output yields a tuple (�𝑥∗, �𝑦∗) +which is the (local) optimal proactive defense strategy. We can only +obtain a local optimal proactive defense strategy here due to the +transferred constrained optimization problem having a nonconvex +constraint set. However, we can start from different initial policies +and select the best one. Moreover, assume the defender is solving her +own problem without considering attacker’s objective. the upper +bound of the defender’s objective can be obtained. We can select +the solution whose objective function is closest to the upper bound. +Remark 1. In our problem, we assume the set 𝐷 is given. If the set +𝐷 is not given, then this problem becomes combinatorial. If the set +𝐷 is not given but to be determined from a candidate set of states. +Then a naive approach is to enumerate all possible combinations +and evaluate the defender’s value for every subset and select the +one that yields the highest defender’s value. It would be interesting +to examine if the combinatorial problem is sub-modular or super- +modular, but it is beyond the scope of this work. +In summary, the proposed algorithm starts with an initial pol- +icy ˜𝜋0, and use the IRL to find the projection 𝜋0 as well as their +corresponding vectors (�𝑥0, �𝑦0) that shape the attacker’s perceptual +reward function for which 𝜋0 is optimal. Then a policy improve- +ment is performed to update 𝜋0 to ˜𝜋1. By alternating the projection +and policy improvement, the process terminates until the stopping +criteria |𝑉 𝑘+1 +1 +(𝜈) − 𝑉 𝑘 +1 (𝜈)| ≤ 𝜖 is satisfied. +4 +EXPERIMENT +We illustrate the proposed methods with two sets of examples, one +is a probabilistic attack graph and another is an attack planning +problem formulated in a stochastic gridworld. For all case studies, +the workstation used is powered by Intel i7-11700K and 32GB RAM. +0 +start +2 +3 +1 +4 +5 +6 +7 +8 +9 +10 +13 +12 +11 +Figure 1: A probabilistic attack graph. +Figure 1 shows a probabilistic attack graph with the target set +𝐹 = {10} and the action set {𝑎,𝑏,𝑐,𝑑}. For clarity, the graph only +shows the transition given action 𝑎 where a thick (resp. thin) arrow +represents a high (resp. low) transition probability. For example, +𝑃(0,𝑎) = {1 : 0.7, 2 : 0.1, 3 : 0.1, 4 : 0.1} 1. +Consider the set D = {11, 13} of decoy states. Recall the defender’s +reward function is 𝑅1(𝑠) = 1, for all 𝑠 ∈ 𝐷. Assuming no resource is +1The exact transition function is provided in the supplementary file. + +allocated to 𝐷 and all states in 𝐷 are sink states, then the attacker +has a 60.33% probability of reaching the target set 𝐹 from the initial +state 0. In the meantime, the defender’s expected value is 0.149. +That is, with probability 14.9%, the attacker will reach a decoy state +in 𝐷 and the attack is terminated. +Given limited resource 1T�𝑦 ≤ 3, the decoy resource allocation +yields �𝑦(11) = �𝑦(13) = 1.313. Based on the given decoy resource +allocation, the attacker has an 8.63% probability of reaching the +target set 𝐹 and the defender’s expected reward is 0.653 at initial +state 0. Thus, by assigning resources to decoys to attract the attacker, +the defender reduces the attacker’s probability of reaching the target +state significantly (85% reduction) and improves the defender’s +value by 3.38 times. +Figure 2: 6 × 6 gridworld example. +Next, we consider a robot motion planning problem in a stochas- +tic 6 × 6 gridworld shown in Figure 2. The attacker/robot aims to +reach a set of goal states while avoiding detection from the defender. +The attacker can move in four compass directions. Given an action, +say, “N”, the attacker enters the intended cell with 1−2𝛼 probability, +and enters the neighboring cells, which are west and east cells with +𝛼 probability. In our experiments, 𝛼 is selected to be 0.1. A state +(𝑖, 𝑗) means the cell at row 𝑖 and column 𝑗. +The defender has deployed sensors shown in Figure 2 to detect +the presence of an attacker. Thus, once the attacker enters a sensor +state, his task fails. The decoy set 𝐷 is given as blue cells and the +target set 𝐹 is given as green cells. +Given the initial state is at (2, 0), which is indicated by the ro- +bot in the figure. We test the following three scenarios: No de- +coy resource allocation, decoy resource allocation only, decoy re- +source allocation together with reward modification. The result is +shown in table 1. When we do not allocate resources to decoys, +the attacker has a 98.98% probability of reaching the target set 𝐹 +while avoiding sensor states. And the defender’s expected value +is 3.56 × 10−6. When the defender is allowed to allocate resources +with a total budget of 4 to decoys, the decoy resource allocation +yields �𝑦((1, 4)) = 2.016, �𝑦((4, 5)) = 1.826. The defender does not +spend all decoy resources because of the use of the logarithmic +barrier function to enforce the constraint, when it is close to the +upper bound, the log barrier function will work as a large penalty +in gradient ascent. +Under the given resource allocation, the attacker has a 9.9% +probability of reaching the target set 𝐹, and the defender’s expected +value at the initial state is 0.3877. In the decoy resource allocation +(a) Converge using different initial policies. +(b) Converge with decoy resource allocation and action reward +modification. +Figure 3: Defender’s value converge trend in 6 × 6 gridworld +example given 𝐷 = {(1, 4), (4, 5)}. +Figure 4: 10 × 10 gridworld example. +and action reward modification experiment, the defender is allowed +to modify all action rewards at state (4, 4) and the action ‘N’ reward +at state (4, 0), (4, 1) and (4, 2). It turns out the defender allocates +1.938 to decoy (1, 4) and 1.734 to decoy (2, 5). Meanwhile, action +‘N’ reward at (4, 0) is modified to −1 and the same action at (4, 1) is + +(5,5) +(0,0)0.40 +0.35 +0.30 +val +Defender's +0.25 +0.20 +0.15 +0.10 +0.05 +Initial policy 1 +0.00 +Initial policy 2 +2 +3 +5 +Iteration0.40 +0.35 +value +0E0 +Defender's +0.25 +0.20 +0.15 +0.10 +0.05 +Initial policy 1 +0.00 +Initial policy 2 +1 +2 +Fm +4 +5 +Iteration(6'6) +(0,0)Figure 5: Defender’s value converge trend in 10 × 10 grid- +world. +modified to −0.94 and the action "N" reward at (4, 2) is modified to +−0.904, the defender will also modify the action reward of "W", "S", +"N" at (4, 4) to −1. Compare the decoy resource allocation result +with the decoy resource allocation and action reward modification +result. We find that by allowing action reward modification, the +defender reduces the attacker’s probability of reaching the target +(13.13% reduction). In the meantime, the defender’s expected value +increases by 1.62%. +It is noted that due to the nonlinearity in the optimization prob- +lem, the algorithm converges to different solutions under different +initial conditions, as shown in Figures 3a and 3a. In the figures, +the initial policy 1 is generated by assuming the attacker receives +the reward of 1 if he reaches the decoy and receives a reward of 0 +when he reaches the target state. This is ideal for the defender’s +objective but is infeasible for the optimization problem because +in the attacker’s perceptual planning problem, reaching the true +target will always provide a reward of 1 regardless of how many +resources are allocated to decoys. The initial policy 2 is randomly +generated. In this experiment, the value of the objective function +given different initial policies is close. +In order to test how the decoy set 𝐷 influences the result. We re- +allocate the position of decoys to {(0, 2), (5, 3)}. The result is shown +in Table 2. Based on the new configuration, if we do not allocate +decoy resources, the attacker reaches the target set with 98.97% +probability and the defender’s value is 7.61×10−8 at the initial state. +If the defender can allocate resources to the decoys, our method +yields �𝑦((0, 2)) = 1.141 and �𝑦((5, 3)) = 1.0. The attacker’s probabil- +ity of reaching the target set is 3.99% and the defender’s expected +value is 0.6991. If the defender is allowed to modify the same set of +state-action rewards as she is in the previous example, our algorithm +yields �𝑦((0, 2)) = 0.985 and �𝑦((5, 3)) = 1.068. Action ‘N’ reward +at (4, 0) is modified to −1 and the same action at (4, 1) is modified +to −0.85 and the action "N" reward at (4, 2) is modified to −0.081, +the defender will also modify the all action reward at (4, 4) to −1. +Under this configuration, the attacker’s probability of reaching the +target set is 0.286% (93% reduction compared to only allocating +decoy resources) and the defender’s expected value is 0.7301 (4.4% +increase compared to only allocate decoy resources). By changing +the configuration of set 𝐷, we show that the configuration of set 𝐷 +influences the attacker’s probability of reaching the target set and +the defender’s expected value: the second set 𝐷 = {(0, 2), (5, 3)} +appears to outperform the first set 𝐷 = {(1, 4), (4, 5)}. +Table 1: Experiment result in 6 × 6 gridworld given 𝐷 = +{(1, 4), (4, 5)}. +No decoy +Decoy only +Decoy and action reward +Attacker’s value +98.98% +9.9% +8.6% +Defender’s value +3.56 × 10−6 +0.3877 +0.394 +Table 2: Experiment result in 6 × 6 gridworld given 𝐷 = +{(0, 2), (5, 3)}. +No decoy +Decoy only +Decoy and action reward +Attacker’s value +98.97% +3.99% +0.286% +Defender’s value +7.61 ×10−8 +0.6991 +0.7301 +Next, in order to test the scalability, we increase the gridworld +size to 10 × 10 as shown in Figure 4. In the large gridworld example, +we only do decoy resource allocation. The sensors, decoy set, and +target set are represented using the same notation as the 6 × 6 +gridworld. The defender’s reward function is still 𝑅1(𝑠) = 1, for all +𝑠 ∈ 𝐷. Assume the initial state is at (5, 1). When the defender does +not allocate decoy resources, the attacker’s probability of reaching +the target is 82.43% and the defender’s expected value at the initial +state is 0.0024. When the defender is allowed to allocate resources +to decoys, our algorithm yields �𝑦((2, 8)) = 1.350, �𝑦((6, 8)) = 1.235. +Under the given decoy resources, the attacker’s probability of reach- +ing the target decreases to 5.1% (94% reduction), and the defender’s +expected value at the initial state increases to 0.4034. We also test +the defender’s converging trend using different initial policies as +shown in Figure 5. Initial policy 1 is obtained similarly to initial +policy 1 in the 6 × 6 example. Initial policy 2 and 3 are randomly +generated policies. From Figure 5, we observe that different initial +policies result in a similar converged value for the objective func- +tion. Considering the scalability of our algorithm, the computation +time for the 10 × 10 gridworld example is 185.94 seconds, while the +computation time of the 6×6 example is 22.51 seconds. The running +time shows our algorithm can be extended to moderate problem +sizes. It is noted that not only the state space size influences the +running time but also the selection of decoys, the number of decoys +influences the running time. +5 +CONCLUSION AND FUTURE WORK +We present a mathematical framework and algorithm for decoy +allocation and reward modification in a proactive defense system. +Our technical approach can be applied to many safety-critical sys- +tems where the probabilistic attack graphs are constructed from +known vulnerabilities in a system. The formulation and solutions +can be extended to a broad set of adversarial interactions in which +proactive defense with deception can be deployed. In the future, we +will consider more complex attack and defense objectives and inves- +tigate the decoy allocation given the uncertainty in the attacker’s +goal or capability. Apart from “revealing the fake” studied herein, +we will also investigate how to “conceal the truth” by manipulating +the attacker’s perceptual reward of compromising true targets. + +0.40 +lue +0.35 +val +0.30 +Defender's +0.25 +0.20 +0.15 +0.10 +Initial policy 1 +0.05 +Initial policy 2 ++ +Initial policy 3 +1 +2 +E +4 +5 +6 +7 +IterationREFERENCES +[1] A. H. Anwar, C. Kamhoua, and N. Leslie. 2020. Honeypot Allocation over Attack +Graphs in Cyber Deception Games. In 2020 International Conference on Computing, +Networking and Communications (ICNC). 502–506. +[2] Stephan Dempe and Alain Zemkoho. 2020. Bilevel optimization. Springer. +[3] Karel Durkota, Viliam Lis`y, Branislav Bošansk`y, and Christopher Kiekintveld. +2015. Optimal network security hardening using attack graph games. In Twenty- +Fourth International Joint Conference on Artificial Intelligence. +[4] Pierre Hansen, Brigitte Jaumard, and Gilles Savard. 1992. New branch-and-bound +rules for linear bilevel programming. SIAM Journal on scientific and Statistical +Computing 13, 5 (1992), 1194–1217. +[5] Jin Hong and Dong-Seong Kim. 2012. HARMs: Hierarchical Attack Representa- +tion Models for Network Security Analysis. In Australian Information Security +Management Conference. SRI Security Research Institute, Edith Cowan University, +Perth, Western Australia, 9. +[6] Jin B. Hong and Dong Seong Kim. 2016. Assessing the Effectiveness of Moving +Target Defenses Using Security Models. IEEE Transactions on Dependable and +Secure Computing 13, 2 (March 2016), 163–177. +[7] S. Jha, O. Sheyner, and J. Wing. 2002. Two Formal Analyses of Attack Graphs. In +Proceedings 15th IEEE Computer Security Foundations Workshop. CSFW-15. 49–63. +[8] Harjinder Singh Lallie, Kurt Debattista, and Jay Bal. 2020. A review of attack +graph and attack tree visual syntax in cyber security. Computer Science Review +35 (Feb. 2020), 100219. https://doi.org/10.1016/j.cosrev.2019.100219 +[9] Omid Madani. 2002. On Policy Iteration as a Newton’s Method and Polyno- +mial Policy Iteration Algorithms. In Eighteenth National Conference on Artificial +Intelligence (Edmonton, Alberta, Canada). American Association for Artificial +Intelligence, USA, 273–278. +[10] Stephanie Milani, Weiran Shen, Kevin S. Chan, Sridhar Venkatesan, Nandi O. +Leslie, Charles Kamhoua, and Fei Fang. 2020. Harnessing the Power of Decep- +tion in Attack Graph-Based Security Games. In Decision and Game Theory for +Security (Lecture Notes in Computer Science), Quanyan Zhu, John S. Baras, Radha +Poovendran, and Juntao Chen (Eds.). Springer International Publishing, Cham, +147–167. https://doi.org/10.1007/978-3-030-64793-3_8 +[11] Stephanie Milani, Weiran Shen, Kevin S Chan, Sridhar Venkatesan, Nandi O +Leslie, Charles Kamhoua, and Fei Fang. 2020. Harnessing the power of deception +in attack graph-based security games. In International Conference on Decision and +Game Theory for Security. Springer, 147–167. +[12] Thanh H. Nguyen, Mason Wright, Michael P. Wellman, and Satinder Singh. 2018. +Multistage Attack Graph Security Games: Heuristic Strategies, with Empirical +Game-Theoretic Analysis. Security and Communication Networks 2018 (Dec. +2018), 1–28. +[13] Steven Noel and Sushil Jajodia. 2008. Optimal ids sensor placement and alert +prioritization using attack graphs. Journal of Network and Systems Management +16, 3 (2008), 259–275. +[14] Steven Noel, Sushil Jajodia, Lingyu Wang, and Anoop Singhal. 2010. Measuring +security risk of networks using attack graphs. International Journal of Next- +Generation Computing 1, 1 (2010), 135–147. +[15] Martin L Puterman. 2014. Markov decision processes: discrete stochastic dynamic +programming. John Wiley & Sons. +[16] Oleg Sheyner, Joshua Haines, Somesh Jha, Richard Lippmann, and Jeannette M +Wing. 2002. Automated generation and analysis of attack graphs. In Proceedings +2002 IEEE Symposium on Security and Privacy. IEEE, 273–284. +[17] Anoop Singhal and Xinming Ou. 2017. Security risk analysis of enterprise +networks using probabilistic attack graphs. In Network Security Metrics. Springer, +53–73. +[18] Arunesh SINHA, Fei FANG, Bo AN, Christopher KIEKINTVELD, and Milind +TAMBE. 2018. Stackelberg Security Games: Looking beyond a Decade of Success. +Proceedings of the Twenty-Seventh International Joint Conference on Artificial +Intelligence (IJCAI-18),Stockholm, Sweden, July 13-19 (July 2018), 5494–5501. +[19] Omkar Thakoor, Milind Tambe, Phebe Vayanos, Haifeng Xu, Christopher Kiek- +intveld, and Fei Fang. 2019. Cyber Camouflage Games for Strategic Deception. In +Decision and Game Theory for Security (Lecture Notes in Computer Science), Tansu +Alpcan, Yevgeniy Vorobeychik, John S. Baras, and György Dán (Eds.). Springer +International Publishing, Cham, 525–541. +https://doi.org/10.1007/978-3-030- +32430-8_31 +[20] Guojun Wu, Yanhua Li, Zhenming Liu, Jie Bao, Yu Zheng, Jieping Ye, and Jun +Luo. 2019. Reward Advancement: Transforming Policy under Maximum Causal +Entropy Principle. arXiv preprint arXiv:1907.05390 (2019). +[21] Brian D Ziebart, Andrew L Maas, J Andrew Bagnell, Anind K Dey, et al. 2008. +Maximum entropy inverse reinforcement learning.. In Aaai, Vol. 8. Chicago, IL, +USA, 1433–1438. + diff --git a/W9AzT4oBgHgl3EQfYfyH/content/tmp_files/load_file.txt b/W9AzT4oBgHgl3EQfYfyH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0f9e3a5163607a00c5b79c8c0bc53a3506f47f11 --- /dev/null +++ b/W9AzT4oBgHgl3EQfYfyH/content/tmp_files/load_file.txt @@ -0,0 +1,551 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf,len=550 +page_content='Optimal Decoy Resource Allocation for Proactive Defense in Probabilistic Attack Graphs Haoxiang Ma University of Florida Gainesville, United State hma2@ufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='edu Shuo Han University of Illinois Chicago Chicago, United State hanshuo@uic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='edu Nandi Leslie Raytheon Technologies Arlington County, United State nandi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='leslie@raytheon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='com Charles Kamhoua U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Army Research Laboratory Gainesville, United State charles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='kamhoua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='civ@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='mil Jie Fu University of Florida Gainesville, United State fujie@ufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='edu ABSTRACT This paper investigates the problem of synthesizing proactive de- fense systems in which the defender can allocate deceptive targets and modify the cost of actions for the attacker who aims to com- promise security assets in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' We model the interaction of the attacker and the system using a formal security model– a probabilistic attack graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' By allocating fake targets/decoys, the defender aims to distract the attacker from compromising true tar- gets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' By increasing the cost of some attack actions, the defender aims to discourage the attacker from committing to certain poli- cies and thereby improve the defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' To optimize the defense given limited decoy resources and operational constraints, we for- mulate the synthesis problem as a bi-level optimization problem, while the defender designs the system, in anticipation of the at- tacker’s best response given that the attacker has disinformation about the system due to the use of deception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Though the general formulation with bi-level optimization is NP-hard, we show that under certain assumptions, the problem can be transformed into a constrained optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' We proposed an algorithm to approximately solve this constrained optimization problem us- ing a novel, incentive-design method for projected gradient ascent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' We demonstrate the effectiveness of the proposed method using extensive numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' KEYWORDS Attack Graph, Deception, Markov Decision Process ACM Reference Format: Haoxiang Ma, Shuo Han, Nandi Leslie, Charles Kamhoua, and Jie Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Optimal Decoy Resource Allocation for Proactive Defense in Probabilistic Attack Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' of the 22nd International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2023), London, United Kingdom, May 29 – June 2, 2023, IFAAMAS, 8 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 1 INTRODUCTION Proactive defense refers to a class of defense mechanisms for the defender to detect any ongoing attacks, distract the attacker with deception, or use randomization to increase the difficulty of an attack to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In this paper, we propose a mathematical Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' of the 22nd International Conference on Autonomous Agents and Multiagent Sys- tems (AAMAS 2023), A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Ricci, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Yeoh, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Agmon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' An (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' ), May 29 – June 2, 2023, London, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' © 2023 International Foundation for Autonomous Agents and Multiagent Systems (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='ifaamas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' framework and solution approach for synthesizing a proactive defense system with deception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' We start by formulating the attack planning problem using a probabilistic attack graph, which can be viewed as a Markov de- cision process (MDP) with a set of attack target states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Attack graphs(AGs)[7] can be used in modeling computer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' They are widely used in network security to identify the minimal subset of vulnerability/sensors to be used in order to prevent all known attacks[13, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Probabilistic attack graphs introduce uncertain out- comes of attack actions that account for action failures in a sto- chastic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' For example, in [5, 6], probabilistic transitions in attack graphs capture uncertainties originated from network- based randomization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Under the probabilistic attack graph modeling framework, we investigate how to allocate decoy resources as fake targets to distract the attacker into attacking the fake targets, and how to modify the attack action costs to discourage the attacker from reaching the true targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The joint design of decoy resource allocation and action cost modification can be cast as a bi-level optimization problem, which is generally NP-hard [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Under the assumption that potential de- coy states are predefined and the defender only needs to allocate resources/rewards to decoys, we prove the bi-level optimization can be equivalently expressed as a constrained optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' To solve the constrained optimization problem using a projected gradient ascent efficiently, we build two important relations: First, we show that the projection step of a defender’s desired attack policy to the set of realizable attack policy space can be performed using Inverse Reinforcement Learning (IRL) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Essentially, IRL is to shape the attacker’s perceived reward so that the rational attacker will mimic a strategy chosen by the defender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Second, the gradient ascent step can be performed using policy improvement, which is a subroutine in policy iteration with respect to maximizing the defender’s total reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The project gradient ascent is ensured to converge to a (local) optimal solution to this nonconvex constrained optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The synthesis of proactive defense strategies stud- ied here is closely related to the Stackelberg security game(SSG) (surveyed in [18]) and its solution via bi-level optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In an SSG, the defender is to protect a set of targets with limited re- sources, while the attacker selects the optimal attack strategy given the knowledge of the defender’s strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In [12], the authors study arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='01336v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='MA] 3 Jan 2023 security countermeasure-allocation and use attack graphs to evalu- ate the network’s security given the allocated resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' However, the SSG does not account for the asymmetric information intro- duced by the use of deception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In [20], the authors introduce reward shaping to motivate the agent to behave as the target policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' How- ever, in our setting, the target policy may be infeasible, because the defender aims to lure the attacker to reach a fake target, while the attacker may not intentionally avoid true targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Deceptions create incorrect/incomplete information to the at- tacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In [19], the authors formulate a security game to allocate limited decoy resources to mask a network configuration from the cyber attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The decoy-based deception manipulates the adver- sary’s perception of the payoff matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In [1], the authors study honeypot allocation in deterministic attack graphs and determine the optimal allocation strategy using the minimax theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In [10], the authors study directed acyclic attack graphs that can be modi- fied by the defender using deceptive and protective resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' They propose a mixed-integer linear program (MILP)-based algorithm to determine the allocation of deceptive and protective resources in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In [3], they harden the network by using honeypots so that the attacker can not discriminate between a true target and a fake target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In [11], the authors assign fake edges in the attack graph in order to interdict the attacker and employ MILP to find the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Compared to existing work, our work makes the following con- tributions: First, we do not assume any graph structure in the attack graph and consider probabilistic attack graphs instead of determin- istic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' As the attacker can take a randomized strategy in the probabilistic attack graph, it is not possible to construct a payoff matrix and apply the minimax theorem for decoy resource alloca- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Second, we consider simultaneously allocating limited decoy resources and modifying the cost of attack actions and analyze the best response of the attacker given the disinformation caused by deception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Third, we proposed an efficient incentive-design in- spired algorithm for synthesizing the defense strategy Under the assumption that the attacker is rational and can not distinguish decoys from the true targets, by modifying the action reward and allocating decoy resources properly, we show that it is possible to shape the attacker’s behavior so that the misperceived attacker is incentivized to commit an attack strategy that maximizes the defender’s reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Finally, we test the scalability of our method on different problem sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2 PRELIMINARIES AND PROBLEM FORMULATION Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Let R denote the set of real numbers and R𝑛 the set of real 𝑛-vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Let R𝑛 >0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' R𝑛 <0) be the set of positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' negative) real 𝑛-vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' We use 1 to represent the vector of all ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Given a vector 𝑧 ∈ R𝑛, let 𝑧𝑖 be the 𝑖-th component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Given a finite set 𝑍, the set of probability distributions over 𝑍 is represented as Dist(𝑍).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Given 𝑑 ∈ Dist(𝑍), the support of 𝑑 is denoted as Supp(𝑑) = {𝑧 ∈ 𝑍 | 𝑑(𝑧) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Let 𝐼𝐵 be the indicator function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=', 𝐼𝐵(𝑥) = 1 if 𝑥 ∈ 𝐵, and 𝐼𝐵(𝑥) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' We consider the adversarial interaction between a defender (player 1, pronoun she/her) and an attacker (player 2, pronoun he/him/his) in a system equipped with proactive defense (formally defined later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' We first introduce a formal model, called probabilistic attack graph, to capture how the attacker plans to achieve the attack objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Then, we introduce proactive defense countermeasures with deception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Attack Planning Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The attack planning problem is mod- eled as a probabilistic attack graph, 𝑀 = (𝑆,𝐴, 𝑃,𝜈,𝛾, 𝐹, 𝑅2), where 𝑆 is a set of states (nodes in the attack graph), 𝐴 is a set of attack actions, 𝑃 : 𝑆 × 𝐴 → Dist(𝑆) is a probabilistic transition function such that 𝑃(𝑠′|𝑠,𝑎) is the probability of reaching state 𝑠′ given action 𝑎 being taken at state 𝑠, 𝜈 ∈ Dist(𝑆) is the initial state distribution, 𝛾 ∈ (0, 1] is a discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The attack’s objective is described by a set 𝐹 of target states and a target reward function 𝑅2 : 𝐹 × 𝐴 → R≥0, which assigns each state-action pair (𝑠,𝑎) where 𝑠 ∈ 𝐹 and 𝑎 ∈ 𝐴 to a nonnegative value of reaching that target for the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The reward function can be extended to the entire state space by defining 𝑅2(𝑠,𝑎) = 0 for any 𝑠 ∈ 𝑆 \\ 𝐹,𝑎 ∈ 𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' To capture the termination of attacks, we introduce a unique sink state 𝑠sink ∈ 𝑆 \\ 𝐹 such that 𝑃(𝑠sink|𝑠sink,𝑎) = 1 for all 𝑎 ∈ 𝐴 and 𝑃(𝑠sink|𝑠,𝑎) = 1 for any target 𝑠 ∈ 𝐹 and 𝑎 ∈ 𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The probabilistic attack graph characterizes goal-directed attacks encountered in cyber security [8, 14], in which by reaching a tar- get state, the attacker compromises certain critical network hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Probabilistic attack graphs [10, 17] capture the uncertain outcomes of the attack actions using the probabilistic transition function and generalize deterministic attack graphs [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The attacker is to maximize his discounted total reward, starting from the initial state 𝑆0 ∼ 𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' A randomized, finite-memory attack policy is a function 𝜋 : 𝑆∗ → Dist(𝐴), which maps a finite run 𝜌 ∈ 𝑆∗ into a distribution 𝜋(𝜌) over actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' A policy is called Markovian if it only depends on the most recent state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=', 𝜋 : 𝑆 → Dist(𝐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' We only consider Markovian policies because it suffices to search within Markovian policies for an optimal attack policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Let (Ω, F ) be the canonical sample space for (𝑆0,𝐴0, (𝑆𝑡,𝐴𝑡)𝑡>1) with the Borel 𝜎-algebra F = B(Ω) and Ω = 𝑆 × 𝐴 × �∞ 𝑡=1(𝑆 × 𝐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The probability measure Pr𝜋 on (Ω, F ) induced by a Markov policy 𝜋 satisfies: Pr𝜋 (𝑆0 = 𝑠) = 𝜇0(𝑠), Pr𝜋 (𝐴0 = 𝑎 | 𝑆0 = 𝑠) = 𝜋(𝑠,𝑎), and Pr𝜋 (𝑆𝑡 = 𝑠 | (𝑆𝑘,𝐴𝑘)𝑘<𝑡) = 𝑃(𝑠 | 𝑆𝑘,𝐴𝑘), and Pr𝜋 (𝐴𝑡 = 𝑎 | (𝑆𝑘,𝐴𝑘)𝑘<𝑡,𝑆𝑡) = 𝜋(𝑆𝑡,𝑎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Given a Markovian policy 𝜋 : 𝑆 → Dist(𝐴), we define the at- tacker’s value function 𝑉 𝜋 2 : 𝑆 → R as 𝑉 𝜋 2 (𝑠) = E𝜋 [ ∞ ∑︁ 𝑘=0 𝛾𝑘𝑅2(𝑆𝑘,𝐴𝑘)|𝑆0 = 𝑠], where E𝜋 is the expectation given the probability measure Pr𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Proactive Defense with Deception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' We assume that the defender knows the attacker’s objective given by the tuple ⟨𝐹, 𝑅2⟩, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=', the target states and target reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The defender’s proactive defense mechanisms are the following: Defend by deception: The defender employs a deception method called “revealing the fake”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Specifically, the defender has a set 𝐷 ⊂ 𝑆 \\ 𝐹 of states in the MDP 𝑀 that can be set to be fake target states with fake target rewards �𝑦 ∈ R|𝐷 |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The attacker cannot distinguish the real targets 𝐹 from fake targets 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Defend by state-action reward modification: The defender has a set𝑊 ⊂ (𝑆\\(𝐹∪𝐷))×𝐴 of state action pairs in the MDP 𝑀 whose reward can be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Once the reward of the state action pair (𝑠,𝑎) is modified, the attacker’s perceived reward 𝑅2(𝑠,𝑎) < 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=', the cost of attack action 𝑎 at state 𝑠 is −𝑅2(𝑠,𝑎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The defender can determine how to allocate her decoy resource and limited state-action reward modification ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Definition 1 (Decoy allocation under constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The defender’s decoy allocation design is a nonnegative real-valued vector �𝑦 ∈ R|𝑆 | ≥0 satisfying �𝑦(𝑠) = 0 for any 𝑠 ∈ 𝑆 \\𝐷 and constrained by 1T�𝑦 ≤ ℎ for some ℎ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Given a decoy allocation �𝑦, the attacker’s perceptual reward function is defined by 𝑅 �𝑦 2 (𝑠,𝑎) = � �𝑦(𝑠) if �𝑦(𝑠) > 0, 𝑅2(𝑠,𝑎) if �𝑦(𝑠) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Definition 2 (Action reward modification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Given a set 𝑊 ⊂ (𝑆 \\ (𝐹 ∪ 𝐷)) × 𝐴, the defender’s action reward modification is a nonpositive reward-valued vector �𝑥 ∈ R|𝑆×𝐴| ≤0 satisfying �𝑥(𝑠,𝑎) = 0 for any (𝑠,𝑎) ∉ 𝑊 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Given an action reward modification �𝑥, the attacker’s perceptual reward function is defined by 𝑅 �𝑥 2 (𝑠,𝑎) = � �𝑥(𝑠,𝑎) if �𝑥(𝑠,𝑎) < 0, 𝑅2(𝑠,𝑎) if �𝑥(𝑠,𝑎) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Note that the defender does not consider modifying the state- action reward for (fake or real) target states 𝐹 ∪ 𝐷 because once a state in 𝐹 ∪ 𝐷 is reached, the attack is terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The defender’s proactive defense strategy is a tu- ple (�𝑥, �𝑦) including an action reward modification �𝑥 and a decoy allocation design �𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Because the action reward modification is independent of the decoy allocation design, the reward function given a defender’s strategy (�𝑥, �𝑦) is the composition of 𝑅 �𝑥 2 and 𝑅 �𝑦 2 and thus omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The attack process terminates under two cases: Either the attack succeeds, in which the attacker reaches a target 𝑠 ∈ 𝐹, or the attack is interdicted, in which the attacker reaches a state allocated with a decoy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Our problem can be informally stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In the attack planning scenario we mentioned above, determine the defender’s strategy to allocate decoy resources and modify action reward so as to maximize the probability that the attacker reaches a fake target given the best response of the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 3 MAIN RESULTS In this section, we first define the attacker’s perceptual planning problem for a fixed action reward modification and decoy resource allocation (�𝑥, �𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Then we show that the design of the proactive defense can be formulated as a bi-level optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' We investigate the special property of the formulated bi-level opti- mization problem to develop an optimization-based approach for synthesizing the proactive defense strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='1 A Bi-level Optimization Formulation The defender’s strategy changes how the attacker perceives the attack planning problem as follows: Definition 4 (Perceptual attack planning problem with modified reward and decoys).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Let the action reward modification be �𝑥 and decoy allocation be �𝑦, and the attacker’s original planning problem 𝑀 = (𝑆,𝐴, 𝑃,𝜈,𝛾, 𝐹, 𝑅2), the perceptual planning problem of the attacker is defined by the following MDP with terminating states: 𝑀(�𝑥, �𝑦) = (𝑆,𝐴, 𝑃 �𝑦,𝜈,𝛾, 𝐹 ∪ 𝐷 �𝑦, 𝑅 �𝑥, �𝑦 2 ), where 𝑆,𝐴,𝜈,𝛾 are the same as those in 𝑀, 𝐷 �𝑦 = {𝑠 ∈ 𝐷 | �𝑦(𝑠) ≠ 0} are decoy target states and absorbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The transition function 𝑃 �𝑦 is obtained from the original transition function 𝑃 by only making all states in 𝐷 �𝑦 absorbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The reward 𝑅 �𝑥, �𝑦 2 is defined based on Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 1 and Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The perceptual value for the attacker is 𝑉 𝜋 2 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑥, �𝑦) = E𝜋 � ∞ ∑︁ 𝑘=0 𝛾𝑘𝑅 �𝑥, �𝑦 2 (𝑆𝑘,𝐴𝑘) | 𝑆0 ∼ 𝜈 � , where E𝜋 is the expectation given the probability measure Pr𝜋 in duced by 𝜋 from the MDP 𝑀(�𝑥, �𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The defender’s deception objective is given by a reward function 𝑅 �𝑦 1 : 𝑆 → R, defined by 𝑅 �𝑦 1 (𝑠) = � 1 if 𝑠 ∈ 𝐷 �𝑦, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' (1) Given the probability measure Pr𝜋, we denote the defender’s value by 𝑉 𝜋 1 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑦) = E𝜋 � ∞ ∑︁ 𝑘=0 𝛾𝑘𝑅1(𝑆𝑘) | 𝑆0 ∼ 𝜈 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' With this reward definition, the value 𝑉 𝜋 1 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑦) is the probability of the attacker reaching a fake target in 𝐷 �𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' To formalize the deception objective, we introduce the notion of a defender’s preferred attack policy as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Definition 5 (A defender’s preferred attack policy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Given the perceptual planning problem of the attacker 𝑀(�𝑥, �𝑦) where (�𝑥, �𝑦) is a fixed proactive defense strategy, let 𝜋 and 𝜋 ′ be two attack policies that achieve the same value for the attacker, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=', 𝑉 𝜋 2 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑥, �𝑦) = 𝑉 𝜋′ 2 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑥, �𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Policy 𝜋 is strictly preferred to 𝜋 ′ by the defender if and only if 𝑉 𝜋 1 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑦) > 𝑉 𝜋′ 1 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In words, if two policies are equally good for the attacker, the one with a higher probability to reach a fake target is preferred by the defender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Then the problem of synthesizing an optimal proactive defense strategy (�𝑥, �𝑦) can be mathematically formulated as Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑥 ∈𝑋, �𝑦∈𝑌 𝑉 𝜋∗ 1 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑦) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 𝜋∗ ∈ argmax 𝜋 𝑉 𝜋 2 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑥, �𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' where 𝑋 = R|𝑊 | ≤0 and 𝑌 = {�𝑦 | ∀𝑠 ∈ 𝑆 \\ 𝐷, �𝑦(𝑠) = 0 and 1T�𝑦 ≤ ℎ} are the ranges for variables �𝑥 and �𝑦 correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In words, the defender decides (�𝑥, �𝑦) so that the attacker’s best response in his perceptual attack planning problem turns out to be an attack policy most preferred by the defender, as it maximizes the defender’s value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='2 Transforming into a Constrained Optimization Problem The bi-level optimization problem is known to be strongly NP-hard [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' However, under certain conditions, the bi-level optimization problem can be shown to be equivalent to a constrained optimiza- tion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Let Π(�𝑥, �𝑦) = {𝜋 | 𝑉 𝜋 2 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑥, �𝑦) = max𝜋 𝑉 𝜋 2 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑥, �𝑦)} , which is the set of optimal policies in the attacker’s perceived planning problem with respect to a choice of variables �𝑥 and �𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The bi-level optimization problem is then equivalently written as the following constrained optimization problem: max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 𝜋∗,�𝑥 ∈𝑋, �𝑦∈𝑌 𝑉 𝜋∗ 1 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑦) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 𝜋∗ ∈ Π(�𝑥, �𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' (2) This, in turn, is equivalent to max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 𝜋∗ 𝑉 𝜋∗ 1 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑦) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 𝜋∗ ∈ � �𝑥 ∈𝑋, �𝑦∈𝑌 Π(�𝑥, �𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' (3) Here, the constraint means the attacker’s response 𝜋∗ can be se- lected from the collection of optimal attack policies given all possi- ble values for �𝑥, �𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' By the definition of the defender’s value function, it is noted that 𝑉 𝜋 1 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑦) does not depend on the exact value of �𝑦 but only depends on whether �𝑦(𝑠) > 0 for each state 𝑠 ∈ 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Formally, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' For any �𝑦1, �𝑦2 ∈ 𝑌, if �𝑦1(𝑠) = 0 =⇒ �𝑦2(𝑠) = 0 and vice versa, then 𝑉 𝜋 1 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑦1) = 𝑉 𝜋 1 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑦2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Given two different vectors �𝑦1 and �𝑦2, we can construct two MDPs: 𝑀1 � 𝑀(�𝑥, �𝑦1) = (𝑆,𝐴, 𝑃 �𝑦1,𝜈,𝛾, 𝐹, 𝑅1) and 𝑀2 � 𝑀(�𝑥, �𝑦2) = (𝑆,𝐴, 𝑃 �𝑦2,𝜈,𝛾, 𝐹, 𝑅1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' If �𝑦1(𝑠) = 0 if and only if �𝑦2(𝑠) = 0, then the transition functions 𝑃 �𝑦1 of 𝑀1 and 𝑃 �𝑦2 of 𝑀2 are the same (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Further, the defender’s reward function 𝑅 �𝑦1 1 also equals to 𝑅 �𝑦2 1 (see (1)), given both the transition dynamics and reward are the same, we have 𝑉 𝜋 1 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑦1) = 𝑉 𝜋 1 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑦2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' □ Next, to remove the dependency of 𝑉 𝜋 1 (𝜈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑦) on �𝑦, we make the following assumption: Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The set 𝐷 �𝑦 = {𝑠 ∈ 𝐷 | �𝑦(𝑠) ≠ 0} of states where decoys are allocated is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Under this assumption, we simply assume all states in the given set 𝐷 have to be assigned with nonzero decoy resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' That is 𝐷 �𝑦 = 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' This assumption further reduces the defender’s synthesis prob- lem into a constrained optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 𝜋∗ 𝑉 𝜋∗ 1 (𝜈) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 𝜋∗ ∈ Π ≜ � �𝑦∈𝑌,�𝑥 ∈𝑋 Π(�𝑥, �𝑦), �𝑦(𝑠) > 0, ∀𝑠 ∈ 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' (4) Because the above problem is a standard constrained optimiza- tion problem, one can obtain a locally optimal solution using the projected gradient method: 𝜋𝑘+1 = projΠ (𝜋𝑘 + 𝜂∇𝑉 𝜋𝑘 1 (𝜈)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' where projΠ (𝜋) denotes projecting policy 𝜋 onto the policy space Π and 𝜂 is the step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='3 Connecting Inverse-reinforcement Learning with Project Gradient Ascent A key step in performing projected gradient ascent is to evaluate, for any policy ˆ𝜋, the projection projΠ ( ˆ𝜋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' However, this is nontrivial because the set ¯Π includes a set of attack policies, each of which corresponds to a choice of vectors (�𝑥, �𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' As a result, ¯Π does not have a compact representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Next, we propose a novel algorithm that computes the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' First, by the definition of projection, it is noted that this pro- jection step is equivalent to solving the following optimization problem: min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 𝜋 D(𝜋, ˆ𝜋) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 𝜋 ∈ Π, �𝑦(𝑠) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='∀𝑠 ∈ 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' (5) where D(𝜋, ˆ𝜋) is the distance between the two policies 𝜋, ˆ𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The distance function D can be chosen to be the Kullback–Leibler (KL)-divergence between policy-induced Markov chains, defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Given an MDP 𝑀 = (𝑆,𝐴, 𝑃,𝜈) and two Markovian policies 𝜋1, 𝜋2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Let 𝑀𝜋1 = (𝑆, 𝑃1,𝜈) and 𝑀𝜋2 = (𝑆, 𝑃2,𝜈) be two Markov chains induced from 𝑀 under 𝜋1 and 𝜋2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The KL divergence DKL �𝑀𝜋1 ∥𝑀𝜋2 � (relative entropy from 𝑀𝜋2 to 𝑀𝜋1) is defined by DKL �𝑀𝜋1 ∥𝑀𝜋2 � = ∑︁ 𝜌 ∈𝑆∗ Pr1(𝜌) log Pr1(𝜌) Pr2(𝜌) , where Pr𝑖 (𝜌) is the probability of a path 𝜌 in the Markov chain 𝑀𝜋𝑖 for 𝑖 = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The KL divergence in (5) can be expressed as DKL �𝑀𝜋 (�𝑥, �𝑦)∥𝑀�𝜋 (�𝑥, �𝑦)� = ∑︁ 𝜌 � Pr(𝜌) log � Pr(𝜌) Pr(𝜌|�𝑥, �𝑦) = ∑︁ 𝜌 � Pr(𝜌) log � Pr(𝜌) − ∑︁ 𝜌 � Pr(𝜌) log Pr(𝜌|�𝑥, �𝑦), (6) where � Pr(𝜌) is the probability of path 𝜌 in the Markov chain 𝑀�𝜋 (�𝑥, �𝑦), and Pr(𝜌|�𝑦) is the probability of path 𝜌 in the Markov chain 𝑀𝜋 (�𝑥, �𝑦) induced by a policy 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Because the first term in the sum in (6) is a constant for ˆ𝜋 is fixed, the KL divergence minimization problem is equivalent to the following maximization problem: max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑥 ∈𝑋, �𝑦∈𝑌 ∑︁ 𝜌 � Pr(𝜌) log Pr(𝜌|�𝑥, �𝑦) (7) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' �𝑦(𝑠) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='∀𝑠 ∈ 𝐷, (8) 1T�𝑦 ≤ ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' (9) Problem (7) can be solved by an extension of the Maximum Entropy (MAXENT) IRL algorithm [21], which was originally developed in the absence of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' It is well-known that IRL is to infer, from the expert demonstration, a reward function for which the expert policy generating the demonstrations is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The use of IRL to perform the projection is intuitively understood as follows: The goal is to compute a pair of vectors (�𝑥, �𝑦) that alters the attacker’s perceived reward function so that the attacker’s optimal policy given (�𝑥, �𝑦) is closed to the “expert policy” ˆ𝜋, under the constraints of �𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' To handle the decoy resource constraint (9), we approximate the constraint using a logarithmic barrier function and compute the optimal solution �𝑦∗ using gradient-based numerical optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Considering the constraint 1T�𝑦 ≤ ℎ, we implement the barrier function in order to approximate the inequality constraints and rewrite the optimization problem as: max �𝑥, �𝑦 ∑︁ 𝜌 � Pr(𝜌) log Pr(𝜌|�𝑥, �𝑦) + 1 𝑡 log(ℎ − 1T�𝑦) subject to: �𝑦(𝑠) = 0, ∀𝑠 ∈ 𝑆 \\ 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' where 𝑡 is the weighting parameter of the logarithmic barrier func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In our experiment, 𝑡 is fixed to be 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Since constraint �𝑦(𝑠) = 0, ∀𝑠 ∈ 𝑆 \\𝐷, can be incorporated into the domain of decision variables �𝑦, we can use gradient ascent to obtain the optimal �𝑥∗, �𝑦∗ that maximizes the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Specifi- cally, �𝑥 and �𝑦 can be updated via �𝑥𝑘+1 = projX (�𝑥𝑘 + 𝜂𝑥∇𝐿(�𝑥, �𝑦)), �𝑦𝑘+1 = projY (�𝑦𝑘 + 𝜂𝑦∇𝐿(�𝑥, �𝑦)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='4 Policy Improvement for Gradient Ascent Step After the projection step to obtain a policy 𝜋𝑘 and the corresponding vector (�𝑥, �𝑦), we aim to compute a one-step gradient ascent to improve the objective function’s value 𝑉 𝑘+1 1 (𝜈) = 𝑉 𝑘 1 (𝜈) + ∇𝑉 𝑘 1 (𝜈), where 𝑉 𝑘 1 (𝜈) is the defender’s value evaluated given the attack policy 𝜋𝑘 at the 𝑘-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' For this step, we perform a policy improvement step with respect to the defender’s reward function 𝑅 �𝑦 1 , which now is independent of �𝑦 because the set 𝐷 �𝑦 is fixed to be a constant set 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' It is shown in [9, 15] that policy improvement is a one-step Newton update of optimizing the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Specifically, the policy improvement is to compute ˜𝜋𝑘+1(𝑠,𝑎) = exp ((𝑅1(𝑠,𝑎) + 𝛾𝑉 𝑘 1 (𝑠′))/𝜏) � 𝑎∈𝐴 exp ((𝑅1(𝑠,𝑎) + 𝛾𝑉 𝑘 1 (𝑠′))/𝜏) , The policy at iteration 𝑘 + 1 is obtained by performing the pro- jection step ((5)) in which ˆ𝜋 ≜ ˜𝜋𝑘+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The iteration stops when |𝑉 𝑘+1 1 (𝜈) − 𝑉 𝑘 1 (𝜈)| ≤ 𝜖 where 𝜖 is a manually defined threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The output yields a tuple (�𝑥∗, �𝑦∗) which is the (local) optimal proactive defense strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' We can only obtain a local optimal proactive defense strategy here due to the transferred constrained optimization problem having a nonconvex constraint set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' However, we can start from different initial policies and select the best one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Moreover, assume the defender is solving her own problem without considering attacker’s objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' the upper bound of the defender’s objective can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' We can select the solution whose objective function is closest to the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In our problem, we assume the set 𝐷 is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' If the set 𝐷 is not given, then this problem becomes combinatorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' If the set 𝐷 is not given but to be determined from a candidate set of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Then a naive approach is to enumerate all possible combinations and evaluate the defender’s value for every subset and select the one that yields the highest defender’s value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' It would be interesting to examine if the combinatorial problem is sub-modular or super- modular, but it is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In summary, the proposed algorithm starts with an initial pol- icy ˜𝜋0, and use the IRL to find the projection 𝜋0 as well as their corresponding vectors (�𝑥0, �𝑦0) that shape the attacker’s perceptual reward function for which 𝜋0 is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Then a policy improve- ment is performed to update 𝜋0 to ˜𝜋1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' By alternating the projection and policy improvement, the process terminates until the stopping criteria |𝑉 𝑘+1 1 (𝜈) − 𝑉 𝑘 1 (𝜈)| ≤ 𝜖 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 4 EXPERIMENT We illustrate the proposed methods with two sets of examples, one is a probabilistic attack graph and another is an attack planning problem formulated in a stochastic gridworld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' For all case studies, the workstation used is powered by Intel i7-11700K and 32GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 0 start 2 3 1 4 5 6 7 8 9 10 13 12 11 Figure 1: A probabilistic attack graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Figure 1 shows a probabilistic attack graph with the target set 𝐹 = {10} and the action set {𝑎,𝑏,𝑐,𝑑}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' For clarity, the graph only shows the transition given action 𝑎 where a thick (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' thin) arrow represents a high (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' low) transition probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' For example, 𝑃(0,𝑎) = {1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='7, 2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='1, 3 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='1, 4 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='1} 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Consider the set D = {11, 13} of decoy states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Recall the defender’s reward function is 𝑅1(𝑠) = 1, for all 𝑠 ∈ 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Assuming no resource is 1The exact transition function is provided in the supplementary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' allocated to 𝐷 and all states in 𝐷 are sink states, then the attacker has a 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='33% probability of reaching the target set 𝐹 from the initial state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In the meantime, the defender’s expected value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' That is, with probability 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='9%, the attacker will reach a decoy state in 𝐷 and the attack is terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Given limited resource 1T�𝑦 ≤ 3, the decoy resource allocation yields �𝑦(11) = �𝑦(13) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Based on the given decoy resource allocation, the attacker has an 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='63% probability of reaching the target set 𝐹 and the defender’s expected reward is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='653 at initial state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Thus, by assigning resources to decoys to attract the attacker, the defender reduces the attacker’s probability of reaching the target state significantly (85% reduction) and improves the defender’s value by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='38 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Figure 2: 6 × 6 gridworld example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Next, we consider a robot motion planning problem in a stochas- tic 6 × 6 gridworld shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The attacker/robot aims to reach a set of goal states while avoiding detection from the defender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The attacker can move in four compass directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Given an action, say, “N”, the attacker enters the intended cell with 1−2𝛼 probability, and enters the neighboring cells, which are west and east cells with 𝛼 probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In our experiments, 𝛼 is selected to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' A state (𝑖, 𝑗) means the cell at row 𝑖 and column 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The defender has deployed sensors shown in Figure 2 to detect the presence of an attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Thus, once the attacker enters a sensor state, his task fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The decoy set 𝐷 is given as blue cells and the target set 𝐹 is given as green cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Given the initial state is at (2, 0), which is indicated by the ro- bot in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' We test the following three scenarios: No de- coy resource allocation, decoy resource allocation only, decoy re- source allocation together with reward modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The result is shown in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' When we do not allocate resources to decoys, the attacker has a 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='98% probability of reaching the target set 𝐹 while avoiding sensor states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' And the defender’s expected value is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='56 × 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' When the defender is allowed to allocate resources with a total budget of 4 to decoys, the decoy resource allocation yields �𝑦((1, 4)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='016, �𝑦((4, 5)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The defender does not spend all decoy resources because of the use of the logarithmic barrier function to enforce the constraint, when it is close to the upper bound, the log barrier function will work as a large penalty in gradient ascent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Under the given resource allocation, the attacker has a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='9% probability of reaching the target set 𝐹, and the defender’s expected value at the initial state is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='3877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In the decoy resource allocation (a) Converge using different initial policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' (b) Converge with decoy resource allocation and action reward modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Figure 3: Defender’s value converge trend in 6 × 6 gridworld example given 𝐷 = {(1, 4), (4, 5)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Figure 4: 10 × 10 gridworld example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' and action reward modification experiment, the defender is allowed to modify all action rewards at state (4, 4) and the action ‘N’ reward at state (4, 0), (4, 1) and (4, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' It turns out the defender allocates 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='938 to decoy (1, 4) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='734 to decoy (2, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Meanwhile, action ‘N’ reward at (4, 0) is modified to −1 and the same action at (4, 1) is (5,5) (0,0)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content="30 val Defender's 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='05 Initial policy 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='00 Initial policy 2 2 3 5 Iteration0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content="35 value 0E0 Defender's 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='05 Initial policy 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content="00 Initial policy 2 1 2 Fm 4 5 Iteration(6'6) (0,0)Figure 5: Defender’s value converge trend in 10 × 10 grid- world." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' modified to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='94 and the action "N" reward at (4, 2) is modified to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='904, the defender will also modify the action reward of "W", "S", "N" at (4, 4) to −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Compare the decoy resource allocation result with the decoy resource allocation and action reward modification result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' We find that by allowing action reward modification, the defender reduces the attacker’s probability of reaching the target (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='13% reduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In the meantime, the defender’s expected value increases by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='62%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' It is noted that due to the nonlinearity in the optimization prob- lem, the algorithm converges to different solutions under different initial conditions, as shown in Figures 3a and 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In the figures, the initial policy 1 is generated by assuming the attacker receives the reward of 1 if he reaches the decoy and receives a reward of 0 when he reaches the target state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' This is ideal for the defender’s objective but is infeasible for the optimization problem because in the attacker’s perceptual planning problem, reaching the true target will always provide a reward of 1 regardless of how many resources are allocated to decoys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The initial policy 2 is randomly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In this experiment, the value of the objective function given different initial policies is close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In order to test how the decoy set 𝐷 influences the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' We re- allocate the position of decoys to {(0, 2), (5, 3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The result is shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Based on the new configuration, if we do not allocate decoy resources, the attacker reaches the target set with 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='97% probability and the defender’s value is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='61×10−8 at the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' If the defender can allocate resources to the decoys, our method yields �𝑦((0, 2)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='141 and �𝑦((5, 3)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The attacker’s probabil- ity of reaching the target set is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='99% and the defender’s expected value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='6991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' If the defender is allowed to modify the same set of state-action rewards as she is in the previous example, our algorithm yields �𝑦((0, 2)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='985 and �𝑦((5, 3)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Action ‘N’ reward at (4, 0) is modified to −1 and the same action at (4, 1) is modified to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='85 and the action "N" reward at (4, 2) is modified to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='081, the defender will also modify the all action reward at (4, 4) to −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Under this configuration, the attacker’s probability of reaching the target set is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='286% (93% reduction compared to only allocating decoy resources) and the defender’s expected value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='7301 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='4% increase compared to only allocate decoy resources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' By changing the configuration of set 𝐷, we show that the configuration of set 𝐷 influences the attacker’s probability of reaching the target set and the defender’s expected value: the second set 𝐷 = {(0, 2), (5, 3)} appears to outperform the first set 𝐷 = {(1, 4), (4, 5)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Table 1: Experiment result in 6 × 6 gridworld given 𝐷 = {(1, 4), (4, 5)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' No decoy Decoy only Decoy and action reward Attacker’s value 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='98% 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='9% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='6% Defender’s value 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='56 × 10−6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='3877 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='394 Table 2: Experiment result in 6 × 6 gridworld given 𝐷 = {(0, 2), (5, 3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' No decoy Decoy only Decoy and action reward Attacker’s value 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='97% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='99% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='286% Defender’s value 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='61 ×10−8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='6991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='7301 Next, in order to test the scalability, we increase the gridworld size to 10 × 10 as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In the large gridworld example, we only do decoy resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The sensors, decoy set, and target set are represented using the same notation as the 6 × 6 gridworld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The defender’s reward function is still 𝑅1(𝑠) = 1, for all 𝑠 ∈ 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Assume the initial state is at (5, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' When the defender does not allocate decoy resources, the attacker’s probability of reaching the target is 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='43% and the defender’s expected value at the initial state is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='0024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' When the defender is allowed to allocate resources to decoys, our algorithm yields �𝑦((2, 8)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='350, �𝑦((6, 8)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Under the given decoy resources, the attacker’s probability of reach- ing the target decreases to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='1% (94% reduction), and the defender’s expected value at the initial state increases to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='4034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' We also test the defender’s converging trend using different initial policies as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Initial policy 1 is obtained similarly to initial policy 1 in the 6 × 6 example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Initial policy 2 and 3 are randomly generated policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' From Figure 5, we observe that different initial policies result in a similar converged value for the objective func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Considering the scalability of our algorithm, the computation time for the 10 × 10 gridworld example is 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='94 seconds, while the computation time of the 6×6 example is 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='51 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The running time shows our algorithm can be extended to moderate problem sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' It is noted that not only the state space size influences the running time but also the selection of decoys, the number of decoys influences the running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 5 CONCLUSION AND FUTURE WORK We present a mathematical framework and algorithm for decoy allocation and reward modification in a proactive defense system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Our technical approach can be applied to many safety-critical sys- tems where the probabilistic attack graphs are constructed from known vulnerabilities in a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' The formulation and solutions can be extended to a broad set of adversarial interactions in which proactive defense with deception can be deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In the future, we will consider more complex attack and defense objectives and inves- tigate the decoy allocation given the uncertainty in the attacker’s goal or capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Apart from “revealing the fake” studied herein, we will also investigate how to “conceal the truth” by manipulating the attacker’s perceptual reward of compromising true targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='40 lue 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='35 val 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content="30 Defender's 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='10 Initial policy 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='05 Initial policy 2 + Initial policy 3 1 2 E 4 5 6 7 IterationREFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Anwar, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Kamhoua, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Leslie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Honeypot Allocation over Attack Graphs in Cyber Deception Games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In 2020 International Conference on Computing, Networking and Communications (ICNC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 502–506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' [2] Stephan Dempe and Alain Zemkoho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Bilevel optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' [3] Karel Durkota, Viliam Lis`y, Branislav Bošansk`y, and Christopher Kiekintveld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Optimal network security hardening using attack graph games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In Twenty- Fourth International Joint Conference on Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' [4] Pierre Hansen, Brigitte Jaumard, and Gilles Savard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' New branch-and-bound rules for linear bilevel programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' SIAM Journal on scientific and Statistical Computing 13, 5 (1992), 1194–1217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' [5] Jin Hong and Dong-Seong Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' HARMs: Hierarchical Attack Representa- tion Models for Network Security Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In Australian Information Security Management Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' SRI Security Research Institute, Edith Cowan University, Perth, Western Australia, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' [6] Jin B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Hong and Dong Seong Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Assessing the Effectiveness of Moving Target Defenses Using Security Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' IEEE Transactions on Dependable and Secure Computing 13, 2 (March 2016), 163–177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Jha, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Sheyner, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Two Formal Analyses of Attack Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In Proceedings 15th IEEE Computer Security Foundations Workshop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' CSFW-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 49–63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' [8] Harjinder Singh Lallie, Kurt Debattista, and Jay Bal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' A review of attack graph and attack tree visual syntax in cyber security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Computer Science Review 35 (Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2020), 100219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='cosrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='100219 [9] Omid Madani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' On Policy Iteration as a Newton’s Method and Polyno- mial Policy Iteration Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In Eighteenth National Conference on Artificial Intelligence (Edmonton, Alberta, Canada).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' American Association for Artificial Intelligence, USA, 273–278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' [10] Stephanie Milani, Weiran Shen, Kevin S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Chan, Sridhar Venkatesan, Nandi O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Leslie, Charles Kamhoua, and Fei Fang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Harnessing the Power of Decep- tion in Attack Graph-Based Security Games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In Decision and Game Theory for Security (Lecture Notes in Computer Science), Quanyan Zhu, John S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Baras, Radha Poovendran, and Juntao Chen (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Springer International Publishing, Cham, 147–167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='1007/978-3-030-64793-3_8 [11] Stephanie Milani, Weiran Shen, Kevin S Chan, Sridhar Venkatesan, Nandi O Leslie, Charles Kamhoua, and Fei Fang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Harnessing the power of deception in attack graph-based security games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In International Conference on Decision and Game Theory for Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Springer, 147–167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' [12] Thanh H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Nguyen, Mason Wright, Michael P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Wellman, and Satinder Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Multistage Attack Graph Security Games: Heuristic Strategies, with Empirical Game-Theoretic Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Security and Communication Networks 2018 (Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2018), 1–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' [13] Steven Noel and Sushil Jajodia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Optimal ids sensor placement and alert prioritization using attack graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Journal of Network and Systems Management 16, 3 (2008), 259–275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' [14] Steven Noel, Sushil Jajodia, Lingyu Wang, and Anoop Singhal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Measuring security risk of networks using attack graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' International Journal of Next- Generation Computing 1, 1 (2010), 135–147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' [15] Martin L Puterman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Markov decision processes: discrete stochastic dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' John Wiley & Sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' [16] Oleg Sheyner, Joshua Haines, Somesh Jha, Richard Lippmann, and Jeannette M Wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Automated generation and analysis of attack graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In Proceedings 2002 IEEE Symposium on Security and Privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' IEEE, 273–284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' [17] Anoop Singhal and Xinming Ou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Security risk analysis of enterprise networks using probabilistic attack graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In Network Security Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Springer, 53–73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' [18] Arunesh SINHA, Fei FANG, Bo AN, Christopher KIEKINTVELD, and Milind TAMBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Stackelberg Security Games: Looking beyond a Decade of Success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence (IJCAI-18),Stockholm, Sweden, July 13-19 (July 2018), 5494–5501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' [19] Omkar Thakoor, Milind Tambe, Phebe Vayanos, Haifeng Xu, Christopher Kiek- intveld, and Fei Fang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Cyber Camouflage Games for Strategic Deception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' In Decision and Game Theory for Security (Lecture Notes in Computer Science), Tansu Alpcan, Yevgeniy Vorobeychik, John S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Baras, and György Dán (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Springer International Publishing, Cham, 525–541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='1007/978-3-030- 32430-8_31 [20] Guojun Wu, Yanhua Li, Zhenming Liu, Jie Bao, Yu Zheng, Jieping Ye, and Jun Luo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Reward Advancement: Transforming Policy under Maximum Causal Entropy Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' arXiv preprint arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='05390 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' [21] Brian D Ziebart, Andrew L Maas, J Andrew Bagnell, Anind K Dey, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Maximum entropy inverse reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content='. In Aaai, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} +page_content=' Chicago, IL, USA, 1433–1438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf'} diff --git a/WNAzT4oBgHgl3EQfmP1C/content/2301.01559v1.pdf b/WNAzT4oBgHgl3EQfmP1C/content/2301.01559v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9395874f56589715feadbcefa4a6a737fb62535f --- /dev/null +++ b/WNAzT4oBgHgl3EQfmP1C/content/2301.01559v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9f5e114b54ddb741ebdfd8545fd37c4773c82a3ee362bdbf6b3e734a235eb5d9 +size 4053853 diff --git a/WNE0T4oBgHgl3EQfmAF1/content/tmp_files/2301.02493v1.pdf.txt b/WNE0T4oBgHgl3EQfmAF1/content/tmp_files/2301.02493v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d750a706790afaec50acc9ab050c3cbdf9140ecd --- /dev/null +++ b/WNE0T4oBgHgl3EQfmAF1/content/tmp_files/2301.02493v1.pdf.txt @@ -0,0 +1,717 @@ +Muon Beam for Neutrino CP Violation: connecting energy and neutrino frontiers +Alim Ruzi,∗ Tianyi Yang,† Dawei Fu,‡ Sitian Qian,§ Leyun Gao, and Qiang Li¶ +State Key Laboratory of Nuclear Physics and Technology, +School of Physics, Peking University, Beijing, 100871, China +We propose here a proposal to connect neutrino and energy frontiers, by exploiting collimated +muon beams for neutrino oscillations, which generate symmetric neutrino and antineutrino sources: +µ+ → e+ ¯νµ νe and µ− → e− νµ ¯νe. +Interfacing with long baseline neutrino detectors such as +DUNE and T2K, this experiment can be applicable to measure tau neutrino properties, and also to +probe neutrino CP phase, by measuring muon electron (anti-)neutrino mixing or tau (anti-)neutrino +appearance, and differences between neutrino and antineutrino rates. There are several significant +benefits leading to large neutrino flux and high sensitivity on CP phase, including 1) collimated and +manipulable muon beams, which lead to a larger acceptance of neutrino sources in the far detector +side; 2) symmetric µ+ and µ− beams, and thus symmetric neutrino and antineutrino sources, which +make this proposal ideally useful for measuring neutrino CP violation. More importantly, ¯νe,µ → ¯ντ +and νe,µ → ντ, and, ¯νe → ¯νµ and νe → νµ oscillation signals can be collected simultaneously, with +no needs for separate specific runs for neutrinos or antineutrinos. Furthermore, it is possible to +exchange µ+ and µ− flying routes, thus further reducing possible bias or systematic uncertainties. +In an optimistic way, we estimate 104 tau (anti-) neutrinos can be collected per year thus this +proposal can serve as a brighter tau neutrino factory. Moreover, 5 standard deviations of sensitivity +can be easily reached for CP phase as |π/2|, with only 1–2 years of data taking, by combining tau +and muon (anti-) neutrino appearances. With the development of a more intensive muon beam +targeting future muon collider, the neutrino potential of the current proposal will surely be further +improved. +Novel collision methods and rich phenomena are crucial to keeping high-energy collision physics more robust and +attractive [1]. Recent years have witnessed vast development towards next generation high energy colliders, including +various proposals on Higgs factory [2, 3], revived interest in Muon collider [4–8], etc. +As for the muon collider design, we take positron on target method (LEMMA) as an example, which has been +proposed for high quality muon beam production [9, 10]. Although it is still quite challenging to achieve enough high +luminosity for muon beam collisions [6, 7], we find it quite promising for neutrino oscillation studies, with comparable +or even larger neutrino flux than other long baseline neutrino experiments, with more details to be discussed below. +In the LEMMA approach, the incident positron energy is around 45 GeV, producing collimated muon pairs with +opening angles of around 0.005 rad. and a large boost about γ ∼ 200, which extends the muon lifetime by the order +of O(102). Generally, the number of muon pairs produced per positron bunch on target can be expressed as +n(µ+µ−) = n+ρe−lσ(µ+µ−) +(1) +where n+ is the number of e+ in each positron bunch, ρe− is the electron density in the medium, l is the thickness +of the the target, and σ(µ+µ−) being the cross section of the muon pair production. The number of muon pairs per +positron bunch on target can be maximally estimated as n(µ+µ−)max ≈ n+ × 10−5. +On the other hand, neutrinos are among the most abundant and least understood of all particles in the SM that +make up our universe. The history of neutrino physics was full of novel discoveries. One of them is the observation +of neutrino oscillations, confirming that at least two types of SM neutrinos have a tiny, but strictly nonzero mass. +Nowadays, the neutrino system is described by nine parameters, the masses m1, m2 and m3 of the three mass +eigenstates, three mixing angles, θ12, θ23, and θ13, and three phases, one Dirac phase, δCP and two Majorana phases. +The Majorana phase only plays a part in neutrinoless double beta decay [11]. The mixing angles and the phases are +the elements of a unitary matrix called Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [12, 13]. The available +experiments on neutrino oscillations to date have measured five of the neutrino mixing parameters, three mixing +angles θ12, θ23, θ13, and the two squared-mass differences ∆m2 +21, ∆m2 +32 within 3σ range [14–18]. Neutrino oscillation +probabilities from one flavor into another are functions of these mixing angles and the squared-mass differences. The +mass of the individual neutrino and the mass hierarchies are not known. +The determination of the CP-violating phase, the Dirac phase, has been the core research program in neutrino +physics for decades because it provides a potential source of CP violation in the SM lepton sector. It has been known +∗ alim.ruzi@pku.edu.cn +† tyyang99@pku.edu.cn +‡ fudw@pku.edu.cn +§ stqian@pku.edu.cn +¶ qliphy0@pku.edu.cn +arXiv:2301.02493v1 [hep-ph] 6 Jan 2023 + +2 +that the leptonic CP violation could generate the matter-antimatter asymmetry through leptogenesis [19]. CP violation +in neutrino oscillation can be measured through the difference between the oscillation probability of the neutrino and +antineutrino, expressed as ∆P CP +αβ += Pαβ − P αβ, which is well quantified by δCP. +There are several experiments +worldwide dedicated to the measurements of the neutrino parameters, especially the CP phase, performing searches +of short-baseline and long-baseline neutrino oscillation. To ensure that there are enough neutrino flavors oscillated +from source neutrino and to be detected by Far Detector (FD), a long-baseline neutrino oscillation experiment is +preferable rather than a short-baseline. Recently, the long-baseline experiments, T2K (Tokai to Kamioka) [20, 21] and +NOvA [22] report their results. T2K reports a measured value for CP phase, δCP = −1.97+0.97 +−0.70 while excluding δCP += 0 and π at 90% CL, indicating CP violation in the lepton sector. The FD in this case is the Super-Kamiokande, +a 50 Kton water Cherenkov detector. A narrow band neutrino beam is produced at an angle of 2.5◦ by a 30 GeV +proton beam hitting on graphite target. With this off-axis method, the narrow band neutrino energy has a peak at +0.6 GeV. The secondary neutrino produced from decays of Kaon or Pion travels a distance of 295 Km to reach the +Super-Kamiokande detector. T2K plans to extend its term to 2026, followed by the Hyper-K project with the mass +of the far detector to be increased by a factor of 10, and will offer a broad science program [23]. +On the other hand, the NOvA experiment [22] in the US is also a long-baseline accelerator-based neutrino oscillation +experiment. It uses the upgraded Fermilab NuMI beam and measures electron-neutrino appearance and muon-neutrino +disappearance at its far detector in Ash River, Minnesota. The reported NOvA result shows no strong preference for +any particular value of the neutrino CP phase and has a slight tension with T2K’s. +Another promising long-baseline neutrino experiment under construction is DUNE (Deep Underground Neutrino +Experiment), whose goals are the determination of the neutrino mass ordering, observation of CP violation (up to 50% +level), and precise measurements of oscillation parameters, such as δCP, sin2(2θ21). The idea is to send a wide-band +high-intensity muon neutrino beam from Fermilab to the Sanford Underground Facility in Homestake at the 1300 Km +distance. The detector technology of DUNE experiment is based on building liquid argon time projection chambers +(LArTPC). Unlike the T2K experiment, the neutrino beam energy has a peak at 2.5 GeV with a broad range of +neutrino energies. The neutrino beam is produced from proton collision on the graphite target. In the corresponding +DUNE TDR report, it is shown that favorable values for δCP with 3σ (5σ) can be achieved after five (ten) years of +running. +In this letter, we are interested in applying collimated muon beams into neutrino mixing and CP phase measure- +ments. Although the beam density is lower than the proton-on-target scenario, there are several significant benefits +leading to large neutrino flux and high sensitivity on CP phase, including 1) collimated and manipulable muon beams, +which lead to a larger acceptance of neutrino sources in the far detector side; 2) symmetric µ+ and µ− beams, and thus +symmetric neutrino and antineutrino sources, which make this proposal ideally for measuring neutrino CP violation. +What is more important is that antineutrino and neutrino flux here are similar, and thus for example, ¯νe → ¯νµ and +νe → νµ oscillation signals can be collected simultaneously, with no needs for separate specific runs for neutrinos or +antineutrinos as done in the DUNE experiment [24, 25]. +As to be discussed below, the estimated neutrino flux in our proposal is comparable to or even larger than the +DUNE experiment [24, 25]. The neutrino energy has wide distributions in the 1–15 GeV region, and peaks at around +7 GeV (neutrino energy can be further tuned with on-axis and off-axis techniques), suggesting our proposal is also +suitable for tau neutrino studies. Taking into account both muon and electron neutrinos and antineutrinos, the signal +yields indeed can be doubled or more. Finally, we point out that it is possible to exchange µ+ and µ− flying routes, +thus further reducing possible bias or systematic uncertainties. +Fig. 1 shows a proposed neutrino oscillation experiment to probe neutrino CP phase by measuring muon electron +(anti-)neutrino mixing and their differences. +We are especially interested in the oscillation modes of νe,µ → ντ, +νe → νµ and their antineutrino correspondents, with more details to be given later in this paper. This proposal is +based on collimated muon beams achieved from e.g., Positron on Target method, where 45 GeV positron beams are +shed on the target. Dipoles are used to separate µ+ and µ− with an angle around 0.01 rad., with direction changeable. +Muon beams fly about 10 Km and radiate neutrinos before being swept away. Neutrinos then further fly e.g., 1300 +Km to reach DUNE or T2K type of detectors [26]. Fig. 2 shows a similar but simpler proposal focusing mainly on +tau (anti-)neutrino appearance and related physics studies. +• Muon production rates n(µ+µ−)max ≈ n+ × 10−5 [10]. Assuming positron bunch density as 1012/bunch and +bunch crossing frequency as 105/sec, we get muon production rates dNµ/dt ∼ 1012/sec (or 1019/year). +(Notice the future TeV scale muon collider is indeed targeting a much larger intensity beam by more than 1-2 +orders of magnitudes [5–7].) +• For muons with the energy around 20 GeV, the mean flying distance is around 100 Km. If there is a straight tube +with a length around 5-10 Km to let muons go through with quadrupoles to keep them merged, the decayed +fraction can reach 10−1 in realistic. On the other hand, we can also refer to a muon complex as discussed in +Ref. [8, 27], where the muon beam is accelerated in a circular section and then extracted into the rectangular + +3 +FIG. 1. A proposed neutrino oscillation experiment to probe neutrino violation CP phase by measuring muon electron (anti-) +neutrino oscillation and the differences of resulted νe,µ → ντ, νe → νµ, and their antineutrino correspondents. This proposal +is based on collimated muon beams achieved from e.g., Positron on Target method, where 45 GeV positron beams are fired. +Dipoles are used to separate µ+ and µ− with an angle around 0.01 rad., with direction changeable. Muon beams fly about 10 +Km and radiate neutrinos before being swept away. Neutrinos then further fly 1300 Km to reach DUNE type of detectors. +FIG. 2. A proposed neutrino oscillation experiment to probe tau neutrino physics by measuring tau (anti-)neutrino appearance. +This proposal is based on collimated muon beams achieved from e.g., Positron on Target method, where 45 GeV positron +beams are shed. Neutrinos are radiated along the muon beam line and fly 100 to 1300 Km to reach DUNE type of detectors. +Quadrupoles can be applied to keep muon beams more collimated. +section for decays. The intensity of the neutrino beam compared with the incoming muon beam is suppressed +by a ratio around 10−1, i.e., the fraction of the collider ring circumference occupied by the production straight +section. +• The opening angles between muons and decay products are around 0.005 rad. as shown in Fig. 3 and may be +kept smaller with quadrupoles. For neutrinos traveling 1300 Km to reach far detectors, the spread size can be +around 1-5 Km. For a DUNE-like detector with a cubic size of about 20 m [26], the neutrino acceptance is +then 10−4. +• Muon/electron neutrinos and antineutrinos interacting with detectors. With a L = 20 m long detector (DUNE +far detector indeed has a length around 50m [26]), the expected event yield rate can be roughly estimated +with: dNµ/dt × L × σnν × ρNA · dE ∼ 10−9 × dNµ/dt, where NA is the Avogadro constant, ρ ∼ 2 g/cm3, σnν +symbols the neutrino nucleon cross sections and is around 10−37cm2 for a 10 GeV neutrino [28, 29]. +Combining all above numbers, with the conservative option (e.g., 20-meter cubic size detector), we get the +muon/electron (anti-)neutrino Charged Current (CC) events per year as +N cc +νµ,e ∼ 1019 × 10−1 × 10−4 × 10−9 = 105/year, +(2) +On top of measuring muon (anti-)neutrino rates, our proposal can also be used to probe tau neutrino appearance. + +Muon neutrino and Electron neutrino oscillation +Vμ → Ve, Vt, +De → u, +(Or proton on target) +Dipoles +→ Vμ,De +e+ (45GeV) +Sweeper +～ 0.01 rad +Detectors +→ μ, Ve +μ→e,， +Ve - +～ 10km +～ 1300kmTau neutrino appearance +Far detector(s) +ee→μμ +(Or proton on target) +μ →Ve,μ,De,μ +quadrupole(s) +Ve, μ → Vr +De, μ→- +e+ ( +(45GeV) +2 mrad +Sweeper + 10km +～ 1300km4 +0 +5 +10 +15 +20 +E +e[GeV] +0.0000 +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +0.0150 +0.0175 +0.0200 +e[rad] +0.0000 +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +0.0150 +0.0175 +0 +5 +10 +15 +20 +E [GeV] +0.0000 +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +0.0150 +0.0175 +0.0200 +[rad] +0.0000 +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +0.0150 +0.0175 +0.0200 +FIG. 3. 2D distributions of energy and angle in respect to muon flying direction, for muon and electron neutrinos from 22.5 +GeV µ+ → e+ ¯νµ νe (similarly for µ− decay). +One additional factor needs to be considered, i.e., the fraction of muon neutrino oscillated into tau neutrino [30]: +P(νµ → ντ) ≃ sin2 (2θ23) cos4(θ13) sin2 +� +1.27∆m2 +32L +Eν +� +± 1.27∆m2 +21 +L +Eν +sin2 +� +1.27∆m2 +32L +Eν +� +× 8JCP +(3) +where the “Jarlskog invariant” [31, 32], +JCP ≡ sin θ13 cos2 θ13 sin θ12 cos θ12 sin θ23 cos θ23 sin δCP += 0.03359 ± 0.0006(±0.0019) sin δCP +(4) +is a function of sin δCP. The oscillation probability difference between P(νµ → ντ) and P(¯νµ → ¯ντ) reads as +∆P(νµ → ντ) = 16JCP × 1.27∆m2 +12 +L +Eν +sin2 +�∆m2 +32L +Eν +� +(5) +Thus we can estimate tau neutrino CC events per year as (for simplicity we ignore the cross section difference of +neutrino and antineutrino nucleon cross sections) +N cc +ντ ∼ [(3 × 104) ± (2.6 × 102)]/year, +(6) +where (2.6 × 102) corresponds to the CP violation term. Notice the yearly expected tau neutrino yields is already +comparable or even surpass the rates at the SHiP experiment at CERN [33]. +Thus our proposal can serve as a +‘brighter’ factory for tau neutrinos. +The oscillation probability of P(νe → ντ) can be estimated as +P(νe → ντ) ≃ sin2(2θ13) cos2(θ23) sin2 +� +1.27∆m2 +32 +L +Eν +� +∓ 1.27∆m2 +21 +L +Eν +sin2 +� +1.27∆m2 +32 +L +Eν +� +× 8JCP +(7) +Because of its heavy mass and very short lifetime, the tau neutrino production in abundance in conventional +accelerator is almost not possible. On the contrary, we have rich tau neutrino flux because of the higher P(νµ → ντ) +oscillation. The tau neutrino flux can be further strengthened by P(νe → ντ) oscillation . With this promising feature +in this proposal, some of the new physics models, such as charged Higgs doublet [34] or leptoquarks [35] maybe tested +through neutrino-type collision [36]. + +5 +1 +10 + (GeV) +ν +E +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +) +τ +ν +→ +µ +ν +P( + = 0 +CP +δ +2 +π + = +CP +δ +2 +π + = - +CP +δ +) +τ +ν +→ +µ +ν +P( +1 +10 + (GeV) +ν +E +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +) +e +ν +→ +µ +ν +P( + = 0 +CP +δ +2 +π + = +CP +δ +2 +π + = - +CP +δ +) +e +ν +→ +µ +ν +P( +1 +10 + (GeV) +ν +E +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +) +τ +ν +→ +e +ν +P( + = 0 +CP +δ +2 +π + = +CP +δ +2 +π + = - +CP +δ +) +τ +ν +→ +e +ν +P( +1 +10 + (GeV) +ν +E +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +) +µ +ν +→ +e +ν +P( + = 0 +CP +δ +2 +π + = +CP +δ +2 +π + = - +CP +δ +) +µ +ν +→ +e +ν +P( +FIG. 4. The oscillation probability of νµ → ντ, νµ → νe, νe → ντ, and νe → ντ at three δCP angles as function of νµ energy, +for a long baseline of L =1300 Km. +Apart from tau neutrino appearance, the oscillation rates with CP phase dependence for νµ → νe is shown as +below [20, 21]: +P(νµ → νe) ≃ sin2(2θ13) sin2(θ23) sin2 +� +1.27∆m2 +32 +L +Eν +� +∓ 1.27∆m2 +21 +L +Eν +sin2 +� +1.27∆m2 +32 +L +Eν +� +× 8JCP +(8) +The corresponding oscillation probability for νe → νµ only differs a minus sign from νµ → νe oscillation in CP violating +term: +P(νe → νµ) ≃ sin2(2θ13) sin2(θ23) sin2 +� +1.27∆m2 +32 +L +Eν +� +± 1.27∆m2 +21 +L +Eν +sin2 +� +1.27∆m2 +32 +L +Eν +� +× 8JCP +(9) +Using the current measured values of the mixing angles and squared mass differences [28] and taking the distance +of neutrino propagation as L =1300 Km, we have the numeric values for the neutrino oscillations at Eν = 7 (5) GeV +(see more in Fig. 4): +P(νµ → ντ) = 0.2916 ± 0.0026 sin δCP (0.5093 ± 0.0048 sin δCP), +(10) +P(νµ → νe) = 0.0151 ∓ 0.0026 sin δCP (0.0264 ∓ 0.0048 sin δCP), +(11) +P(νe → νµ) = 0.0151 ± 0.0026 sin δCP (0.0264 ± 0.0048 sin δCP), +(12) +P(νe → ντ) = 0.0119 ∓ 0.0026 sin δCP (0.0209 ∓ 0.0048 sin δCP). +(13) + +6 +We will now evaluate the sensitivities on neutrino CP violation, taking δCP = ±π/2 and Eν = 7 GeV as benchmark +parameters: +• 1) Firstly, we consider the tau (anti-) neutrino appearance from muon and electron neutrino oscillations: +µ+ → e+ ¯νµ1 νe1 =⇒ ¯νµ1 → ¯ντ1, νe1 → ντ1, +(14) +µ− → e− νµ2 ¯νe2 =⇒ νµ2 → ντ2, ¯νe2 → ¯ντ2, +(15) +where ‘1’ and ‘2’ symbol the two far detectors as shown in Fig. 1. +Notice that the CP dependence of P(νe → ντ) and P(¯νµ → ¯ντ) as shown in Eq. 10 vary in the same direction. +If we count on tau-related events in the far detector inclusively, this means our signal doubles. The sensitivity +can be estimated then as +¯ντ2 + ντ2 − ¯ντ1 − ντ1 +√¯ντ2 + ντ2 + ¯ντ1 + ντ1 +(16) +which is around 4 × 260/ +√ +60000 ∼ 4.2 standard deviations (σ) in one year, and can reach near 13.4σ in 10 +years. Although only statistics are taken into account here, systematics should be able to be reduced efficiently +due to the symmetric property of the proposed device. Furthermore, it is possible to exchange µ+ and µ− flying +routes, thus further reducing possible bias or systematic. +• 2) Secondly, if the far detector can distinguish tau neutrino from antineutrino such as the CERN SHiP ex- +periment [33], then with only P(νe → ντ), we can already have higher CP sensitivity. The sensitivity can be +estimated then as +¯ντ2 − ντ1 +√¯ντ2 + ντ1 +(17) +which is around 2 × 260/ +√ +2200 ∼ 11 σ in one year. +• 3) Finally, one can also exploit electron to muon oscillation which has also clear sensitivity on neutrino CP phase, +if the far detector can distinguish muon neutrino from antineutrino, possibly can be achieved with moderate +magnets. The sensitivity can be estimated then as 2 × 260/ +√ +3000 ∼ 9.5 σ in one year. +By combing all three options and taking into account statistic errors, one can achieve a sensitivity of far more than +5 σ in one year, for δCP = ±π/2. Option 1) corresponds to a most conservative scenario, where one can reuse such as +the DUNE-like detectors. Options 2) and 3) lead to much larger sensitivities on neutrino CP violation, yet putting +additional requirements on the experimental side, such as to distinguish µ+ from µ−, or τ + from τ − under the help +of moderate magnet field. +In Summary, we propose here a new idea to exploit collimated muon beams which generate symmetric neutrino and +antineutrino sources: µ+ → e+ ¯νµ νe and µ− → e− νµ ¯νe. Interfacing with long baseline neutrino detectors such as +in DUNE or T2K, this experiment can be useful to measure tau neutrino properties, and also to probe neutrino CP +phase, by measuring muon (anti-) neutrino disappearance or tau (anti-)neutrino appearance, and differences between +neutrino and antineutrino rates. +There are several significant benefits leading to large neutrino flux and high sensitivity on CP phase, including 1) +collimated and manipulable muon beams, which lead to a larger acceptance of neutrino sources in the far detector +side; 2) symmetric µ+ and µ− beams, and thus symmetric neutrino and antineutrino sources, which make this +proposal ideally for measuring neutrino CP violation. More importantly, ¯νe,µ → ¯ντ and νe,µ → ντ, and, ¯νe → ¯νµ +and νe → νµ oscillation signals can be collected simultaneously, with no needs for separate specific runs for neutrinos +or antineutrinos. +It is also possible to exchange µ+ and µ− flying routes, thus further reducing possible bias or +systematic. In an optimistic way, we estimate 104 tau (anti-) neutrinos can be collected per year thus our proposal +can serve as a brighter tau neutrino factor. Moreover, 5 standard deviations of sensitivity can be reached easily for +CP phase as |π/2|, with only 1 year of data taking, by combining tau and muon (anti-) neutrino appearances. +Notice for tau (anti-)neutrino appearance, the CP dependence of P(νe → ντ) and P(¯νµ → ¯ντ), or P(¯νe → ¯ντ) and +P(νµ → ντ) vary in the same direction, thus signal rates double and one can reuse directly DUNE-like far detector. +In this draft, we mainly provide a preliminary estimation of the feasibility study. +A detailed study is surely +necessary to follow up. On the other hand, there exist also rich potential to be further explored with such a proposal +that connects energy and neutrino frontiers. Especially, one can imagine a post-DUNE (or in parallel to DUNE as the +probe channels are indeed orthogonal and thus complementary) experiment with neutrinos from a strong muon source +located at the Fermilab site. This connection between energy and neutrino frontiers can also serve as a precursor for + +7 +future high-energy muon colliders. Notice that a muon collider requires a 1–2 orders of magnitude more intense beam +as compared with the number (dNµ/dt ∼ 1012/sec ) listed above as our benchmark. Thus with the development of +a more intensive muon beam targeting future muon colliders, it surely will improve further the neutrino potential of +the current proposal. +ACKNOWLEDGMENTS +This work is supported in part by the National Natural Science Foundation of China under Grants No. 12150005, +No. 12075004, and No. 12061141002, by MOST under grant No. 2018YFA0403900. +[1] M. Lu et al. [PKU SYSU Novel Collider Group], [arXiv:2210.06690 [hep-ph]]. +[2] R. K. Ellis, B. Heinemann, J. de Blas, M. Cepeda, C. Grojean, F. Maltoni, A. Nisati, E. Petit, R. Rattazzi and W. Verkerke, +et al. [arXiv:1910.11775 [hep-ex]]. +[3] J. A. Bagger, B. C. Barish, S. Belomestnykh, P. C. Bhat, J. E. Brau, M. Demarteau, D. Denisov, S. C. Eno, C. G. R. Geddes +and P. D. Grannis, et al. [arXiv:2203.06164 [hep-ex]]. +[4] C. Aime, A. Apyan, M. A. Mahmoud Mohammed, N. Bartosik, F. Batsch, A. Bertolin, M. Bonesini, S. Bottaro, D. Buttazzo +and R. M. Capdevilla Roldan, et al. [arXiv:2203.07256 [hep-ph]]. +[5] D. Stratakis et al. [Muon Collider], [arXiv:2203.08033 [physics.acc-ph]]. +[6] S. Jindariani et al. [Muon Collider], [arXiv:2203.07224 [physics.ins-det]]. +[7] J. de Blas et al. [Muon Collider], [arXiv:2203.07261 [hep-ph]]. +[8] C. Quigg, AIP Conf. Proc. 435, no.1, 242-257 (1998) doi:10.1063/1.56235 [arXiv:hep-ph/9803326 [hep-ph]]. +[9] M. +Antonelli, +M. +Boscolo, +R. +Di +Nardo +and +P. +Raimondi, +Nucl. +Instrum. +Meth. +A +807, +101-107 +(2016) +doi:10.1016/j.nima.2015.10.097 [arXiv:1509.04454 [physics.acc-ph]]. +[10] D. Alesini, M. Antonelli, M. E. Biagini, M. Boscolo, O. R. Blanco-Garc´ıa, A. Ciarma, R. Cimino, M. Iafrati, A. Giribono +and S. Guiducci, et al. [arXiv:1905.05747 [physics.acc-ph]]. +[11] J. D. Vergados, H. Ejiri and F. Simkovic, Rept. Prog. Phys. 75, 106301 (2012) doi:10.1088/0034-4885/75/10/106301 +[arXiv:1205.0649 [hep-ph]]. +[12] Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28, 870-880 (1962) doi:10.1143/PTP.28.870 +[13] B. Pontecorvo, Zh. Eksp. Teor. Fiz. 53, 1717-1725 (1967) +[14] F. P. An et al. [Daya Bay], [arXiv:2211.14988 [hep-ex]]. +[15] P. +Adamson +et al. +[NOvA], +Phys. +Rev. +Lett. +118, +no.15, +151802 +(2017) +doi:10.1103/PhysRevLett.118.151802 +[arXiv:1701.05891 [hep-ex]]. +[16] K. Abe et al. [T2K], Phys. Rev. Lett. 121, no.17, 171802 (2018) doi:10.1103/PhysRevLett.121.171802 [arXiv:1807.07891 +[hep-ex]]. +[17] K. +Abe +et +al. +[Super-Kamiokande], +Phys. +Rev. +D +97, +no.7, +072001 +(2018) +doi:10.1103/PhysRevD.97.072001 +[arXiv:1710.09126 [hep-ex]]. +[18] M. A. Acero et al. [NOvA], Phys. Rev. Lett. 123, +no.15, +151803 (2019) doi:10.1103/PhysRevLett.123.151803 +[arXiv:1906.04907 [hep-ex]]. +[19] M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45-47 (1986) doi:10.1016/0370-2693(86)91126-3 +[20] K. Abe et al. [T2K], Nature 580, no.7803, 339-344 (2020) [erratum: Nature 583, no.7814, E16 (2020)] doi:10.1038/s41586- +020-2177-0 +[21] J. G. Walsh [T2K], [arXiv:2208.01164 [hep-ex]]. +[22] M. A. Acero et al. [NOvA], Phys. Rev. D 106, no.3, 032004 (2022) doi:10.1103/PhysRevD.106.032004 [arXiv:2108.08219 +[hep-ex]]. +[23] K. Abe et al. [Hyper-Kamiokande], [arXiv:1805.04163 [physics.ins-det]]. +[24] B. Abi et al. [DUNE], [arXiv:2002.03005 [hep-ex]]. +[25] B. Abi et al. [DUNE], [arXiv:2103.04797 [hep-ex]]. +[26] B. Abi et al. [DUNE], JINST 15, +no.08, +T08009 (2020) doi:10.1088/1748-0221/15/08/T08009 [arXiv:2002.03008 +[physics.ins-det]]. +[27] B. J. King, AIP Conf. Proc. 530, no.1, 142-164 (2000). +[28] P. A. Zyla et al. [Particle Data Group], PTEP 2020, no.8, 083C01 (2020) doi:10.1093/ptep/ptaa104 +[29] J. A. Formaggio and G. P. Zeller, Rev. Mod. Phys. 84, 1307-1341 (2012) doi:10.1103/RevModPhys.84.1307 [arXiv:1305.7513 +[hep-ex]]. +[30] A. +De +Gouvˆea, +K. +J. +Kelly, +G. +V. +Stenico +and +P. +Pasquini, +Phys. +Rev. +D +100, +no.1, +016004 +(2019) +doi:10.1103/PhysRevD.100.016004 +[31] C. Jarlskog, Phys. Rev. Lett. 55, 1039 (1985) doi:10.1103/PhysRevLett.55.1039 +[32] R. L. Workman et al. [Particle Data Group], PTEP 2022, 083C01 (2022) doi:10.1093/ptep/ptac097 + +8 +[33] C. Ahdida et al. [SHiP], Eur. Phys. J. C 82, no.5, 486 (2022) doi:10.1140/epjc/s10052-022-10346-5 [arXiv:2112.01487 +[physics.ins-det]]. +[34] G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher and J. P. Silva, Phys. Rept. 516, 1-102 (2012) +doi:10.1016/j.physrep.2012.02.002 [arXiv:1106.0034 [hep-ph]]. +[35] I. de Medeiros Varzielas and J. Talbert, Eur. Phys. J. C 79, no.6, 536 (2019) doi:10.1140/epjc/s10052-019-7047-2 +[arXiv:1901.10484 [hep-ph]]. +[36] S. Qian, T. Yang, S. Deng, J. Xiao, L. Gao, A. M. Levin, Q. Li, M. Lu and Z. You, [arXiv:2205.15350 [hep-ph]]. + diff --git a/WNE0T4oBgHgl3EQfmAF1/content/tmp_files/load_file.txt b/WNE0T4oBgHgl3EQfmAF1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..40ad87a843ee60babfd0d547b95121accd8a4dd2 --- /dev/null +++ b/WNE0T4oBgHgl3EQfmAF1/content/tmp_files/load_file.txt @@ -0,0 +1,623 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf,len=622 +page_content='Muon Beam for Neutrino CP Violation: connecting energy and neutrino frontiers Alim Ruzi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='∗ Tianyi Yang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='† Dawei Fu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='‡ Sitian Qian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='§ Leyun Gao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' and Qiang Li¶ State Key Laboratory of Nuclear Physics and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' China We propose here a proposal to connect neutrino and energy frontiers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' by exploiting collimated muon beams for neutrino oscillations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' which generate symmetric neutrino and antineutrino sources: µ+ → e+ ¯νµ νe and µ− → e− νµ ¯νe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Interfacing with long baseline neutrino detectors such as DUNE and T2K, this experiment can be applicable to measure tau neutrino properties, and also to probe neutrino CP phase, by measuring muon electron (anti-)neutrino mixing or tau (anti-)neutrino appearance, and differences between neutrino and antineutrino rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' There are several significant benefits leading to large neutrino flux and high sensitivity on CP phase, including 1) collimated and manipulable muon beams, which lead to a larger acceptance of neutrino sources in the far detector side;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 2) symmetric µ+ and µ− beams, and thus symmetric neutrino and antineutrino sources, which make this proposal ideally useful for measuring neutrino CP violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' More importantly, ¯νe,µ → ¯ντ and νe,µ → ντ, and, ¯νe → ¯νµ and νe → νµ oscillation signals can be collected simultaneously, with no needs for separate specific runs for neutrinos or antineutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Furthermore, it is possible to exchange µ+ and µ− flying routes, thus further reducing possible bias or systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' In an optimistic way, we estimate 104 tau (anti-) neutrinos can be collected per year thus this proposal can serve as a brighter tau neutrino factory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Moreover, 5 standard deviations of sensitivity can be easily reached for CP phase as |π/2|, with only 1–2 years of data taking, by combining tau and muon (anti-) neutrino appearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' With the development of a more intensive muon beam targeting future muon collider, the neutrino potential of the current proposal will surely be further improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Novel collision methods and rich phenomena are crucial to keeping high-energy collision physics more robust and attractive [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Recent years have witnessed vast development towards next generation high energy colliders, including various proposals on Higgs factory [2, 3], revived interest in Muon collider [4–8], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' As for the muon collider design, we take positron on target method (LEMMA) as an example, which has been proposed for high quality muon beam production [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Although it is still quite challenging to achieve enough high luminosity for muon beam collisions [6, 7], we find it quite promising for neutrino oscillation studies, with comparable or even larger neutrino flux than other long baseline neutrino experiments, with more details to be discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' In the LEMMA approach, the incident positron energy is around 45 GeV, producing collimated muon pairs with opening angles of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='005 rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' and a large boost about γ ∼ 200, which extends the muon lifetime by the order of O(102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Generally, the number of muon pairs produced per positron bunch on target can be expressed as n(µ+µ−) = n+ρe−lσ(µ+µ−) (1) where n+ is the number of e+ in each positron bunch, ρe− is the electron density in the medium, l is the thickness of the the target, and σ(µ+µ−) being the cross section of the muon pair production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The number of muon pairs per positron bunch on target can be maximally estimated as n(µ+µ−)max ≈ n+ × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' On the other hand, neutrinos are among the most abundant and least understood of all particles in the SM that make up our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The history of neutrino physics was full of novel discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' One of them is the observation of neutrino oscillations, confirming that at least two types of SM neutrinos have a tiny, but strictly nonzero mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Nowadays, the neutrino system is described by nine parameters, the masses m1, m2 and m3 of the three mass eigenstates, three mixing angles, θ12, θ23, and θ13, and three phases, one Dirac phase, δCP and two Majorana phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The Majorana phase only plays a part in neutrinoless double beta decay [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The mixing angles and the phases are the elements of a unitary matrix called Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The available experiments on neutrino oscillations to date have measured five of the neutrino mixing parameters, three mixing angles θ12, θ23, θ13, and the two squared-mass differences ∆m2 21, ∆m2 32 within 3σ range [14–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Neutrino oscillation probabilities from one flavor into another are functions of these mixing angles and the squared-mass differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The mass of the individual neutrino and the mass hierarchies are not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The determination of the CP-violating phase, the Dirac phase, has been the core research program in neutrino physics for decades because it provides a potential source of CP violation in the SM lepton sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' It has been known ∗ alim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='ruzi@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='cn † tyyang99@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='cn ‡ fudw@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='cn § stqian@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='cn ¶ qliphy0@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='cn arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='02493v1 [hep-ph] 6 Jan 2023 2 that the leptonic CP violation could generate the matter-antimatter asymmetry through leptogenesis [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' CP violation in neutrino oscillation can be measured through the difference between the oscillation probability of the neutrino and antineutrino, expressed as ∆P CP αβ = Pαβ − P αβ, which is well quantified by δCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' There are several experiments worldwide dedicated to the measurements of the neutrino parameters, especially the CP phase, performing searches of short-baseline and long-baseline neutrino oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' To ensure that there are enough neutrino flavors oscillated from source neutrino and to be detected by Far Detector (FD), a long-baseline neutrino oscillation experiment is preferable rather than a short-baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Recently, the long-baseline experiments, T2K (Tokai to Kamioka) [20, 21] and NOvA [22] report their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' T2K reports a measured value for CP phase, δCP = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='97+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='70 while excluding δCP = 0 and π at 90% CL, indicating CP violation in the lepton sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The FD in this case is the Super-Kamiokande, a 50 Kton water Cherenkov detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' A narrow band neutrino beam is produced at an angle of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='5◦ by a 30 GeV proton beam hitting on graphite target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' With this off-axis method, the narrow band neutrino energy has a peak at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='6 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The secondary neutrino produced from decays of Kaon or Pion travels a distance of 295 Km to reach the Super-Kamiokande detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' T2K plans to extend its term to 2026, followed by the Hyper-K project with the mass of the far detector to be increased by a factor of 10, and will offer a broad science program [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' On the other hand, the NOvA experiment [22] in the US is also a long-baseline accelerator-based neutrino oscillation experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' It uses the upgraded Fermilab NuMI beam and measures electron-neutrino appearance and muon-neutrino disappearance at its far detector in Ash River, Minnesota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The reported NOvA result shows no strong preference for any particular value of the neutrino CP phase and has a slight tension with T2K’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Another promising long-baseline neutrino experiment under construction is DUNE (Deep Underground Neutrino Experiment), whose goals are the determination of the neutrino mass ordering, observation of CP violation (up to 50% level), and precise measurements of oscillation parameters, such as δCP, sin2(2θ21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The idea is to send a wide-band high-intensity muon neutrino beam from Fermilab to the Sanford Underground Facility in Homestake at the 1300 Km distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The detector technology of DUNE experiment is based on building liquid argon time projection chambers (LArTPC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Unlike the T2K experiment, the neutrino beam energy has a peak at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='5 GeV with a broad range of neutrino energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The neutrino beam is produced from proton collision on the graphite target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' In the corresponding DUNE TDR report, it is shown that favorable values for δCP with 3σ (5σ) can be achieved after five (ten) years of running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' In this letter, we are interested in applying collimated muon beams into neutrino mixing and CP phase measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Although the beam density is lower than the proton-on-target scenario, there are several significant benefits leading to large neutrino flux and high sensitivity on CP phase, including 1) collimated and manipulable muon beams, which lead to a larger acceptance of neutrino sources in the far detector side;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 2) symmetric µ+ and µ− beams, and thus symmetric neutrino and antineutrino sources, which make this proposal ideally for measuring neutrino CP violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' What is more important is that antineutrino and neutrino flux here are similar, and thus for example, ¯νe → ¯νµ and νe → νµ oscillation signals can be collected simultaneously, with no needs for separate specific runs for neutrinos or antineutrinos as done in the DUNE experiment [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' As to be discussed below, the estimated neutrino flux in our proposal is comparable to or even larger than the DUNE experiment [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The neutrino energy has wide distributions in the 1–15 GeV region, and peaks at around 7 GeV (neutrino energy can be further tuned with on-axis and off-axis techniques), suggesting our proposal is also suitable for tau neutrino studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Taking into account both muon and electron neutrinos and antineutrinos, the signal yields indeed can be doubled or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Finally, we point out that it is possible to exchange µ+ and µ− flying routes, thus further reducing possible bias or systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 1 shows a proposed neutrino oscillation experiment to probe neutrino CP phase by measuring muon electron (anti-)neutrino mixing and their differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' We are especially interested in the oscillation modes of νe,µ → ντ, νe → νµ and their antineutrino correspondents, with more details to be given later in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' This proposal is based on collimated muon beams achieved from e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=', Positron on Target method, where 45 GeV positron beams are shed on the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Dipoles are used to separate µ+ and µ− with an angle around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='01 rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=', with direction changeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Muon beams fly about 10 Km and radiate neutrinos before being swept away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Neutrinos then further fly e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=', 1300 Km to reach DUNE or T2K type of detectors [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 2 shows a similar but simpler proposal focusing mainly on tau (anti-)neutrino appearance and related physics studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Muon production rates n(µ+µ−)max ≈ n+ × 10−5 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Assuming positron bunch density as 1012/bunch and bunch crossing frequency as 105/sec, we get muon production rates dNµ/dt ∼ 1012/sec (or 1019/year).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' (Notice the future TeV scale muon collider is indeed targeting a much larger intensity beam by more than 1-2 orders of magnitudes [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=') For muons with the energy around 20 GeV, the mean flying distance is around 100 Km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' If there is a straight tube with a length around 5-10 Km to let muons go through with quadrupoles to keep them merged, the decayed fraction can reach 10−1 in realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' On the other hand, we can also refer to a muon complex as discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [8, 27], where the muon beam is accelerated in a circular section and then extracted into the rectangular 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' A proposed neutrino oscillation experiment to probe neutrino violation CP phase by measuring muon electron (anti-) neutrino oscillation and the differences of resulted νe,µ → ντ, νe → νµ, and their antineutrino correspondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' This proposal is based on collimated muon beams achieved from e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=', Positron on Target method, where 45 GeV positron beams are fired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Dipoles are used to separate µ+ and µ− with an angle around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='01 rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=', with direction changeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Muon beams fly about 10 Km and radiate neutrinos before being swept away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Neutrinos then further fly 1300 Km to reach DUNE type of detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' A proposed neutrino oscillation experiment to probe tau neutrino physics by measuring tau (anti-)neutrino appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' This proposal is based on collimated muon beams achieved from e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=', Positron on Target method, where 45 GeV positron beams are shed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Neutrinos are radiated along the muon beam line and fly 100 to 1300 Km to reach DUNE type of detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Quadrupoles can be applied to keep muon beams more collimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' section for decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The intensity of the neutrino beam compared with the incoming muon beam is suppressed by a ratio around 10−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=', the fraction of the collider ring circumference occupied by the production straight section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The opening angles between muons and decay products are around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='005 rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 3 and may be kept smaller with quadrupoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' For neutrinos traveling 1300 Km to reach far detectors, the spread size can be around 1-5 Km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' For a DUNE-like detector with a cubic size of about 20 m [26], the neutrino acceptance is then 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Muon/electron neutrinos and antineutrinos interacting with detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' With a L = 20 m long detector (DUNE far detector indeed has a length around 50m [26]), the expected event yield rate can be roughly estimated with: dNµ/dt × L × σnν × ρNA · dE ∼ 10−9 × dNµ/dt, where NA is the Avogadro constant, ρ ∼ 2 g/cm3, σnν symbols the neutrino nucleon cross sections and is around 10−37cm2 for a 10 GeV neutrino [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Combining all above numbers, with the conservative option (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=', 20-meter cubic size detector), we get the muon/electron (anti-)neutrino Charged Current (CC) events per year as N cc νµ,e ∼ 1019 × 10−1 × 10−4 × 10−9 = 105/year, (2) On top of measuring muon (anti-)neutrino rates, our proposal can also be used to probe tau neutrino appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Muon neutrino and Electron neutrino oscillation Vμ → Ve, Vt, De → u, (Or proton on target) Dipoles → Vμ,De e+ (45GeV) Sweeper ～ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='01 rad Detectors → μ, Ve μ→e,， Ve - ～ 10km ～ 1300kmTau neutrino appearance Far detector(s) ee→μμ (Or proton on target) μ →Ve,μ,De,μ quadrupole(s) Ve, μ → Vr De, μ→- e+ ( (45GeV) 2 mrad Sweeper 10km ～ 1300km4 0 5 10 15 20 E e[GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0000 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0200 e[rad] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0175 0 5 10 15 20 E [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0200 [rad] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0200 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 2D distributions of energy and angle in respect to muon flying direction, for muon and electron neutrinos from 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='5 GeV µ+ → e+ ¯νµ νe (similarly for µ− decay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' One additional factor needs to be considered, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=', the fraction of muon neutrino oscillated into tau neutrino [30]: P(νµ → ντ) ≃ sin2 (2θ23) cos4(θ13) sin2 � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='27∆m2 32L Eν � ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='27∆m2 21 L Eν sin2 � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='27∆m2 32L Eν � × 8JCP (3) where the “Jarlskog invariant” [31, 32], JCP ≡ sin θ13 cos2 θ13 sin θ12 cos θ12 sin θ23 cos θ23 sin δCP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='03359 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0006(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0019) sin δCP (4) is a function of sin δCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The oscillation probability difference between P(νµ → ντ) and P(¯νµ → ¯ντ) reads as ∆P(νµ → ντ) = 16JCP × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='27∆m2 12 L Eν sin2 �∆m2 32L Eν � (5) Thus we can estimate tau neutrino CC events per year as (for simplicity we ignore the cross section difference of neutrino and antineutrino nucleon cross sections) N cc ντ ∼ [(3 × 104) ± (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='6 × 102)]/year, (6) where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='6 × 102) corresponds to the CP violation term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Notice the yearly expected tau neutrino yields is already comparable or even surpass the rates at the SHiP experiment at CERN [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Thus our proposal can serve as a ‘brighter’ factory for tau neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The oscillation probability of P(νe → ντ) can be estimated as P(νe → ντ) ≃ sin2(2θ13) cos2(θ23) sin2 � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='27∆m2 32 L Eν � ∓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='27∆m2 21 L Eν sin2 � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='27∆m2 32 L Eν � × 8JCP (7) Because of its heavy mass and very short lifetime, the tau neutrino production in abundance in conventional accelerator is almost not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' On the contrary, we have rich tau neutrino flux because of the higher P(νµ → ντ) oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The tau neutrino flux can be further strengthened by P(νe → ντ) oscillation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' With this promising feature in this proposal, some of the new physics models, such as charged Higgs doublet [34] or leptoquarks [35] maybe tested through neutrino-type collision [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 5 1 10 (GeV) ν E 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='9 1 ) τ ν → µ ν P( = 0 CP δ 2 π = CP δ 2 π = - CP δ ) τ ν → µ ν P( 1 10 (GeV) ν E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='08 ) e ν → µ ν P( = 0 CP δ 2 π = CP δ 2 π = - CP δ ) e ν → µ ν P( 1 10 (GeV) ν E 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='08 ) τ ν → e ν P( = 0 CP δ 2 π = CP δ 2 π = - CP δ ) τ ν → e ν P( 1 10 (GeV) ν E 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='08 ) µ ν → e ν P( = 0 CP δ 2 π = CP δ 2 π = - CP δ ) µ ν → e ν P( FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The oscillation probability of νµ → ντ, νµ → νe, νe → ντ, and νe → ντ at three δCP angles as function of νµ energy, for a long baseline of L =1300 Km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Apart from tau neutrino appearance, the oscillation rates with CP phase dependence for νµ → νe is shown as below [20, 21]: P(νµ → νe) ≃ sin2(2θ13) sin2(θ23) sin2 � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='27∆m2 32 L Eν � ∓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='27∆m2 21 L Eν sin2 � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='27∆m2 32 L Eν � × 8JCP (8) The corresponding oscillation probability for νe → νµ only differs a minus sign from νµ → νe oscillation in CP violating term: P(νe → νµ) ≃ sin2(2θ13) sin2(θ23) sin2 � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='27∆m2 32 L Eν � ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='27∆m2 21 L Eν sin2 � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='27∆m2 32 L Eν � × 8JCP (9) Using the current measured values of the mixing angles and squared mass differences [28] and taking the distance of neutrino propagation as L =1300 Km, we have the numeric values for the neutrino oscillations at Eν = 7 (5) GeV (see more in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 4): P(νµ → ντ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='2916 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0026 sin δCP (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='5093 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0048 sin δCP), (10) P(νµ → νe) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0151 ∓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0026 sin δCP (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0264 ∓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0048 sin δCP), (11) P(νe → νµ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0151 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0026 sin δCP (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0264 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0048 sin δCP), (12) P(νe → ντ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0119 ∓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0026 sin δCP (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0209 ∓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0048 sin δCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' (13) 6 We will now evaluate the sensitivities on neutrino CP violation, taking δCP = ±π/2 and Eν = 7 GeV as benchmark parameters: 1) Firstly, we consider the tau (anti-) neutrino appearance from muon and electron neutrino oscillations: µ+ → e+ ¯νµ1 νe1 =⇒ ¯νµ1 → ¯ντ1, νe1 → ντ1, (14) µ− → e− νµ2 ¯νe2 =⇒ νµ2 → ντ2, ¯νe2 → ¯ντ2, (15) where ‘1’ and ‘2’ symbol the two far detectors as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Notice that the CP dependence of P(νe → ντ) and P(¯νµ → ¯ντ) as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 10 vary in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' If we count on tau-related events in the far detector inclusively, this means our signal doubles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The sensitivity can be estimated then as ¯ντ2 + ντ2 − ¯ντ1 − ντ1 √¯ντ2 + ντ2 + ¯ντ1 + ντ1 (16) which is around 4 × 260/ √ 60000 ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='2 standard deviations (σ) in one year, and can reach near 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='4σ in 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Although only statistics are taken into account here, systematics should be able to be reduced efficiently due to the symmetric property of the proposed device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Furthermore, it is possible to exchange µ+ and µ− flying routes, thus further reducing possible bias or systematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 2) Secondly, if the far detector can distinguish tau neutrino from antineutrino such as the CERN SHiP ex- periment [33], then with only P(νe → ντ), we can already have higher CP sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The sensitivity can be estimated then as ¯ντ2 − ντ1 √¯ντ2 + ντ1 (17) which is around 2 × 260/ √ 2200 ∼ 11 σ in one year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 3) Finally, one can also exploit electron to muon oscillation which has also clear sensitivity on neutrino CP phase, if the far detector can distinguish muon neutrino from antineutrino, possibly can be achieved with moderate magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' The sensitivity can be estimated then as 2 × 260/ √ 3000 ∼ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='5 σ in one year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' By combing all three options and taking into account statistic errors, one can achieve a sensitivity of far more than 5 σ in one year, for δCP = ±π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Option 1) corresponds to a most conservative scenario, where one can reuse such as the DUNE-like detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Options 2) and 3) lead to much larger sensitivities on neutrino CP violation, yet putting additional requirements on the experimental side, such as to distinguish µ+ from µ−, or τ + from τ − under the help of moderate magnet field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' In Summary, we propose here a new idea to exploit collimated muon beams which generate symmetric neutrino and antineutrino sources: µ+ → e+ ¯νµ νe and µ− → e− νµ ¯νe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Interfacing with long baseline neutrino detectors such as in DUNE or T2K, this experiment can be useful to measure tau neutrino properties, and also to probe neutrino CP phase, by measuring muon (anti-) neutrino disappearance or tau (anti-)neutrino appearance, and differences between neutrino and antineutrino rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' There are several significant benefits leading to large neutrino flux and high sensitivity on CP phase, including 1) collimated and manipulable muon beams, which lead to a larger acceptance of neutrino sources in the far detector side;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 2) symmetric µ+ and µ− beams, and thus symmetric neutrino and antineutrino sources, which make this proposal ideally for measuring neutrino CP violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' More importantly, ¯νe,µ → ¯ντ and νe,µ → ντ, and, ¯νe → ¯νµ and νe → νµ oscillation signals can be collected simultaneously, with no needs for separate specific runs for neutrinos or antineutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' It is also possible to exchange µ+ and µ− flying routes, thus further reducing possible bias or systematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' In an optimistic way, we estimate 104 tau (anti-) neutrinos can be collected per year thus our proposal can serve as a brighter tau neutrino factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Moreover, 5 standard deviations of sensitivity can be reached easily for CP phase as |π/2|, with only 1 year of data taking, by combining tau and muon (anti-) neutrino appearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Notice for tau (anti-)neutrino appearance, the CP dependence of P(νe → ντ) and P(¯νµ → ¯ντ), or P(¯νe → ¯ντ) and P(νµ → ντ) vary in the same direction, thus signal rates double and one can reuse directly DUNE-like far detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' In this draft, we mainly provide a preliminary estimation of the feasibility study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' A detailed study is surely necessary to follow up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' On the other hand, there exist also rich potential to be further explored with such a proposal that connects energy and neutrino frontiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Especially, one can imagine a post-DUNE (or in parallel to DUNE as the probe channels are indeed orthogonal and thus complementary) experiment with neutrinos from a strong muon source located at the Fermilab site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' This connection between energy and neutrino frontiers can also serve as a precursor for 7 future high-energy muon colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Notice that a muon collider requires a 1–2 orders of magnitude more intense beam as compared with the number (dNµ/dt ∼ 1012/sec ) listed above as our benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Thus with the development of a more intensive muon beam targeting future muon colliders, it surely will improve further the neutrino potential of the current proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work is supported in part by the National Natural Science Foundation of China under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 12150005, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 12075004, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 12061141002, by MOST under grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 2018YFA0403900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [PKU SYSU Novel Collider Group], [arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='06690 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Ellis, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Heinemann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' de Blas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Cepeda, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Grojean, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Maltoni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Nisati, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Petit, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Rattazzi and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Verkerke, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='11775 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Bagger, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Barish, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Belomestnykh, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Bhat, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Brau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Demarteau, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Denisov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Eno, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Geddes and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Grannis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='06164 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Aime, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Apyan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Mahmoud Mohammed, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Bartosik, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Batsch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Bertolin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Bonesini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Bottaro, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Buttazzo and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Capdevilla Roldan, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='07256 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Stratakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [Muon Collider], [arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='08033 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='acc-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Jindariani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [Muon Collider], [arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='07224 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='ins-det]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' de Blas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [Muon Collider], [arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='07261 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Quigg, AIP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 435, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1, 242-257 (1998) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='56235 [arXiv:hep-ph/9803326 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Antonelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Boscolo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Di Nardo and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Raimondi, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' A 807, 101-107 (2016) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='097 [arXiv:1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='04454 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='acc-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Alesini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Antonelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Biagini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Boscolo, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Blanco-Garc´ıa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Ciarma, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Cimino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Iafrati, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Giribono and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Guiducci, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='05747 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='acc-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Vergados, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Ejiri and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Simkovic, Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 75, 106301 (2012) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1088/0034-4885/75/10/106301 [arXiv:1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0649 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [12] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Maki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Nakagawa and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Sakata, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 28, 870-880 (1962) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1143/PTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='870 [13] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Pontecorvo, Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 53, 1717-1725 (1967) [14] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [Daya Bay], [arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='14988 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [15] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Adamson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [NOvA], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 118, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='15, 151802 (2017) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='151802 [arXiv:1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='05891 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [16] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [T2K], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 121, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='17, 171802 (2018) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='171802 [arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='07891 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [17] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [Super-Kamiokande], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' D 97, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='7, 072001 (2018) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='072001 [arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='09126 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Acero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [NOvA], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 123, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='15, 151803 (2019) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='151803 [arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='04907 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Fukugita and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Yanagida, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' B 174, 45-47 (1986) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1016/0370-2693(86)91126-3 [20] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [T2K], Nature 580, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='7803, 339-344 (2020) [erratum: Nature 583, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='7814, E16 (2020)] doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1038/s41586- 020-2177-0 [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Walsh [T2K], [arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='01164 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Acero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [NOvA], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' D 106, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='3, 032004 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='032004 [arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='08219 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [23] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [Hyper-Kamiokande], [arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='04163 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='ins-det]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [24] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Abi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [DUNE], [arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='03005 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [25] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Abi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [DUNE], [arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='04797 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [26] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Abi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [DUNE], JINST 15, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='08, T08009 (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1088/1748-0221/15/08/T08009 [arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='03008 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='ins-det]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [27] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' King, AIP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 530, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1, 142-164 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [28] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Zyla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [Particle Data Group], PTEP 2020, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='8, 083C01 (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1093/ptep/ptaa104 [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Formaggio and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Zeller, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 84, 1307-1341 (2012) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1103/RevModPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1307 [arXiv:1305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='7513 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' De Gouvˆea, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Kelly, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Stenico and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Pasquini, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' D 100, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1, 016004 (2019) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='016004 [31] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Jarlskog, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 55, 1039 (1985) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1039 [32] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Workman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [Particle Data Group], PTEP 2022, 083C01 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1093/ptep/ptac097 8 [33] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Ahdida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [SHiP], Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' C 82, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='5, 486 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1140/epjc/s10052-022-10346-5 [arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='01487 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='ins-det]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [34] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Branco, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Ferreira, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Lavoura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Rebelo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Sher and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Silva, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' 516, 1-102 (2012) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='physrep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='002 [arXiv:1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='0034 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [35] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' de Medeiros Varzielas and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Talbert, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' C 79, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='6, 536 (2019) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='1140/epjc/s10052-019-7047-2 [arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='10484 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' [36] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Qian, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Deng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Xiao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Gao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Levin, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' Lu and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content=' You, [arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} +page_content='15350 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf'} diff --git a/X9E5T4oBgHgl3EQfdQ8b/vector_store/index.pkl b/X9E5T4oBgHgl3EQfdQ8b/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..be696110dff21eb6f9b5da8af527a59cc1b532e8 --- /dev/null +++ b/X9E5T4oBgHgl3EQfdQ8b/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3f28271ec3a975c7cd2c00c161f548adce2092e7560d58f693e9840c2bc4b4c8 +size 124221 diff --git a/XtAzT4oBgHgl3EQfYvw0/content/tmp_files/2301.01339v1.pdf.txt b/XtAzT4oBgHgl3EQfYvw0/content/tmp_files/2301.01339v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0541ac20b78b03da7bf24adc907bdedc4bb9235c --- /dev/null +++ b/XtAzT4oBgHgl3EQfYvw0/content/tmp_files/2301.01339v1.pdf.txt @@ -0,0 +1,2578 @@ +arXiv:2301.01339v1 [math.NA] 3 Jan 2023 +DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL +COMPONENT ANALYSIS +JIAN-GUO LIU AND ZIBU LIU +Abstract. Oja’s algorithm of principal component analysis (PCA) has been one of the +methods utilized in practice to reduce dimension. In this paper, we focus on the convergence +property of the discrete algorithm. To realize that, we view the algorithm as a stochastic +process on the parameter space and semi-group. We approximate it by SDEs, and prove +large time convergence of the SDEs to ensure its performance. This process is completed in +three steps. First, the discrete algorithm can be viewed as a semigroup: Skϕ = E[ϕ(W(k))]. +Second, we construct stochastic differential equations (SDEs) on the Stiefel manifold, i.e. the +diffusion approximation, to approximate the semigroup. By proving the weak convergence, +we verify that the algorithm is ’close to’ the SDEs. Finally, we use reversibility of the SDEs +to prove long time convergence. +1. Introduction +Principal component anlysis (PCA) is a basic tool in dimension reduction. Due to explo- +sion of data, command of efficient PCA algorithms is increasing. In this paper, we focus on +the online PCA algorithm proposed by Oja in [14], which is also named as the stochastic +gradient ascent (SGA) method. +Suppose that x ∈ Rn is a mean zero random variable (R.V.). Let +A := E[xxT ] +(1.1) +be the covariance matrix. Traditional PCA algorithms diagonalize A to derive principal +eigenvectors (i.e. the principal components) of A. However, due to limitation of storage +and high dimension of data in recent fields such as deep learning, explicit form of the dense +matrix A may not be available. Therefore, practitioners prefer ’online’ algorithms: it only +requires a limited amount of samples of x in each iteration. To solve this problem, Oja +proposed the following SGA method in [14]: +(1.2) +wj(k) = wj(k − 1) + η(k)xT(k)wj(k − 1)[x(k) − (xT(k)wj(k − 1))wj(k − 1) +− 2 +j−1 +� +i=1 +(xT(k)wi(k − 1))wi(k − 1)], j = 1, 2, ..., p. +This algorithm iterates the first p principal components wj ∈ Rn, j = 1, 2, ..., p. +Here +x(k), k = 1, 2, ... are independent samples of x, η(k), k = 1, 2, ... are learning rates. +Algorithm (1.2) determines a discrete time Markovian process, i.e. W(k) = [w1(k), w2(k), ..., wp(k)]. +The main goal of this paper is to gain a good understanding of this random process (R.P.) +from the view of semigroups, diffusion approximations and SDEs. +Key words and phrases. machine learning, dimensionality reduction, online principal component analysis, +gradient flow, stochastic differential equations, random matrix. +1 + +2 +J.-G. LIU AND Z. LIU +First of all, as η → 0, replacing xxT by A in (1.2), we derive the corresponding ODE: + + + + + + + + + + + +˙q1 = Aq1 − (q1 · Aq1)q1, +˙qj = Aqj − (qj · Aqj)qj − 2 +j−1 +� +i=1 +(qi · Aqj)qi, j = 2, 3, ..., n. +qi(0) = qi,0, i = 1, 2, ..., p. +(1.3) +Convergence properties including global convergence, stable manifolds and exponential con- +vergence were thoroughly investigated in our previous work [10]. In particular, we proved +that for almost every initial value Q0 ∈ O(n), the solution exponentially converges to the +eigenbasis (up to a sign). Moreover, the eigenvectors are aligned in a descending order of +the eigenvalues. See Theorem 5.2 in [10]. As far as we know, this is the first complete result +providing global exponential convergence and closed formula for stable manifolds of a PCA +flow [1]. +Given convergence of the corresponding ODE, we aim at proving similar result for the +discrete algorithm (1.2) in this paper. We consider this problem in three steps: Viewing +the SGA iteration as a semigroup, we construct proper diffusion approximations and prove +convergence of diffusion approximations to ensure the performance of the algorithm. +First, we view the SGA method as a semigroup. It can be reformulated in the following +form: +W(k + 1) = W(k) + η · G(x(k + 1)xT(k + 1), W(k)). +Here G ∈ Rn×p is defined in (2.1). For arbitrary test function ϕ ∈ C(Rn×p), define +Sϕ(W) := Eϕ(W + ηG(xxT, W)). +(1.4) +Under this notation, if the initial datum of the SGA method is W0, then the Markovian +property yields +Skϕ(W0) = Eϕ(W(k)). +(1.5) +Thus, convergence of the SGA method can also be interpreted as the convergence of the +semigroup {Sk}, k = 1, 2, .... +Second, we construct appropriate diffusion approximations. Although the SGA method +does not preserve W(k) to stay on the Stiefel manifold O(n × p), the desired result, i.e. the +eigenbasis, is in O(n × p). Thus, we aim at deriving a good diffusion approximation of the +semigroup S and the SGA method. It should be an SDE that stays on the Stiefel manifold. +The classical method to derive an SDE on a certain manifold is to project a Stratanovich +SDE onto the desired manifold [5]: +dW = PTWM(G(A, W)dt + σ ◦ dB). +Here PTZM is the projection operator onto the tangent space at W on M. If the semigroup +is close to the diffusion process, then by proving convergence of the diffusion process in some +sense, we can also guarantee the performance of the algorithm. +Finally, we prove convergence of the diffusion process. The way to prove it is by seeking +’reversibility’. In fact, if the Fokker-Planck equation of the SDE can be recast in the following +form: +∂tρ = ∇ · (ρ∇U + ∇ρ) = ∇ · +� +e−U∇ +� ρ +eU +�� +for some potential U, then the diffusion process satisfies detailed-balance condition, i.e., the +process is reversible. Then, Poincare’s inequality can ensure the exponential convergence + +DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS +3 +of ρ in a certain L2 sense. This proves the convergence of SDEs, which also finishes our +analysis. +Under this framework of analysis, we will provide our main results and revise previous +literature. +1.1. Previous results and unsolves problems. One of the important features of (1.2) +which other algorithms do not possess is its semi-decoupling feature: iteration of wj does +not depend on wi, i > j. This feature facilitates its implementation in neural networks [13], +thus researchers focus on it. This feature was also extended to the corresponding ODE, i.e. +(2.5). Based on this semi-decoupling property, we also proved all convergence results of (2.5) +in [10]. +However, a satisfying convergence result for (1.2) is still wanting. Since Oja and Karhunen +proposed (1.2) in [14], its convergence behavior has always been a focus in analysis of online +PCA. Oja and Karhunen used stochastic approximation to derive almost sure convergence of +(1.2) under an implicit condition on the distribution of x [14, 12, 13]. This implicit condition +requires the iteration to visit a compact set containing the equilibrium for infinitely many +times. However, this condition is difficult to verify in practice. +To improve Oja’s result, more recently, authors in [7] derived weak convergence of the first +component of (1.2) to a multidimensional Ornstein-Uhlenbeck process. Following [7], the +algorithm conducting full orthonormalization was considered and the weak convergence of +all components was derived [9]. +The diffusion approximation of the first component of (1.2) was also considered in our +previous work [3] in which both first -order and second-order approximation were derived. +As a corollary, the weak convergence of the first component of the SGA method was verified. +However, a diffusion approximation of the whole SGA iteration method (1.2) is still an open +problem. +The main tool utilized in [3] is the Lax equivalence theorem. An alternative stochastic +analysis approach to prove the convergence is developed by Milstein [11], which was adopted +to derive diffusion approximations of the stochastic gradient descent (SGD) method [8]. +1.2. Main results. First of all, we investigated properties of the semigroup Sk, k = 1, 2...,. +In particular, we proved the stability and the regularity of it. For the stability, we proved +that for a fixed terminal time T, if ∥W0∥F ≤ r, then there exist constants C and η0 that +depend on r, T and the distribution of x such that +∥W(k)∥F ≤ C +holds for all k = 1, 2, ..., +�T +η +� +. Here η ∈ (0, η0) is the learning rate in (1.2). See Lemma 3.1. +We proved the stability because it is necessary for the application of the Lax equivalence +theorem. For the regularity, we prove that Skϕ, k = 1, 2, ... admit the same order regularity +as ϕ, i.e. +∥Skϕ∥Cm(B(0,r)) ≤ C∥ϕ∥Cm(B(0,r′)). +Here C and r′ are constants that depend on m, r, T and the distribution of x. See Theorem +3.1 for details. +Second, we constructed the desired diffusion approximations. We proved that the following +family of SDEs +˙W = G(A, W) + ηPTWO(n)F(W) + √ηPTWO(n) ˙Z, W(0) = W0 ∈ O(n) + +4 +J.-G. LIU AND Z. LIU +will stay on the Stiefel manifold for all t > 0. See Lemma 3.5. Here +˙Z = ( ˙Zij)n×n, +˙Zij(W) = Hijkl(W) ◦ ˙Bkl, +where H = (Hijkl)n×n×n×n are coefficients and ˙Bkl is the white noise. See (3.13) for detail. +In fact, the SDE (3.13) serves as the first-order diffusion approximation of the SGA +method. we proved that under proper regularity conditions of the test function ϕ, there +exists a constant C1 = C1(x, T, η) such that +sup +W∈O(n),kη≤T +|Skϕ(W) − u(W, kη)| ≤ C1(x, T, ϕ)η. +Here u(W, t) is the solution to the Kolmogorov equation determined by (3.13), with the +initial value ϕ. See Theorem 3.2. The main idea of the proof comes from the Lax equivalence +theorem [6]: stability and consistence is equivalent to convergence. The consistence is ensured +by Taylor’s expansion, see Section 6 for details. +A natural question is that whether higher- order approximation exists. Unfortunately, the +answer is no. We proved that the possible second order approximation, which is an SDE, +does not stay on the Stiefel manifold. See Lemma 3.6. This instability is probably due to +the omitted higher-order terms in the SGA algorithm: second and higher-order (w.r.t. η) +terms were neglected in (1.2) when conducting the Gram-Schmidt orthogonalization. +Finally, for two special cases, we proved the exponential convergence of the SDE. As we +introduced before, we seek for reversibility to prove the exponential convergence. +First, we consider the overdamped Langevin equation on the Stiefel manifold: +dQ(t) = PTQO(n) ◦ (−∇U(Q)dt + σdW(t)). +The exponential convergence of it is proved in Section 3. If we select the potential U as +the weighted Rayleigh quotient (see [10]) and let σ = 0, then the Oja-Brockett flow [2] is +recovered. We have to emphasize that the overdamped Langevin equation is not a special +case of (3.13) since G in (1.2) is not a gradient of a certain potential. +Second, for n = 2 of (3.13), the SDE is rewritten as +dW = F1(W)dt + √η · c(W)W ◦ dZ. +Here F1 is defined in (2.5) and c(W) is a scalar. See details in (4.11). Exponential conver- +gence of this case is proved in Theorem 4.1. The main approach is to consider the dynamics +of the rotational angle of W ∈ O(2), which is a one-dimensional SDE, and the reversibility +automatically holds. +2. Premier +First, we rewrite (1.2) by matrices. For Λ, Q ∈ Rn×n, define +(2.1) +Σ(Λ, Q) := +n +� +j=1 +j−1 +� +k=1 +EjQTΛQEk − EkQTΛQEj, +G(Λ, Q) := ΛQ − QQTΛQ + QΣ(Λ, Q). +Then (1.2) also reads as +� +A(k) = x(k)xT (k), +W(k) = W(k − 1) + ηkG(A(k), W(k − 1)). +(2.2) + +DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS +5 +In our previous paper [10], we thoroughly investigated corresponding ODE, which can be +written as: + + + + + + + +˙Q = Q +n +� +j=1 +j−1 +� +k=1 +(EjQTAQEk − EkQTAQEj), +Q(0) = Q0 ∈ O(n). +(2.3) +We define +F1(Q) := QΣ(A, Q). +(2.4) +Thus one can rewrite (2.3) as +� +˙Q = F1(Q), +Q(0) = Q0 ∈ O(n). +(2.5) +From now on, we will use (2.5) in all proofs. +2.1. Notations and assumptions. We will follow the convention of notations in our pre- +vious paper [10]. +We assume that ν is compact, i.e., there exists a constant M > 0 such that +∥x∥2 ≤ M. +(2.6) +In the following sections, we will adopt both the matrix representation and the component- +wise representation of (2.5), thus we clarify the notation here. qi, i = 1, 2, ..., n represent +the column vectors of Q in order, i.e. +Q = [q1, q2, ..., qn], +(2.7) +while ˜qi, i = 1, 2, ..., n represent the row vectors of Q in order, i.e. +QT = [ ˜q1 +T, ˜q2 +T, ..., +˜qn +T]. +(2.8) +For each entry, qi,j, i, j = 1, 2, ..., n represent the entries at ith row, jth column of the matrix +Q, i.e. +qj = (q1,j, q2,j, ..., qn,j)T. +(2.9) +The canonical orthonormal basis in Rn is denoted as ej, j = 1, 2, ..., n, which are written in +column vectors, i.e. +In = [e1, e2, ..., en]. +(2.10) +Here In is the identity matrix of size n. +For M, N ∈ Rn×n, ∥M∥F represents the Frobenius norm of M and ⟨M, N⟩F represents +the inner product in the Frobenius sense: +∥M∥ = +� +tr(MMT), ⟨M, N⟩F = tr(MTN). +(2.11) +For x = (x1, x2, ..., xn) ∈ Rn, ∥x∥2 represents the ℓ2 norm of x, i.e. +∥x∥2 = +� +� +� +� +n +� +j=1 +|xj|2 +(2.12) +Suppose that the eigenvalues of A are all single, i.e. of multiplicity one. Denote them as +λ1 > λ2 > ... > λn > 0 +(2.13) + +6 +J.-G. LIU AND Z. LIU +in descending order. Without loss of generality, we assume that A is diagonal: +A = diag{λ1, λ2, ..., λn}. +By default, omitted proofs of Lemmas and other important but complicated computations +are available Section 6. +3. Diffusion approximation of the online PCA algorithm +In this section, we consider the iteration scheme (1.2). We also assume that the learning +rates are constant, i.e. ηk = η > 0, k = 1, 2, .... Under these assumptions, we derived the +diffusion approximation of (1.2). Our main results imply that (2.5) is the weak limit of (1.2) +as η approaches 0. We also derived families of matrix-valued SDEs (invariant in the Steifel +manifold) which serve as first order weak approximations. In particular, (2.5) is understood +as a special case of these SDEs by taking the time step size η = 0. +3.1. The semigroup. The matrix-valued discrete time Markov process defined in (1.2), i.e. +W(k), k = 1, 2, ..., is time homogeneous because x(k) share the same distribution. +Following the notations in [3], we denote the expectation under the distribution of this +Markov chain starting from W0 as EW0. In our discussion, W0 is assumed to be deterministic +though it could be a random variable in general contexts. Denote the law of W(k) (starting +from W0) as µk(·; W0) and the transition probability as µ(V, ·). +Then by the Markov +property, for any Borel set E ⊂ Rn×n, +µk+1(E; W0) = +� +Rn×n µ(V, E)µk(dV; W0) = +� +Rn×n µk(E; U)µ(W0, dU). +For a fixed test function ϕ ∈ L∞(Rn×n), define +uk(W0) = EW0 [ϕ(W(k))] = +� +Rn×n ϕ(V)µk(dV; W0), k = 0, 1, 2, ... +(3.1) +Here W(k) is defined in (1.2). The Markov property yields +uk+1(W0) = EW0[EW0[ϕ(W(k + 1))|W(1)]] += EW0[uk(W(1))] += +� +Rn×n µ(dW1, W0) +� +Rn×n ϕ(V)µk(dV; W1). +Then by (1.2), we derive that for any W ∈ Rn×n, +uk+1(W) = Euk(W + ηG(xxT, W)) := Suk(W), +(3.2) +hence u0(W) = ϕ(W) and {Sk}k≥0 forms a semigroup. +Before discussing the diffusion approximation, we derive some basic properties of the +Markov chain and the semigroup. +Lemma 3.1. (stability) Fix a real number r > 0 and a terminal time T > 0. Let W(k), k = +0, 1, 2, ... be the Markov chain generated by (1.2) with an initial datum W0 satisfying ∥W0∥F ≤ +r. Then there exist constants η(r, M, T) > 0 and C(r, M, T) > 0 which depend on r, T and +M in (2.6) such that for any 0 < η ≤ η(r, M, T) and k = 0, 1, 2, ..., +�T +η +� +, +P +� +∥W(k)∥2 +F ≤ C(r, M, T) +� += 1, +(3.3) +i.e., W(k) is uniformly bounded for any time discretization with time step size less than +η(r, M, T). + +DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS +7 +See Section 6 for the proof of this lemma. Based on Lemma 3.1, we prove that uk(W) +possesses the same regularity as the test function ϕ. Admissible sets of test functions are +Cm +b (Rn×n) := + + +f ∈ Cm(Rn×n) +��∥f∥Cm(Rn×n) := +� +|α|≤m +|Dαf|∞ + + + , m = 1, 2, .... +(3.4) +We denote the open ball in (Rn×n, ∥ · ∥F) centered at M ∈ Rn×n with radius r by B(M, r), +i.e. +B(M, r) := {N ∈ Rn×n : ∥N − M∥F < r}. +(3.5) +Theorem 3.1. (properties of the semigroup) Fix a real number r > 0 and a terminal time +T > 0. Let {uk}k≥0, {Sk}k≥0 be the semigroup defined in (3.2) and M be the upper bound +in (2.6). Suppose that ϕ ∈ Cm +b (Rn×n) where m ≥ 1. Then: +(1) (L∞ contraction) S : L∞(Rn×n) → L∞(Rn×n) is a contraction; +(2) (regularity) there exist constants η(r, m, M, T) > 0 and C(r, m, M, T) > 0 such that +for any 0 < η ≤ η(r, m, M, T): +∥uk∥Cm(B(0,r)) ≤ C(r, m, M, T)∥ϕ∥Cm(B(0,eT (2M2+1)/2r)). +(3.6) +Proof. By Lemma 3.1, for any η ≤ η(r, M, T), there exists C(r, M, T) > 0 such that ∥W∥F ≤ +C(r, M, T) holds almost surely for k = 0, 1, 2, ..., [T/η]. Notice that G is a polynomial w.r.t. +W, so there exists a constant C′(r, M, T) > 0 such that for any W ∈ B(0, r) and indices +(i, j) and (i′, j′), +���� +∂(G(x(k)x(k)T , W))i′,j′ +∂wi,j +���� ≤ C′(r, M, T). +(3.7) +Here (G)i′,j′ represents the entry at the i′th row and j′th column in the matrix G. +According to the proof of Lemma 3.1, we know that +∥W(k)∥2 +F ≤ (1 + (2M2 + 1)η)∥W(k − 1)∥2 +F, k = 1, 2, ..., [T/η]. +Thus if W ∈ B(0, r), then +W + ηG(x(k)x(k)T, W) ∈ B(0, r +� +1 + (2M2 + 1)η) ⊂ B +� +0, +� +1 + (2M2 + 1)η +2 +� +r +� +. +(3.8) +Denote C′′ = (2M2 + 1)/2. Now we proceed to prove the theorem. +(1) Suppose that ψ ∈ L∞(Rn×n), then the L∞ contraction is derived directly by (3.2): +(3.9) +∥Sψ∥L∞(Rn×n) = +sup +W∈Rn×n |Eψ(W + ηG(xxT, W))| +≤ +sup +W∈Rn×n |ψ(W)| +≤ ∥ψ∥L∞(Rn×n). +(2) We prove the case m = 1 by induction. The proof for the case of m ≥ 1 is similar. +Because u0 = ϕ, so the conclusion holds for k = 0. +Now for k ≥ 0, by the dominant + +8 +J.-G. LIU AND Z. LIU +convergence theorem (DCT), we have for any W ∈ B(0, r) and 1 ≤ i, j ≤ n, +(3.10) +∂uk+1 +∂wi,j +(W) = +� +1≤i′,j′≤n +E +� ∂uk +∂wi′,j′ (W + ηG(x(k)x(k)T, W)) · ∂(W + ηG(x(k)x(k)T, W))i′,j′ +∂wi,j +� += E +� ∂uk +∂wi,j +(W + ηG(x(k)x(k)T, W)) +� ++ η +� +1≤i′,j′≤n +E +� ∂uk +∂wi′,j′ (W + ηG(x(k)x(k)T, W)) · ∂(G(x(k)x(k)T , W))i′,j′ +∂wi,j +� +. +Remember that (3.8) holds, so +����E +� ∂uk +∂wi,j +(W + ηG(x(k)x(k)T, W)) +����� ≤ +���� +∂uk +∂wi,j +���� +L∞(B(0,(1+C′′η)r)) +Substituting this into (3.10) yields +���� +∂uk+1 +∂wi,j +(W) +���� ≤ +���� +∂uk +∂wi,j +���� +L∞(B(0,(1+C′′η)r)) ++ ηC′(r, M, T) +� +1≤i′,j′≤n +���� +∂uk +∂wi′,j′ (W + ηG(x(k)x(k)T, W)) +���� . +Then using (3.7), taking L∞ norm on both sides and summing up over 1 ≤ i, j ≤ n, we +derive +� +1≤i,j≤n +���� +∂uk+1 +∂wi,j +���� +L∞(B(0,r)) +≤ +� +1≤i,j≤n +���� +∂uk +∂wi,j +���� +L∞(B(0,(1+C′′η)r)) ++ n2C′(r, M, T)η∥uk∥C1(B(0,(1+C′′η)r)). +This inequality yields +∥uk+1∥C1(B(0,r)) ≤ (1 + n2C′(r, M, T)η)∥uk∥C1(B(0,(1+C′′η)r)). +(3.11) +By induction, we have +(3.12) +∥uk∥C1(B(0,r)) ≤ (1 + n2C′(r, M, T)η)k∥u0∥C1(B(0,(1+C′′η)kr)) +≤ (1 + n2C′(r, M, T)η)T/η∥ϕ∥C1(B(0,(1+C′′η)T/ηr)) +≤ en2C′(r,M,T)T∥ϕ∥C1(B(0,eT (2M2+1)/2r)). +□ +3.2. The diffusion approximation. In this section, we discuss the diffusion approximation +of the semigroup (3.2). +3.2.1. SDEs on the Stiefel manifold. Because the SGA method aims to derive the correct +unit eigenbasis, so we expect that our SDE should stay on the Stiefel manifold. Therefore, +we consider the following family of SDEs: +˙W = G(A, W) + ηPTWO(n)F(W) + √ηPTWO(n) ˙Z, W(0) = W0 ∈ O(n). +(3.13) +Here PTWO(n) is the projection onto the tangent space of O(n) at W, see (6.5); ˙Z is defined +as +˙Z(W) = ( ˙Zij)n×n, +˙Zij(W) = Hijkl(W) ◦ ˙Bkl; +(3.14) + +DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS +9 +H(W) = (Hijkl(W))n×n×n×n is the coefficient tensor of the Brownian motion; B = (Bij)n×n +is the standard Brownian motion in Rn×n; The notation ’◦’ represents that (3.13) is an SDE +in the Stratonovich sense. +By letting η = 0 in (3.13), the SDE degenerates to the ODE +˙W = G(A, W), W(0) = W0 ∈ O(n). +This is exactly (2.5). Thus we expect that (3.13) serves as the diffusion approximation of +the SGA method. +We first check that for arbitrary F(W) ∈ Rn×n and H(W) ∈ Rn×n×n×n, (3.13) stays on +the Stiefel manifold if W0 ∈ O(n). As preparation, we first rewrite the projection operator. +Lemma 3.2. Let PTWO(n) defined as in (6.5) for some W = (wij)n×n ∈ O(n). Let +P := (Pijkl)n×n×n×n, Pijkl := 1 +2(wiswksδjl − wkjwil). +(3.15) +Then we have: +(i) (projection) for any F = (fij)n×n, +(PTWO(n)F)ij = Pijklfkl = 1 +2(fij − wilfklwkj); +(3.16) +(ii) (symmetry) Pijkl = Pklij, i.e. P is symmetric; +(iii) (idempotence) PijklPklrs = Pijrs, i.e. P2 = P. +Proof. Because W ∈ O(n), so wiswks = δik and +Pijkl = 1 +2(δikδjl − wkjwil). +(3.17) +We first prove (i). Because PTWO(n)F = 1 +2(F − WFTW), we have +(PTWO(n)F)ij = 1 +2(fij − wilfklwkj). +By (3.17), we derive +Pijklfkl = 1 +2(δikδjl − wkjwil)fkl = 1 +2(fij − wilfklwkj). +Thus +(PTWO(n)F)ij = Pijklfkl = 1 +2(fij − wilfklwkj). +(ii) is obvious: +Pijkl = 1 +2(δikδjl − wkjwil) = 1 +2(δkiδlj − wilwkj) = Pklij. +For (iii), we have +PijklPklrs = 1 +4(δikδjl − wkjwil)(δkrδls − wrlwks) += 1 +4(δirδjs − wrjwis − wrjwis + wkjwilwrlwks). +Because W ∈ O(n), we have +wilwrl = δir, wkjwks = δjs. + +10 +J.-G. LIU AND Z. LIU +Thus +PijklPklrs = 1 +2(δirδjs − wrjwis) = Pijrs. +□ +Remark 3.1. We define Pijkl as in (3.15) instead of (3.17), because in this case, even though +W /∈ O(n), the following equality still holds: +wijPtjkl + wtjPijkl = 0. +(3.18) +This is helpful in proving invariance of Stiefel manifold, i.e. Lemma 3.3. +Now we are ready to check that the Stiefel manifold is invariant for (3.13). +Lemma 3.3. Consider (3.13). If W0 ∈ O(n), then W(t) ∈ O(n) holds for all t > 0. +Proof. By Lemma 3.2, we can rewrite (3.13) as +dwij = gij(A, W)dt + ηPijkl(W)fkl(W)dt + Pijkl(W)Hklrs(W) ◦ dBrs. +Then for 1 ≤ i, t ≤ n, remember that the SDE is in the Stratonovich sense, we have +d(wijwtj) = wij ◦ dwtj + wtj ◦ dwij += (wijgtj(A, W) + wtjgij(A, W))dt + (wijPtjkl + wtjPijkl)(fkldt + Hklrs ◦ dBrs). +Notice that +wijPtjkl + wtjPijkl = 1 +2[wij(wtswksδjl − wkjwtl) + wtj(wiswksδjl − wkjwil)] += 1 +2(wilwtswks − wijwkjwtl + wiswkswtl − wilwtjwkj) = 0. +Thus +d(wijwtj) = (wijgtj(A, W) + wtjgij(A, W))dt, +which is an ODE system. We can rewrite this in matrices as +d(WWT) +dt += WGT(A, W) + G(A, W)WT += WWTA + AWWT − 2WWTAWWT. +Thus X = WWT satisfies the algebraic Riccati equation: +dX +dt = AX + XA − 2XAX, +with initial value X = In. Thus by results in [18], we know that X(t) = In holds. Therefore, +W(t) ∈ O(n). +□ +3.2.2. Fokker-Planck equation and Kolmogorov equation. In the Itˆo sense, (3.13) reads as +˙W = G(A, W) + ηPTWO(n)F(W) + η +2J(W) + √ηPTWO(n) ˙Y(W), +(3.19) +Here Y is defined as +˙Y(W) = ( ˙Yij)n×n, ˙Yij(W) = Hijkl(W) ˙Bkl. +(3.20) +The correction term J is then given by +(3.21) +K = (Kijkl)n×n×n×n, Kijkl = PijrsHrskl +J = (Jij)n×n, Jij(W) := Krsml +∂Kijml +∂wrs +. + +DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS +11 +Then (3.19) could be rewritten as +˙wij = gij(A, W) + ηPijklfkl + η +2Jij + √ηKijkl ˙ +Bkl += gij(A, W) + ηPijklfkl + η +2Jij + √ηPijklHklrs ˙ +Brs +The corresponding backward Kolmogorov equation of (3.19) is +∂tu(W, t) = Lu(W, t), u(W, 0) = ϕ(W), +(3.22) +where L is an elliptic operator defined as +(3.23) +Lu := +� +gij(A, W) + ηPijklfkl + η +2Jij +� ∂u +∂wij ++ η +2KijrsKklrs +∂2u +∂wijwkl += +� +gij(A, W) + ηPijklfkl + η +2Jij +� ∂u +∂wij ++ η +2PijzvHzvrsPklxyHxyrs +∂2u +∂wijwkl +. +The solution of (3.22) is given by +u(W, t) = etLϕ(W) = EW[ϕ(X(t))], +(3.24) +here X(t) ∈ O(n) is the random process (before vectorization) determined by (3.13) (or +equivalently (3.19)) with starting point X(0) = W ∈ O(n). +For more details, we refer +readers to [15]. +On the other hand, the solution of the forward Kolmogorov equation (the Fokker-Planck +equation) is denoted as ρ(W, t) = etL∗ρ0(W): +∂tρ = L∗ρ, ρ(W, t) = ρ0(W), +(3.25) +where ρ0 is the initial distribution of W and L∗ is defined as +(3.26) +L∗ρ := − ∂ +∂wij +�� +gij(A, W) + ηPijklfkl + η +2Jij +� +ρ +� ++ η +2 +∂2 +∂wijwkl +(ρKijrsKklrs) += − ∂ +∂wij +�� +gij(A, W) + ηPijklfkl + η +2Jij +� +ρ +� ++ η +2 +∂2 +∂wijwkl +(ρPijzvHzvrsPklxyHxyrs) . +The solution ρ(W, t) = etL∗ρ0(W) is interpreted as the probability distribution of the +random process X in (3.13) with the initial distribution ρ0. +Notice that {etL}t≥0 and {etL∗}t≥0 form two semigroups, we have the following basic +properties: +Lemma 3.4. Consider {etL}t≥0 in (3.23) and {etL∗}t≥0 in (3.26). Then: +(1) (probability and positivity preserving) if ρ0 ∈ L1(O(n)) and ρ0 ≥ 0, then etL∗ρ0 ≥ 0 +and +∥etL∗ρ∥L1(O(n)) = ∥ρ0∥L1(O(n)). +(3.27) +In particular, etL∗ is a contraction from L1(O(n)) to L1(O(n)). +(2) (L∞ contraction) Suppose that ϕ : O(n) → R is a continuous function. +etL : +L∞(O(n)) → L∞(O(n)) is a contraction, i.e. +∥etLϕ∥L∞(O(n)) ≤ ∥ϕ∥L∞(O(n)). +(3.28) + +12 +J.-G. LIU AND Z. LIU +Proof. The proof of (1) can be found in [3], here we just prove (2). Because ϕ is continuous +and O(n) is compact, we have ϕ ∈ L∞(O(n)). +Suppose W0 ∈ O(n), let W(t) be the +stochastic process generated by (3.13) starting from W0. Then we have +etLϕ(W0) = EW0 [ϕ(W(t))] . +(3.29) +By Lemma 3.3, we know that W(t) ∈ O(n) for all t ≥ 0, thus +|etLϕ(W0)| ≤ EW0|ϕ(W(t))| ≤ ∥ϕ∥L∞(O(n)). +(3.30) +This holds for any W0 ∈ O(n), thus (3.28) holds. +□ +More discussion on contraction and positivity preserving is available in [16]. +Now we are ready to verify that (3.13) serves as a weak diffusion approximation of the +SGA iteration. +3.2.3. First-order diffusion approximations. The method to validate that (3.13) serves as a +diffusion approximation originates from the idea of the Lax equivalence theorem [6], which +was first adopted in [3] for the same purpose. +Theorem 3.2. (first-order diffusion approximation) Fix time T > 0. Consider {uk(W)}k≥0 +defined in (3.1). Suppose that ϕ ∈ C4 +b (Rn×n) and u(W, t) is the solution to (3.22) with the +initial value u(·, 0) = ϕ(·). Then there exist η1(M, T) > 0 and C1(M, T, ϕ) > 0 such that for +any η ∈ (0, η1(M, T)], +sup +W∈O(n),kη≤T +|uk(W) − u(W, kη)| ≤ C1(M, T, ϕ)η. +(3.31) +Here M is the constant in (2.6). +Proof. In the following proof, C is a general constant that varies among equations. +By Theorem 3.1, there exist constants η(M, T) and C(M, T) such that for all k = +0, 1, 2..., [T/η] and η ∈ (0, η(M, T)], +∥uk∥C4(B(0,√n+1)) ≤ C(M, T)∥ϕ∥C4(Rn×n). +(3.32) +Then by Taylor’s expansion w.r.t. η, we have that for any W ∈ O(n), +(3.33) +����uk+1(W) − uk(W) − ηgij(A, W) ∂uk +∂wij +���� ≤ C(M, T)∥uk∥C4(B(0,√n+1))η2 +≤ C(M, T, ϕ)η2. +Details of this computation can be found in Section 6. +By Lemma 3.4, etL is a contraction from L∞(O(n)) to itself. Therefore for any W ∈ O(n), +there exists η′ ∈ (0, η] such that +|eηLuk(W) − uk(W) − ηLuk(W)| ≤ η2 +2 |eη′LL2uk(W)| +≤ η2 +2 ∥L2uk∥L∞(O(n)) +≤ C(M, T, ∥uk∥C4(B(0,√n+1)))η2 +≤ C(M, T, ϕ)η2. +Recall the definition of L in (3.23), we can rewrite the above estimate as +����eηLuk(W) − uk(W) − ηgij(A, W) ∂uk +∂wij +���� ≤ C(M, T, ϕ)η2 +(3.34) + +DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS +13 +See Section 6 for details of this computation. +Thus by (3.33) and (3.34), we have +∥uk+1(W) − eηLuk(W)∥L∞(O(n)) ≤ C(M, T, ϕ)η2. +(3.35) +Now for k = 0, 1, 2..., [T/η], define +ek := ∥uk(W) − u(W, kη)∥L∞(O(n)), rk := ∥uk+1(W) − eηLuk(W)∥L∞(O(n)). +(3.36) +Thus rk ≤ C(M, T, ϕ)η2 for k = 1, 2, ..., [T/η]. Notice that u(W, (k + 1)η) = eηLu(W, kη), +we have +(3.37) +ek+1 = ∥uk+1(W) − u(W, (k + 1)η)∥L∞(O(n)) += ∥uk+1(W) − eηLu(W, kη)∥L∞(O(n)) +≤ ∥uk+1(W) − eηLuk(W)∥L∞(O(n)) + ∥eηL(u(W, kη) − uk(W))∥L∞(O(n)) +≤ rk + ek. +The last step in (3.37) used that eηL is a contraction on L∞(O(n)). Thus summing up (3.37) +in k yields +ek ≤ +k−1 +� +l=1 +rl ≤ TC′′(M, T, ϕ) +η +· η2 = C1(M, T, ϕ)η. +(3.38) +This holds uniformly in k, so the claim is proven. +□ +3.2.4. Unstable second order approximation. According to the proof of Theorem 3.2, the key +step that ensures first order approximation is the following estimate on the truncation error, +which is of second order: +∥Sϕ(W) − eηLϕ(W)∥L∞(O(n)) ≤ Cη2. +Here L is defined in (3.23). +To derive second order approximation, we need to carefully select the drift term F and H +in (3.13) such that +∥Sϕ(W) − eηLϕ(W)∥L∞(O(n)) ≤ Cη3. +(3.39) +Direct calculation yields +Sϕ(W) = E +� +ϕ(W + ηG(xxT, W)) +� += ϕ(W) + ηgij(A, W) ∂ϕ +∂wij ++ η2 +2 E +� +gij(xxT, W)gkl(xxT, W) +� +∂2ϕ +∂wij∂wkl ++ O(η3). +Meanwhile, +eηLϕ(W) = ϕ(W) + ηLϕ(W) + 1 +2η2L2ϕ(W) + O(η3). +By (3.23), we have +L2ϕ = L +� +gij(A, W) ∂ϕ +∂wij ++ O(η) +� += gkl(A, W) ∂ +∂wkl +� +gij(A, W) ∂ϕ +∂wij +� ++ O(η) += +� +gkl(A, W)∂gij(A, W) +∂wkl +� ∂ϕ +∂wij ++ gij(A, W)gkl(A, W) +∂2ϕ +∂wij∂wkl ++ O(η). +(3.40) + +14 +J.-G. LIU AND Z. LIU +Comparing the coefficients of +∂2ϕ +∂wij∂wkl +and ∂ϕ +∂wij +, we derive +(3.41) +Pijklfkl = −1 +2Jij − 1 +2gkl(A, W)∂gij(A, W) +∂wkl +, +PijuvHuvrsPklxyHxyrs = E[gij(A − xxT , W)gkl(A − xxT , W)]. +Here F = (fij)n×n and H = (Hijkl)n×n×n×n are coefficients in (3.13). +Details of the +derivation of (3.41) is summarized in Section 6. +To interpret (3.41), one can see that the R.H.S. of the second equation in (3.41) is exactly +the covariance tensor of G(xxT, W). We denote it as +M(W) = (Mijkl(W))n×n×n×n, Mijkl = E +� +gij(xxT − A, W)gkl(xxT − A, W) +� +. +(3.42) +Therefore, we desire suitable H such that +PijuvHuvrsPklxyHxyrs = Mijkl. +(3.43) +This can be realized if W ∈ O(n), which is sufficient for our purpose. +Lemma 3.5. Consider M in (3.42). Then there exists a unique N = (Nijkl)n×n×n×n that +satisfy +(i) (symmetry) Nijkl = Nklij; +(ii) (positive semidefinite) for any (mij)n×n, mijNijklmkl ≥ 0; +(iii) (square root of M) NijrsNklrs = Mijkl. +Moreover, if W ∈ O(n), then +PijrsNrskl = Nijkl, +(3.44) +i.e. PN = N. +See Section 6 for the proof of this lemma. In the view of Lemma 3.5, we also denote N as +N = +√ +M. +(3.45) +If we take +H = N = +√ +M, +in (3.13), then +PijuvHuvrsPklxyHxyrs = PijuvNuvrsPklxyNxyrs = NijrsNklrs = Mijkl +holds if W ∈ O(n). Thus (3.43) is satisfied. +Now we take +F = 0, H = N = +√ +M +(3.46) +in (3.13). By Lemma 3.5, we know that (3.13) stays on O(n), so (3.13) could be rewritten +as +(3.47) +dwij = gij(A, W)dt + √ηPijkl(W)Nklrs(W) ◦ dBrs, += gij(A, W)dt + √ηNijrs(W) ◦ dBrs. +This SDE seems to serve as the second order approximation for the SGA method: we have +gij in (1.2) as the drift term; covariance of gij is also reflected in the Brownian motion. How- +ever, because F in (3.46) does not satisfy (3.41), (3.47) is not a second order approximation, +but a first order approximation by Theorem 3.2. +Moreover, there is even no solution to the first equation of (3.41). +This excludes the +possibility of deriving a second order approximation on the Stiefel manifold. + +DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS +15 +Even with the last try, we require H satisfy (3.41) and just replace PTWO(n)F in (3.13) by +L = (Lij)n×n, Lij := −1 +2Jij − 1 +2gkl(A, W)∂gij(A, W) +∂wkl +, +(3.48) +then resulted SDE still does not stay on the Stiefel manifold. Therefore, we have no second +order diffusion approximation that stays on the Stiefel manifold. +Lemma 3.6. (unstable second order approximation) Consider (3.41), L in (3.48), and J = +(Jij)n×n in (3.21). Then +(i) Suppose that H satisfies (3.41). Then there is no F = (fij)n×n that satisfies +Pijklfkl = Lij, +i.e. the first equation of (3.41) admits no solution. +(ii) Consider the solution W(t) to the following SDE: +dwij = (gij(A, W) + ηLij) dt + √ηNijrs(W) ◦ dBrs, W(0) = W0 ∈ O(n). +(3.49) +Then there is no time interval [t0, t1] such that W(t) ∈ O(n) for t ∈ [t0, t1]. Here +N = (Nijkl)n×n×n×n is the square root of the covariance matrix M (see Lemma 3.5). +Proof. We first prove (i) by contradiction. Suppose there is a solution F, then by (3.18), +wijLtj + wtjLij = wijPtjklfkl + wtjPijklfkl = 0. +Notice +−2(wijLtj + wtjLij) = wijJtj + wtjJlj +� +�� +� +I ++ gkl(A, W) +� +wij +∂gtj(A, W) +∂wkl ++ wtj +∂gij(A, W) +∂wkl +� +� +�� +� +II +. +By definition of J, K in (3.21), we have +I = +∂ +∂wrs +((wijPtjuv + wtjPijuv) HuvmlKrsml) − Ktjml +∂ +∂wrs +(wijKrsml) − Kijml +∂ +∂wrs +(wtjKrsml). +By (3.18), the first term of I is zero, so we have +I = −Ktjml +∂ +∂wrs +(wijKrsml) − Kijml +∂ +∂wrs +(wtjKrsml) += −2KtjmlKijml − wijPtjuvHuvmlKrsml − wtjPijuvHuvmlKrsml += −2KtjmlKijml += −2E[gtj(A − xxT, W)gij(A − xxT, W)]. +For II, we have +II = +∂ +∂wkl +[(wijgtj(A, W) + wtjgij(A, W))gkl(A, W)] +− gtj(A, W) ∂ +∂wkl +(wijgkl(A, W)) − gij(A, W) ∂ +∂wkl +(wtjgkl(A, W)) += −2gij(A, W)gtj(A, W) + gkl +∂ +∂wkl +(gtjwij + gijwtj). + +16 +J.-G. LIU AND Z. LIU +Thus +−2(wijLtj + wtjLij) = I + II += −2E[gtj(A − xxT , W)gij(A − xxT , W)] − 2gij(A, W)gtj(A, W) ++ gkl +∂ +∂wkl +(gtjwij + gijwtj) += −2E[gtj(xxT, W)gij(xxT, W)] + gkl +∂ +∂wkl +(gtjwij + gijwtj). +This can not be zero for general R.V. x, because the first term depends on the fourth-order +momentum while the second term only depends on the second-order momentum. This is a +contradiction, so wijLtj + wtjLij ̸= 0 and (3.41) admits no solution. +For (ii), we still prove by contradiction. +Suppose that for some time interval [t0, t1], +W(t) ∈ O(n). Then +d(WWT) +dt += 0, t ∈ (t0, t1). +By Lemma 3.5, we know that if W ∈ O(n), then PijrsNrskl = Nijkl. So we can rewrite (3.49) +as +dwij = (gij(A, W) + ηLij) dt + √ηPijrsNrskl ◦ dBkl +However, +d(wijwtj) = wijdwtj + wtjdwij += (wijgtj(A, W) + wtjgij(A, W))dt + η(wijLtj + wtjLij)dt ++ √η(wijPtjrs + wtjPijrs)Nrskl ◦ Bkl. +Remember that if W ∈ O(n), then G(A, W) = PTWO(n)G(A, W). So gij = Pijklgkl, which +yields that for all t ∈ (t0, t1), +wijgtj + wtjgij = (wijPtjrs + wtjPijrs)grs = 0 +by (3.18). Due to the same reason, +√η(wijPtjrs + wtjPijrs)Nrskl ◦ Bkl = 0. +Thus +d(wijwtj) = η(wijLtj + wtjLij)dt. +However, (wijLtj + wtjLij) ̸= 0 by (i). Thus d(wijwtj) ̸= 0 for t ∈ (t0, t1), which contradicts +with +wijwtj = δit. +This is a contradiction, the proof is finished. +□ +In the view of Lemma 3.6, (3.49) fails to stay on the Stiefel manifold. +In this case, +(3.22) is a degenerate parabolic PDE with unbounded coefficients, which fails to control the +diffusion in the normal direction of the Stiefel manifold. Thus, utilizing solutions of (3.22) +to approximate the behavior of the semigroup (3.2) is meaningless. + +DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS +17 +4. Reversible diffusion approximation: exponential convergence +In Theorem 3.2, we derived diffusion approximations on the Stiefel manifold. A natural +question is that whether the SDE is ergodic and converges. +In fact, reversibility and Poincare’s inequality ensure exponential convergence. To see this, +the Fokker-Planck operator (3.26) can be recast as +L∗ρ = +∂ +∂wij +� +ρ +� +−gij − ηPijklfkl + η +2KijrsKklrs +∂ log ρ +∂wkl ++ η +2Jij +�� += +∂ +∂wij +�η +2KijrsKklrsρ +�ηJij − 2gij − 2ηPijklfkl +ηKijrsKklrs ++ ∂ log ρ +∂wkl +�� +. +If there exists a function U(W) such that +∂U +∂wkl += ηJij − 2gij − 2ηPijklfkl +ηKijrsKklrs +, +(4.1) +then solutions to ∂tρ = L∗ρ satisfies +∂tρ = +∂ +∂wij +�η +2KijrsKklrsρ∂(log ρ + U) +∂wkl +� += +∂ +∂wij +�η +2e−UKijrsKklrs +∂(eUρ) +∂wkl +� +. +Without loss of generality, we assume that +� +O(n) e−U(W)dW = 1. Then multiply eUρ − 1 on +both sides, and by definition of K in (3.21), we derive +d +dt +� +O(n) +e−U|eUρ − 1|2dW = −η +� +O(n) +e−U +� +Kijrs +∂ +∂wij +(eUρ − 1) +� � +Kklrs +∂ +∂wkl +(eUρ − 1) +� +dW += −η +� +O(n) +e−U|HT∇W(eUρ − 1)|2dW. +Here ∇W is the gradient operator on (O(n), ge) where ge is the Euclidean metric. Then by +Poincare’s inequality, we derive exponential convergence. +Therefore, we desire to carefully select H and F such that the potential condition (4.1) is +satisfied. We provide the following two special cases where reversibility is ensured. +First, we consider the overdamped Langevin on O(n). In particular, we select the potential +as U(W; A, N) = tr(NWTAW), where N is a diagonal matrix. Then we recover the Oja- +Brockett flow with Bronwian motion. In fact, the Oja-Brockett flow is the gradient flow of +U(W; A, N) = tr(NWTAW) on (O(n), ge). +Second, we consider the two-dimensional case, i.e. n = 2. In this case, orthogonal matrices +are determined by the rotational angle. +The SDE of the angle is an SDE on R, which +automatically satisfy the potential condition. +In each case, Poincare’s inequality is verified, so they converge to the invariant measure +exponentially fast. +4.1. The overdamped Langevin dynamics on O(n). Suppose that U(Q) : O(n) → R +is a smooth function (which serves as the free energy), consider the overdamped Langevin +dynamics on O(n): +dQ(t) = PTQO(n) ◦ (−∇U(Q)dt + σdW(t)). +(4.2) +Here PTQO(n) is the projection onto TQO(n), ∇ represents the derivative w.r.t. Q, i.e. +(∇U(Q))ij = ∂u(Q) +∂qij +. + +18 +J.-G. LIU AND Z. LIU +W(t) ∈ Rn×n is the standard Brownian motion and σ is a constant. ’◦’ means that the +above SDE is in the Stratonovich sense. In general, σ can be a matrix that depends on Q. +To illustrate the the Langevin dynamics on O(n), we consider the simplest case here which +is sufficient for diffusion approximation. +If we take U(Q) = −tr(NQTAQ) where N is a diagonal matrix with entries on the +diagonal line aligned in the descending order, then (4.2) reads as +dQ(t) = (AQN − QNQTAQ)dt + σPTQO(n) ◦ dW(t). +(4.3) +Then (4.3) should be viewed as the disturbed Oja-Brockett flow [2]. +By the conversion rule, (4.2) should be formulated in Ito’s sense as +dQ(t) = +� +−PTQO(n)∇U(Q) − σ2(n − 1) +4 +Q +� +dt + σPTQO(n)dW(t). +(4.4) +By Ito’s formula, we can derive the Fokker-Planck equation of (4.2) which is +∂ρ(Q, t) +∂t += ∇Q · (ρ(Q, t)∇QU(Q)) + σ2 +2 ∆Qρ(Q, t). +(4.5) +Here ∇Q·, ∇Q and ∆Q are the divergence, gradient and Laplace-Beltrami operator on (O(n), ge) +respectively. See Section 6 for more details. +Compactness of O(n) implies exponential convergence of (4.5). Direct calculation yields +that the invariant measure of (4.5) is given by +ρeq(Q) := 1 +Z e−2U(Q)/σ2, Z := +� +O(n) +e−2U(Q)/σ2dV, +(4.6) +where dV is the volume form on (SO(n), ge). Equation (4.5) can be reformulated as +∂ρ(Q, t) +∂t += σ2 +2 ∇Q · +� +ρeq(Q)∇Q +�ρ(Q, t) +ρeq(Q) +�� +:= L∗ρ(Q, t), +(4.7) +and we denote the Fokker-Planck operator as L∗. This also implies that the invariant measure +of (4.5) is unique since +L∗ρ = 0 ⇐⇒ ∇Q · +� +ρeq(Q)∇Q +�ρ(Q, t) +ρeq(Q) +�� += 0 +⇐⇒ +� +O(n) +ρeq(Q) +����∇Q +�ρ(Q, t) +ρeq(Q) +����� +2 +dV = 0 +⇐⇒ ρ(Q) = cρeq(Q), +where c is a constant. If ρ(Q) is a probability measure on SO(n), then c = 1 and ρ = ρeq. In +the above induction we used the positivity of ρeq(Q), which is a consequence of the continuity +of U(Q) and the compactness of SO(n). +Multiplying ρ(Q, t)/ρeq(Q) − 1 on both sides of (4.5) results in +d +dt +� +O(n) +ρeq(Q) +���� +ρ(Q, t) − ρeq(Q) +ρeq(Q) +���� +2 +dV = −σ2 +� +O(n) +ρeq(Q) +����∇Q +�ρ(Q, t) − ρeq(Q) +ρeq(Q) +����� +2 +dV. +(4.8) + +DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS +19 +If we can prove the Poincare inequality in L2(ρeq(Q)dV), i.e., there exists a constant C > 0 +such that for all f ∈ H1(ρeq(Q)dV) satisfying +� +SO(n) f(Q)ρeq(Q)dV = 0, we have +� +O(n) +ρeq(Q) |∇Q (f(Q))|2 dV ≥ C +� +O(n) +ρeq(Q)|f(Q)|2dV, +(4.9) +then (4.8) yields +d +dt +� +O(n) +ρeq(Q) +���� +ρ(Q, t) − ρeq(Q) +ρeq(Q) +���� +2 +dV ≤ −C +� +O(n) +ρeq(Q) +���� +ρ(Q, t) − ρeq(Q) +ρeq(Q) +���� +2 +dV. +By Gronwall’s inequality, this gives exponential convergence of (4.5) whose initial value is a +probability measure. +Now we prove the Poincare inequality in L2(ρeqdV). +Lemma 4.1. (Poincare’s inequality) Suppose that f ∈ H1(ρeq(Q)dV) satisfying +� +O(n) f(Q)ρeq(Q)dV = +0. Then (4.9) holds. +Proof. To prove this, we prove that for any λ > 0, (λ − L∗)−1 : +L2 (dV/ρeq(Q)) → +L2 (dV/ρeq(Q)) is a compact operator. Suppose that {un}n≥1 and {gn}n≥1 are two sequences +in L2(dV/ρeq(Q)) such that +(λ − L∗)un = gn, n ≥ 1, +and {gn}n≥1 is uniformly bounded in L2(dV/ρeq(Q)). Then multiplying un on both sides +yields +� +O(n) +1 +ρeq +· un · (λ − L∗)undV = +� +O(n) +ρeq +����∇Q +� un +ρeq +����� +2 +dV + λ +� +SO(n) +ρeq +���� +un +ρeq +���� +2 +dV += +� +O(n) +1 +ρeq +· gnundV. +By Cauchy-Schwartz’s inequality, we know that un/ρeq uniformly bounded in H1(ρeqdV). +The compact embedding H1(ρeqdV) ֒→֒→ L2(ρeqdV) (see [17]) implies that up to subse- +quences, there exists u∗ ∈ L2(ρeqdV), +un +ρeq +→ u∗ +ρeq +in L2(ρeqdV), +or equivalently +un → u∗ in L2(dV/ρeq). +Thus (λ − L∗)−1 is compact. Thus 1/λ ̸= 0 is not an accumulation point of σ((λ − L∗)−1), +hence 0 is not an accumulation point of σ(L∗), but the single principal eigenvalue of L∗, +whose eigenvectors are c · ρeq where c is a constant. So for any g ∈ L2(dV/ρeq), we have +− +� +O(n) +g +ρeq +L∗gdV ≥ C +� +O(n) +|g|2 +ρeq +dV +(4.10) +for all g satisfying +� +O(n) gdV = 0. Let g = ρeqf, we have (4.9). +□ + +20 +J.-G. LIU AND Z. LIU +4.2. The case of n = 2. If we ask F, H in (3.13) to satisfy (3.46), then the SDE reads as +dwij = gij(A, W)dt + √η · Nijkl ◦ dBkl. +(4.11) +Here M(W) is the covariance matrix of G(xxT , W) defined in (3.42). By Lemma 3.3, (4.11) +admits the Stiefel manifold as an invariant set. Utilizing this fact, we can reformulate (4.11): +Lemma 4.2. Suppose n = 2. Consider (3.13) where H and H satisfy (3.46). Then (3.13) +(or (4.11)) can be reformulated as +dW = F1(W)dt + √η · c(W)W ◦ dZ. +(4.12) +Here F1 is defined in (2.4), c(W) is a scalar defined as +(4.13) +c(W) := +� +c1(w4 +1,1 + w4 +1,2) + c2w2 +1,1w2 +1,2 + c3w1,1w1,2(w1,1w2,2 + w1,2w2,1), +c1 := E[x2 +1x2 +2] − (E[x1x2])2, +c2 := E[x4 +1 + x4 +2 − 4x2 +1x2 +2] + 2(E[x1x2])2 + 2E[x2 +1]E[x2 +2] − (E[x2 +1])2 − (E[x2 +2])2, +c3 := 2E[x3 +1x2 − x1x3 +2] − E[x2 +1]E[x1x2] + E[x2 +2]E[x1x2] +and Z(t) ∈ R2×2 is defined as +Z(t) := +� +0 +−B(t) +B(t) +0 +� +, +(4.14) +where B(t) is the standard Brownian motion in one dimension. +See Section 6 for the proof of this lemma. Remember that each element in O(2) can be +expressed in either of the following forms: +O1(θ) = +� +cos θ +sin θ +− sin θ +cos θ +� +, θ ∈ [0, 2π), +(4.15) +O−1(θ) = +� +cos θ +sin θ +sin θ +− cos θ +� +, θ ∈ [0, 2π). +(4.16) +Because the orbit of W(t) in (3.13) is continuous a.s. and the determinant is also a continuous +function w.r.t. W, so if W0 = O1(θ) for some θ, then W(t) = O1(θ(t)) for some θ(t) ∈ +[0, 2π). Without loss of generality, we assume W0 = O1(θ0) for some θ0 ∈ [0, 2π), thus for +any t ≥ 0, +w1,1(t) = w2,2(t), w1,2(t) + w2,1(t) = 0, w2 +1,1(t) + w2 +1,2(t) = 1. +(4.17) +To prove the convergence of (4.12), we consider the process of θ(t) instead of W. We +construct the following one dimensional SDE in Ito’s sense: +dθ(t) = f(θ)dt + √ηg(θ)dB, θ(t) = θ0, +(4.18) +where B(t) is the standard Brownian motion, f, g are defined as +(4.19) +g(θ) = −c(θ) +f(θ) = (E[x2 +2] − E[x2 +1]) cos θ sin θ + η(2c1(θ) − c2(θ)) +2 +(cos3 θ sin θ − sin3 θ cos θ) ++ 3ηc3(θ) +2 +cos2 θ sin2 θ. + +DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS +21 +Here c, c1, c2 and c3 are the scalar functions defined in (4.13) and c(θ) is c(W) where W is +replaced by O1(θ), i.e. +c(θ) = c +�� +cos θ +sin θ +− sin θ +cos θ +�� +. +(4.20) +Then we can use the solution of (4.18) to represent the solution of (4.12). +Lemma 4.3. Consider (4.12). Suppose that the initial value W(0) satisfies W(0) = O1(θ0) +(defined in (4.15)) for some θ0 ∈ [0, 2π). Suppose that θ(t) is the solution of (4.18), then +w1,1(t) = w2,2(t) = cos(θ(t)), w1,2(t) = sin(θ(t)), w2,1(t) = − sin(θ(t)) +(4.21) +solves (4.12) with the initial value W(0) = O1(θ0). +See Section 6 for the proof of this lemma. +4.2.1. Convergence analysis. Let T = R/2πZ. Now consider (4.18). We denote the invariant +measure of this SDE as ρ∞(x) ∈ P(T). Then ρ∞ is the stationary solution to the Fokker- +Planck equation with the periodic boundary condition, i.e. +∂ρ(x, t) +∂t += L1ρ(x, t), L1ρ(x, t) := −∂(f(x)ρ(x, t)) +∂x ++ 1 +2 · ∂2(ηg2(x)ρ(x, t)) +∂x2 +. +(4.22) +Here f, g are defined in (4.19) A direct calculation yields +ρ∞(x) = C exp +�� x +0 +2f(s) +η · g2(s)ds +� +, C = +� +exp +�� 2π +0 +2f(s) +η · g2(s)ds +��−1 +. +(4.23) +According to the proof of Lemma 4.3 (see Section 6), we know that g(x) is strictly positive +on T, thus ρ∞(x) > 0 for any x ∈ T, thus the exponential convergence holds by a Poincare’s +inequality. Define the weighted L2 space L2(T, dx/ρ∞) as +L2(T, dx/ρ∞) := +� +p : +� +T +p2(x) +ρ∞(x)dx < ∞ +� +, ⟨p, q⟩dx/ρ∞ = +� +T +p(x)q(x) +ρ∞(x) dx. +(4.24) +Theorem 4.1. Suppose that θ(t), t ≥ 0 solves (4.18) with the initial value θ0, which is a +random variable with density function ρ0 ∈ P(T). Let ρ(x, t) ∈ P(T) be the law of θ(t), +i.e., P(θ(t) ∈ B) = +� +B ρ(x, t)dx for any Borel set B ∈ T. Consider ρ∞(x) ∈ P(T), i.e. the +invariant measure in (4.23). Then there exists a constant c > 0 only depending on η and +momentums of x such that +∥ρ(·, t) − ρ∞∥L2(T,dx/ρ∞) ≤ e−ct∥ρ0 − ρ∞∥L2(T,dx/ρ∞). +(4.25) +See Section 6 for proof of Theorem 4.1. +4.2.2. An example. In this section, we explicitly calculated an example here to illustrate our +main results. Suppose that x = (x1, x2)T where x1 and x2 are independent random variables +such that they possess density function +ρ1 = 1 +4 · +1(−2,2), ρ2 = 1 +2 · +1(−1,1), +i.e. x1 ∼ Uni(−2, 2) and x2 ∼ Uni(−1, 1). Then the covariance matrix of x is +A = +� +4/3 +0 +0 +1/3 +� +. + +22 +J.-G. LIU AND Z. LIU +Then by (4.13), we have +c1 = 4 +9, c2 = 8 +45, c3 = 0, +and +c = +� +4 +9(cos4 θ + sin4 θ) + 8 +45 cos2 θ sin2 θ = +� +4 +9 − 32 +45 cos2 θ sin2 θ. +Thus c ≥ +� +4/15 for any θ ∈ [0, 2π). Then (4.18) reads as +dθ = +� +− cos θ sin θ + 16η +45 (cos3 θ sin θ − sin3 θ cos θ) +� +dt + +� +4 +9 − 32 +45 cos2 θ sin2 θ · dB. +According to (4.23), the invariant measure is +ρ∞(x) = exp +�1 +η +� x +0 +−45 cos θ sin θ + 16η(cos3 θ sin θ − sin3 θ cos θ) +10 − 16 cos2 θ sin2 θ +dθ +� +, x ∈ [0, 2π). +Direct calculation yields +ρ∞(x) = exp +� +−15 +√ +6 +16η arctan 2 +√ +6 sin2 x +5 − 4 sin2 x +� � +5 +8 sin4 x − 8 sin2 x + 5, x ∈ [0, 2π). +(4.26) +The explicit formula of the invariant measure, i.e. (4.26), shows that if η << 1, then the +mass of the invariant measure concentrates around x = 0 and x = π, or in terms of the +matrix, around −I2 and I2, which gives the right principal component decomposition. +5. The case of p < n +All results for the case of p = n can be extended to the case of p < n, by being careful on +the size of tensors and rewrite the projection operator. +First, all regularity and stability results for the semigroup still hold. The proof is exactly +the same as in Lemma 3.1 and Theorem 3.1, by replacing the terminal index n by p for +column indices. +Second, the diffusion approximation is now formulated as +˙W = G(A, W) + ηPTWO(n×p)F(W) + √ηPTWO(n×p) ˙Z, W(0) = W0 ∈ O(n × p). +(5.1) +Here PTWO(n×p) is the projection onto the tangent space of O(n × p) at W, see (5.3); ˙Z is +defined as +˙Z(W) = ( ˙Zij)n×p, +˙Zij(W) = Hijkl(W) ◦ ˙Bkl; +(5.2) +H(W) = (Hijkl(W))n×p×n×p is the coefficient tensor of the Brownian motion; B = (Bij)n×p +is the standard Brownian motion in Rn×p; The notation ’◦’ represents that (5.1) is an SDE +in the Stratonovich sense. +The projection operator onto TWO(n × p) then reads as +PTWO(n×p)M = (In − WWT)M + 1 +2W(WTM − MT W). +(5.3) +When p = n, WWT = In so PTWO(n×p) is exactly the projection on O(n); when p = 1, +WTM = MTW are all scalars, then +PTWO(n×p)M = PTWSn−1M = (In − WWT)M, +(5.4) +i.e. PTWO(n×p) degenerates to the projection onto the unit sphere. + +DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS +23 +Using the same technique and method as in Lemma 3.3, one can prove that (5.1) stays on +O(n × p); moreover, the diffusion approximation is also of first-order as in Theorem 3.2. +For reversibility, the overdamped Langevin is still reversible. All the proof in Section 4.1 +holds generally on O(n × p). +Acknowledgement +Jian-Guo Liu was supported in part by the National Science Foundation (NSF) under +award DMS-2106988. +6. Appendix +6.1. Riemannian manifolds and the Stiefel manifold. We denote the tangent space at +m on manifold M as TmM, the tangent vector field on M as Γ(TM). The tangent bundle +(the disjoint union of the tangent spaces) is denoted as TM. +Definition 6.1. (Riemannian manifolds) Suppose that M is a smooth manifold. A Rie- +mannian manifold (M, g) is a smooth mainfold equipped with an inner product gm on TmM +at each m ∈ M. Moreover, for any tangent vector field ˙x and ˙y, the function +⟨ ˙x(m), ˙y(m)⟩gm : M → R +(6.1) +is smooth. +Given a Riemannian metric g on M, the gradient of a smooth function E on M is defined +as +Definition 6.2. (the gradient on the Riemannian manifold) A tangent vector field ∇gE on +M is called the gradient of E w.r.t. the metric g if for every tangent vector field ˙x on M, +⟨E′, ˙x⟩F = ⟨∇gE, ˙x⟩g. +(6.2) +Here E′ is the derivative of E, which is a cotangent vector field. +Now we consider the Stiefel manifold with the Euclidean metric ge, under the global +coordinate Q ∈ Rn×n. We first introduce several important properties of O(n). See [4] for +more details. +Lemma 6.1. (the Stiefel manifold) The Stiefel manifold O(n) is a smooth, compact manifold +of dimension n(n − 1)/2. The tangent space at Q is given by +TQO(n) = {QΩ | Ω ∈ Rn×n, Ω + ΩT = 0}, +(6.3) +while the normal space at Q is given by +TQO(n)⊥ = {QΩ | Ω ∈ Rn×n, Ω = ΩT}. +(6.4) +See [4] for proof of Lemma 6.1. By Lemma 6.1, we can prove that for any M ∈ Rn×n, the +projection on the tangent spaces and the normal spaces are respectively: +PTQO(n)M := 1 +2(M − QMT Q), PTQO(n)⊥M := 1 +2(M + QMT Q). +(6.5) +Lemma 6.2. (Gradient on (O(n), ge)) Suppose that ϕ(Q) : O(n) → R is a restriction of a +smooth function (still denoted as ϕ : Rn×n → R) on O(n). Then gradient of ϕ w.r.t. ge at +point Q is given by +∇geϕ := PTQO(n)(∇ϕ) = 1 +2(∇ϕ − Q(∇ϕ)TQ), +(6.6) + +24 +J.-G. LIU AND Z. LIU +here ∇ϕ ∈ Rn×n is the gradient of ϕ in Rn×n, i.e. (∇ϕ)ij = ∂ϕ +∂qij +. +Proof. We just need to prove that for any Q ∈ O(n) and any tangent vector at Q, (6.2) holds. +By Lemma 6.1, a tangent vector at Q can be represented as QΩ where Ω is skew-symmetric. +Therefore, for any Ω that is skew-symmetric, we have +⟨∇ϕ, QΩ⟩F = ⟨PTQO(n)(∇ϕ), QΩ⟩F + ⟨PTQO(n)⊥(∇ϕ), QΩ⟩F. +(6.7) +Notice that +⟨PTQO(n)⊥(∇ϕ), QΩ⟩F = ⟨Q(QT∇ϕ + (∇ϕ)TQ), QΩ⟩F += ⟨QT∇ϕ + (∇ϕ)TQ, Ω⟩F += 0, +since ⟨M, N⟩F = 0 if M is symmetric while N is skew-symmetric. Therefore, +⟨∇ϕ, QΩ⟩F = ⟨PTQO(n)(∇ϕ), QΩ⟩F = ⟨PTQO(n)(∇ϕ), QΩ⟩ge. +So ∇geϕ = PTQO(n)(∇ϕ). +□ +Under this global coordinate, the divergence on (O(n), ge) can also be explicitly computed. +The ij entry of ∇geϕ is given by +(∇geϕ)ij = 1 +2 +� +∂ϕ +∂qij +− +� +k,l +qikqlj +∂ϕ +∂qlk +� +. +(6.8) +So given a tangent vector field H(Q) = (hij(Q)), the divergence of it is defined by +∇ge · H(Q) := +� +i,j +(∇ge(hij(Q)))ij. +(6.9) +The Laplace-Beltrami operator is then defined as: +∆geϕ := ∇ge · ∇geϕ. +(6.10) +Explicit expression of the Laplace-Beltrami operator is given by the following lemma: +Lemma 6.3. (Laplace-Beltrami operator) The Laplace-Beltrami operator on (O(n), ge) is +given by +∆geϕ = 1 +2 +�� +i,j +∂2ϕ +∂q2 +ij +− (n − 1) +� +i,j +qij +∂ϕ +∂qij +− +� +i,j,k,l +qikqlj +∂2ϕ +∂qij∂qlk +� +. +(6.11) +Proof. By the expression of the divergence and gradient, we have +∆geϕ = +� +i,j +� +∂hij +∂qij +− +� +k,l +qikqlj +∂hij +∂qlk +� +, hij = +� +i,j +� +∂ϕ +∂qij +− +� +k,l +qikqlj +∂ϕ +∂qlk +� +. +Thus +∂hij +∂qij += 1 +2 +� +∂2ϕ +∂q2 +ij +− +� +k,l +δklqlj +∂ϕ +∂qlk +− +� +k,l +δilqik +∂ϕ +∂qlk +− +� +k,l +qikqlj +∂2ϕ +∂qijqlk +� += 1 +2 +� +∂2ϕ +∂q2 +ij +− +� +l +qlj +∂ϕ +∂qlj +− +� +k +qik +∂ϕ +∂qik +− +� +k,l +qikqlj +∂2ϕ +∂qijqlk +� +, + +DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS +25 +and +∂hij +∂qlk += 1 +2 +� +∂2ϕ +∂qij∂qlk +− +� +k′,l′ +δilδkk′ql′j +∂ϕ +∂ql′k′ − +� +k′,l′ +δll′δkjqik′ ∂ϕ +∂ql′k′ − +� +k′,l′ +qik′ql′j +∂2ϕ +∂ql′k′∂qlk +� +. +Substituting the above formulas into the Laplacian operator, we can derive (6.11). +□ +6.2. Proofs of lemmas and omitted calculations. +6.2.1. Section 3. +Proof of Lemma 3.1. According to (2.1) and (2.2), direct computation yields +(6.12) +∥W(k)∥2 +F = ∥W(k − 1)∥2 +F + 2η · tr(W(k − 1)TA(k)W(k − 1)) +− 2η · tr(W(k − 1)TW(k − 1)W(k − 1)TA(k)W(k − 1)) ++ η2 · tr(W(k − 1)TA(k)2W(k − 1)) +− 2η2 · tr(W(k − 1)TA(k)W(k − 1)W(k − 1)TA(k)W(k − 1)) ++ η2 · tr(W(k − 1)TA(k)W(k − 1)W(k − 1)TW(k − 1)W(k − 1)TA(k)W(k − 1)) +− 2η2 · tr(W(k − 1)TA(k)W(k − 1)W(k − 1)TW(k − 1)Σ(A(k), W(k − 1))) ++ η2 · tr(Σ(A(k), W(k − 1))TW(k − 1)TW(k − 1)Σ(A(k), W(k − 1))). +By definition of Σ in (2.1), we have +(6.13) +∥Σ(A(k), W(k − 1)∥F = +�� +i̸=j +(wi(k − 1)TA(k)wj(k − 1))2 += +�� +i̸=j +(wi(k − 1)Tx(k))2(wj(k − 1)Tx(k))2 +≤ +n +� +i=1 +(wi(k − 1)Tx(k))2 +≤ M2 +n +� +i=1 +∥wi(k − 1)∥2 +2 += M2∥W(k − 1)∥2 +F. +By the following norm inequality: ∥M∥2 ≤ ∥M∥F ≤ √n∥M∥2, (2.6) and (6.13), we derive +the following estimates for each term in the above equality in the almost surely sense: +(6.14) +tr(W(k − 1)TA(k)W(k − 1)) = ∥W(k − 1)Tx(k)∥2 +2 ≤ ∥W(k − 1)T∥2 +2∥x(k)∥2 +2 +≤ M2∥W(k − 1)∥2 +F. +(6.15) +tr(W(k − 1)TA(k)2W(k − 1)) = ∥x(k)∥2 +2tr(W(k − 1)TA(k)W(k − 1)) +≤ M4∥W(k − 1)∥2 +F. + +26 +J.-G. LIU AND Z. LIU +(6.16) +tr(W(k − 1)TA(k)W(k − 1)W(k − 1)TW(k − 1)W(k − 1)TA(k)W(k − 1)) += tr((W(k − 1)Tx(k))(x(k)TW(k − 1)W(k − 1)TW(k − 1)W(k − 1)TA(k)W(k − 1))) +≤ ∥W(k − 1)Tx(k)∥2∥x(k)TW(k − 1)W(k − 1)TW(k − 1)W(k − 1)TA(k)W(k − 1)∥2 +≤ M2∥W(k − 1)∥6 +2∥A(k)∥2 +≤ M4∥W(k − 1)∥6 +F. +(6.17) +|tr(W(k − 1)TA(k)W(k − 1)W(k − 1)TW(k − 1)Σ(A(k), W(k − 1)))| += |tr((W(k − 1)Tx(k))(x(k)TW(k − 1)W(k − 1)TW(k − 1)Σ(A(k), W(k − 1))))| +≤ ∥(W(k − 1)Tx(k)∥2∥x(k)TW(k − 1)W(k − 1)TW(k − 1)Σ(A(k), W(k − 1))∥2 +≤ M2∥W(k − 1)∥4 +2∥Σ(A(k), W(k − 1))∥2 +≤ M2∥W(k − 1)∥4 +F∥Σ(A(k), W(k − 1))∥F +≤ M4∥W(k − 1)∥6 +F. +(6.18) +tr(Σ(A(k), W(k − 1))TW(k − 1)TW(k − 1)Σ(A(k), W(k − 1))) += ∥W(k − 1)Σ(A(k), W(k − 1))∥2 +F +≤ n∥W(k − 1)Σ(A(k), W(k − 1))∥2 +2 +≤ n∥W(k − 1)∥2 +2∥Σ(A(k), W(k − 1))∥2 +2. +≤ nM4∥W(k − 1)∥6 +F. +Substituting (6.14) ∼ (6.18) into (6.12) yields +∥W(k)∥2 +F ≤ (1 + 2M2η)∥W(k − 1)∥2 +F + η2(M4∥W(k − 1)∥2 +F + (n + 3)M4∥W(k − 1)∥6 +F). +(6.19) +Select η(T) as +η(T) = +1 +M4(1 + (n + 3)r4e2T(2M2+1)). +(6.20) +Then we can prove that for any η ≤ η(T), the Markov chain generated by (??) which starts +from W0 satisfies +∥W(k)∥2 +F ≤ r2e2M2+1, k = 1, 2, ..., [T/η], a.s.. +(6.21) +We prove by induction. Given any η ≤ η(T), if ∥W(k − 1)∥2 +F ≤ r2eT(2M2+1), then +∥W(k)∥2 +F ≤ (1 + 2M2η)∥W(k − 1)∥2 +F + η2(M4∥W(k − 1)∥2 +F + (n + 3)M4∥W(k − 1)∥6 +F) +≤ (1 + 2M2η)∥W(k − 1)∥2 +F + (ηM4(1 + (n + 3)r4e2T(2M2+1))) · η · ∥W(k − 1)∥2 +F +≤ (1 + (2M2 + 1)η)∥W(k − 1)∥2 +F. +Remember that ∥W0∥2 +F ≤ r2, so ∥W0∥2 +F ≤ r2e2M2+1, hence for any k = 1, 2, ..., [T/η], +∥W(k)∥2 +F ≤ (1 + (2M2 + 1)η)kr2 ≤ (1 + (2M2 + 1)η)T/ηr2 ≤ r2eT(2M2+1). +Thus taking C(T) = r2eT(2M2+1) concludes the proof. +□ + +DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS +27 +Proof of Lemma 3.5. Because M = (Mijkl) is the covariance matrix of G, it is positive +semidefinite. Therefore, the square root of M is well-defined, which satisfies (i),(ii) and (iii) +above. Denote it as N. +When W ∈ O(n), we have proved that for any symmetric B ∈ Rn×n, G(B, W) ∈ TWO(n), +so +PTWO(n)G(A − xxT, W) = G(A − xxT, W), +i.e. Pijklgkl = gij by Lemma 3.2. Therefore, by linearity of expectation and symmetry +Mijkl = E[gij(A − xxT, W)gkl(A − xxT, W)] += E[Pijrsgrs(A − xxT, W)gxy(A − xxT, W)Pklxy] += PijrsE[grs(A − xxT, W)gxy(A − xxT, W)]Pxykl += PijrsMrsxyPxykl, +or in matrix notation, M = PMP. Thus by (iii) in Lemma 3.2, we have +PijrsMrskl = PijrsPrsuvMuvxyPxykl = PijrsPuvrsMuvxyPxykl = PijuvMuvxyPxykl, +or PM = PMP. Similarly, +MijrsPrskl = PijuvMuvxyPxyrsPrskl = PijuvMuvxyPxykl, +or MP = PMP. Thus +MijrsPrskl = PijrsMrskl. +So P and M are commutative. Because P is symmetric, so P is also commutative with the +square root of M. Thus +NijrsPrskl = PijrsNrskl. +Thus +PijrsNrsxyPxyuvNuvkl = PijrsNrsxyNxyuvPuvkl = PijrsMrsuvPuvkl = Mijkl, +or PNPN = PNNP = PMP = M. So PN is also the square root of M, by uniqueness, +PijrsNrskl = Nijkl, +i.e. PN = N. +□ +Calculations in Theorem 3.2. Direct calculation yields +(6.22) +Sϕ(W) = E +� +ϕ(W + ηG(xxT, W)) +� += ϕ(W) + +� +gij(A, W) ∂ϕ +∂wij +� +η + +�1 +2E +� +gij(xxT , W)gkl(xxT, W) +� +∂2ϕ +∂wij∂wkl +� +η2 + O(η3). +Meanwhile, +eηLϕ(W) = ϕ(W) + ηLϕ(W) + 1 +2η2L2ϕ(W) + O(η3). + +28 +J.-G. LIU AND Z. LIU +By (3.23), we have +L2ϕ = L +� +gij(A, W) ∂ϕ +∂wij ++ O(η) +� += gkl(A, W) ∂ +∂wkl +� +gij(A, W) ∂ϕ +∂wij +� ++ O(η) += +� +gkl(A, W)∂gij(A, W) +∂wkl +� ∂ϕ +∂wij ++ gij(A, W)gkl(A, W) +∂2ϕ +∂wij∂wkl ++ O(η). +(6.23) +Thus +(6.24) +eηLϕ(W) = ϕ(W) + η +�� +gij(A, W) + ηPijklfkl + η +2Jij +� ∂ϕ +∂wij ++ η +2KijrsKklrs +∂2ϕ +∂wijwkl +� ++ η2 +2 +�� +gkl(A, W)∂gij(A, W) +∂wkl +� ∂ϕ +∂wij ++ gij(A, W)gkl(A, W) +∂2ϕ +∂wij∂wkl +� ++ O(η3). +Recast (6.24) in the order of η, we have +(6.25) +eηLϕ(W) = ϕ(W) + +� +gij(A, W) ∂ϕ +∂wij +� +η+ +�� +Pijklfkl + 1 +2Jij + 1 +2gkl(A, W)∂gij(A, W) +∂wkl +� ∂ϕ +∂wij +� +η2 ++ +� +(KijrsKklrs + gij(A, W)gkl(A, W)) +∂2ϕ +∂wij∂wkl +� +η2 + O(η3). +Comparing the η2 term in (6.22) and (6.25), we have +∂ϕ +∂wij +: Pijklfkl + 1 +2Jij + 1 +2gkl(A, W)∂gij(A, W) +∂wkl += 0; +∂2ϕ +∂wij∂wkl +: KijrsKklrs + gkl(A, W)gij(A, W) = E +� +gij(xxT, W)gkl(xxT, W) +� +. +Thus F = (fij)n×n and H = (Hijkl)n×n×n×n should satisfy +Pijklfkl = −1 +2Jij − 1 +2gkl(A, W)∂gij(A, W) +∂wkl +, +PijuvHuvrsPklxyHxyrs = Mijkl = E[gij(A − xxT, W)gkl(A − xxT, W)], +which is (3.41) +□ +6.2.2. Section 4. +Proof of Lemma 4.2. We first compute H(W). By Proposition 3.3, we know that WWT = +I2. Thus +G(xxT − A, W) = (xxT − A)W − WWT(xxT − A)W + WΣ(xxT − A, W) += WΣ(xxT − A, W). +Denote B = xx − A. Since n = 2, we have +Σ(B, W) = +� +0 +−w1 · Bw2 +w1 · Bw2 +0 +� +. + +DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS +29 +Denote b = w1 · Bw2, then +G(xxT − A, W) = W +� +0 +−b +b +0 +� += +� +w1,2b +−w1,1b +w2,2b +−w2,1b +� +. +Thus +vec(G(B, W)) = (w1,2b, −w1,1b, w2,2b, −w2,1b)T. +Denote u = (w1,2, −w1,1, w2,2, −w2,1)T. Then the covariance matrix defined in (??) is +(6.26) +M(W) = E[vec(G(B, W))vec(G(B, W))T] += E[b2]uuT . +Therefore, +H(W) = +� +M(W) = +� +E[b2] +∥u∥2 +uuT . +Remember that W ∈ O(2), thus ∥u∥2 = +√ +2. So we have +H(W) = +� +E[b2] +√ +2 +uuT . +(6.27) +Substituting (6.27) into (4.12), we have +dvec(W) = vec(G(A, W))dt + +� +ηE[b2] +√ +2 +uuT ◦ dB. +(6.28) +Moreover, because ∥u∥2 = +√ +2 and B(t) is the standard Brownian motion in R4, thus uTdB +has the same law with +√ +2dB where B(t) is the standard Brownian motion in one dimension. +Thus (4.12) can also be reformulated as +dvec(W) = vec(G(A, W))dt + +� +ηE[b2]u ◦ dB(t). +(6.29) +Moreover, because W(t) ∈ O(2), thus G(A, W) = F(W) where F is defined in (2.4). Thus +(4.12) can be directly transformed in the form of matrices: +dW = F(W)dt + +� +ηE[b2] · Q ◦ dZ, +(6.30) +where Z is defined in (4.14). +Now we compute E[b2]. Direct computation yields +b = w1 +TBw2 = b1,1w1,1w1,2 + b1,2(w1,1w2,2 + w2,1w1,2) + b2,2w2,1w2,2. +Thus +b2 = b2 +1,1w2 +1,1w2 +1,2 + b2 +1,2(w1,1w2,2 + w1,2w2,1)2 + b2 +2,2w2 +2,1w2 +2,2 + 2b1,1b2,2w1,1w1,2w2,1w2,2 ++ 2b1,1b1,2w1,1w2,2w1,2 + 2b1,1b1,2w1,1w2 +1,2w2,1 + 2b1,2b2,2w1,1w2,1w2 +2,2 + 2b1,2b2,2w1,2w2 +2,1w2,2. +Meanwhile, we know bi,j = xixj − E[xixj] for 1 ≤ i, j ≤ 2, thus for any i, j, i′, j′ = 1, 2, we +have +E[bi,jbi′,j′] = E[xixjxi′xj′] − E[xixj]E[xi′xj′]. +Substituting this into expression of b2, we have +E[b2] = var(x2 +1)w2 +1,1w2 +1,2 + var(x1x2)(w2 +1,1w2 +2,2 + w2 +2,1w2 +1,2 + 2w1,1w1,2w2,1w2,2) + var(x2 +2)w2 +2,1w2 +2,2 ++ 2(E[x2 +1x2 +2] − E[x2 +1]E[x2 +2])w1,1w1,2w2,1w2,2 + 2(E[x3 +1x2] − E[x2 +1]E[x1x2])w1,1w1,2(w1,1w2,2 + w1,2w2,1) ++ 2(E[x1x3 +2] − E[x2 +2]E[x1x2])w2,1w2,2(w1,1w2,2 + w1,2w2,1). + +30 +J.-G. LIU AND Z. LIU +Using w2 +1,1 = w2 +2,2, w2 +1,2 = w2 +2,1, we then derive +E[b2] = c1(W)(w4 +1,1 + w4 +1,2) + c2(W)w2 +1,1w2 +1,2 + c3(W)w1,1w1,2(w1,1w2,2 + w1,2w2,1), +(6.31) +where c1, c2 and c3 is defined in (4.13). Thus by taking c(W) = +� +E[b2], we derive (4.12) +which concludes the proof. +□ +Proof of Lemma 4.3. We first write (4.12) in It´o’s sense. Because we have assumed that +|W(t)| = 1, thus we only need to consider the equation for w1,1 and w1,2. In the It´o sense, +we have +dw1,1 = [(F(W))1,1 + h1(W)]dt + √η · c(W)w1,2dB +dw1,2 = [(F(W))1,2 + h2(W)]dt − √η · c(W)w1,1dB, +where h1 and h2 are defined in 3.21. Denote g1 = √ηc(W)w1,2, g2 = −√ηc(W)w1,1, then h1 +and h2 are computed as +(6.32) +h1(W) = 1 +2 +� +g1 +∂g1 +∂w1,1 ++ g2 +∂g1 +∂w1,2 +� += η +2 +�1 +2w2 +1,2 +∂c(W)2 +∂w1,1 +− 1 +2w1,1w1,2 +∂c(W2) +∂w1,2 +− w1,1c(W)2 +� +h2(W) = 1 +2 +� +g1 +∂g2 +∂w1,1 ++ g2 +∂g2 +∂w1,2 +� += η +2 +�1 +2w2 +1,1 +∂c(W)2 +∂w1,2 +− 1 +2w1,1w1,2 +∂c(W2) +∂w1,1 +− w1,2c(W)2 +� +. +Here we used the chain rule: c ∂c +∂w = 1 +2 +∂c2 +∂w . Substituting the above equation into the SDE +in It´o’s sense yields +(6.33) dw1,1 = +� +(E[x2 +2] − E[x2 +1])w1,1w1,2 + η +4w2 +1,2 +∂c2 +∂w1,1 +− η +4w1,1w1,2 +∂c2 +∂w1,2 +− η +2w1,1c(W)2 +� +dt ++ √ηc(W)w1,2dB. +Now consider the process of θ. Suppose that θ(t) satisfies the following SDE in Ito’s sense: +dθ = f(θ)dt + √η · g(θ)dB. +Then Ito’s isometry yields +d cos θ = − sin θ · dθ − ηg2(θ) +2 +dt += +� +− sin θ · f(θ) − η cos θ +2 +g2(θ) +� +dt − √η · sin(θ)g(θ)dB. +Replacing w1,1 by cos θ, w1,2 by sin θ in (6.33) and comparing the coefficients of it with the +above equation yields +(6.34) +g(θ) = −c(θ), +f(θ) = (E[x2 +2] − E[x2 +1]) sin θ cos θ + η · 2c1(θ) − c2(θ) +2 +(cos3 θ sin θ − cos θ sin3 θ) ++ η · 3c3(θ) +2 +cos2 θ sin2 θ. +Here we used (6.31) since c2(W) = E[b2]. This is exactly the SDE in (4.18). +□ + +DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS +31 +Proof of Theorem 4.1. In the following proof, C is just a general constant that may vary +among equations. Because ρ(x, t), t ≥ 0 is the law of θ(t), so ρ solves (4.22) on T × R+. +Thus by the periodic boundary condition, +d +dt +� +T +ρ(x, t)dx = 1 +2 +� +T +� +−∂(f(x)ρ(x, t)) +∂x ++ 1 +2 · ∂2(ηg2(x)ρ(x, t)) +∂x2 +� +dx = 0. +So +� +T ρ(x, t)dx = 1, t ≥ 0. +Now consider the Fokker-Planck operator L∗ +1 which is self-adjoint in L2(T, dx/ρ∞): +L∗ +1 : D(L∗ +1) ⊂ L2(T, dx/ρ∞) → L2(T, dx/ρ∞), L∗ +1p := − d +dx +� +ρ∞(x) d +dx +� p(x) +ρ∞(x) +�� +. +Here D(L∗ +1) = H2(T, dx/µ). A direct calculation yields +⟨L1∗p, q⟩dx/ρ∞ = +� +T +ρ∞(x) d +dx +� p(x) +ρ∞(x) +� d +dx +� q(x) +ρ∞(x) +� +dx, +thus L∗ +1 is semi-positive definite, and +L∗ +1p = 0 ⇐⇒ p(x) = cρ∞(x). +So 0 is the simple principle eigenvalue of L∗ +1 with ρ∞(x) as the eigenvector. +Moreover, 0 is isolated. +For any λ > 0, we prove that (λ + L∗ +1)−1 is compact. +Let +{gn}∞ +n=1 ∈ L2(T, dx/ρ∞) be a bounded sequence, with +(λ + L∗)un = gn, n = 1, 2, .... +To prove that (λ+L∗)−1 is compact, we just need to prove that there exists a subsequence of +{un}∞ +n=1 which is Cauchy in L2(T, dx/ρ∞). Because L∗ is semi-positive definite, so (λ + L∗) +is bounded, thus {un}∞ +n=1 is bounded in L2(T, dx/ρ∞). By the Cauchy-Schwatz inequality, +we have +⟨L∗ +1un, un⟩dx/ρ∞ = ⟨un, gn⟩dx/ρ∞ − λ∥un∥2 +L2(T,dx/ρ∞) ≤ C. +(6.35) +Here C is a constant. Thus +∥un/ρ∞∥2 +H1(T,ρ∞dx) = +� +T +ρ∞(x) +� d +dx +un(x) +ρ∞(x) +�2 +dx + +� +T +ρ∞(x) +� un(x) +ρ∞(x) +�2 +dx += ∥un∥L2(T,dx/ρ∞) + ⟨L∗ +1un, un⟩dx/ρ∞ ≤ C. +So un/ρ∞ is bounded in H1(T, ρ∞dx). By the compact embedding H1(T, ρ∞dx) ⊂⊂ L2(T, ρ∞dx), +we know that there exists a subsequence of un (still denoted as un) such that +un +ρ∞ +→ u∗ +ρ∞ +in L2(T, ρ∞dx), +or equivalently +un → u∗ in L2(T, dx/ρ∞). +So (λ + L∗)−1 is a compact operator. Thus the spectrum of (λ + L∗)−1 only admits 0 as +an accumulation point. +So 0 is an isolated point in the spectrum of L∗. +Thus for any +p ∈ L2(T, dx/ρ∞) such that +� +T p(x)dx = 0, we have the following Poincare’s inequality: +there exists a constant c > 0 such that +� +T +ρ∞(x) +� d +dx +p(x) +ρ∞(x) +�2 +dx = ⟨L∗ +1p, p⟩L2(T,dx/ρ∞) ≥ c∥p∥2 +L2(T,dx/ρ∞) = c +� +T +|p(x)|2 +ρ∞(x) dx. +(6.36) + +32 +J.-G. LIU AND Z. LIU +Multiplying ρ(x, t)/ρ∞(x) − 1 on both sides of (4.22) and substituting p = ρ(x, t) − ρ∞(x) +in (6.36) yields +1 +2 +d +dt +� +T +(ρ(x, t) − ρ∞(x))2 +ρ∞(x) +dx = − +� +T +ρ∞(x) +� d +dx +ρ(x, t) +ρ∞(x) +�2 +dx ≤ −c +� +T +|ρ(x, t) − ρ∞(x)|2 +ρ∞(x) +dx. +(6.37) +Thus by Gronwall’s inequality, (4.25) holds. +□ +References +[1] Vincent D Blondel, Alexandre Megretski, and Vincent DD Blondel. Unsolved problems in mathematical +systems and control theory. Princeton University Press Princeton, NJ, 2004. +[2] Roger W Brockett. Dynamical systems that sort lists, diagonalize matrices, and solve linear program- +ming problems. Linear Algebra and its applications, 146:79–91, 1991. +[3] Yuanyuan Feng, Lei Li, and Jian-Guo Liu. Semigroups of stochastic gradient descent and online prin- +cipal component analysis: properties and diffusion approximations. Communications in Mathematical +Sciences, 16(3), 2018. +[4] Uwe Helmke and John B Moore. Optimization and dynamical systems. Springer Science & Business +Media, 2012. +[5] Elton P Hsu. Stochastic analysis on manifolds. Number 38. American Mathematical Soc., 2002. +[6] Peter D Lax and Robert D Richtmyer. Survey of the stability of linear finite difference equations. +Communications on pure and applied mathematics, 9(2):267–293, 1956. +[7] Chris Junchi Li, Mengdi Wang, Han Liu, and Tong Zhang. Diffusion approximations for online principal +component estimation and global convergence. Advances in Neural Information Processing Systems, 30, +2017. +[8] Qianxiao Li, Cheng Tai, and Weinan E. Stochastic modified equations and adaptive stochastic gradient +algorithms. In International Conference on Machine Learning, pages 2101–2110. PMLR, 2017. +[9] Xin Liang, Zhen-Chen Guo, Ren-Cang Li, and Wen-Wei Lin. Nearly optimal stochastic approximation +for online principal subspace estimation. arXiv preprint arXiv:1711.06644, 2017. +[10] Jian-Guo Liu and Zibu Liu. Convergence of oja’s online principal component flow. arXiv preprint +arXiv:2202.11308, 2022. +[11] Grigorii Noikhovich Milstein. Numerical integration of stochastic differential equations, volume 313. +Springer Science & Business Media, 1994. +[12] Erkki Oja. Simplified neuron model as a principal component analyzer. Journal of mathematical biology, +15(3):267–273, 1982. +[13] Erkki Oja. Principal components, minor components, and linear neural networks. Neural networks, +5(6):927–935, 1992. +[14] Erkki Oja and Juha Karhunen. On stochastic approximation of the eigenvectors and eigenvalues of the +expectation of a random matrix. Journal of mathematical analysis and applications, 106(1):69–84, 1985. +[15] Bernt Oksendal. Stochastic differential equations: an introduction with applications. Springer Science & +Business Media, 2013. +[16] Christian Soize. The Fokker-Planck equation for stochastic dynamical systems and its explicit steady +state solutions, volume 17. World Scientific, 1994. +[17] Michael Eugene Taylor. Partial differential equations. 1, Basic theory. Springer, 1996. +[18] Wei-Yong Yan, Uwe Helmke, and John B Moore. Global analysis of oja’s flow for neural networks. IEEE +Transactions on Neural Networks, 5(5):674–683, 1994. +Department of Mathematics and Department of Physics, Duke University, Durham, NC +Email address: jliu@math.duke.edu +Department of Mathematics, Duke University, Durham, NC +Email address: zibu.liu@duke.edu + diff --git a/XtAzT4oBgHgl3EQfYvw0/content/tmp_files/load_file.txt b/XtAzT4oBgHgl3EQfYvw0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..661c23cfc28fc8aaa5847612116a2b085b192adc --- /dev/null +++ b/XtAzT4oBgHgl3EQfYvw0/content/tmp_files/load_file.txt @@ -0,0 +1,1330 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf,len=1329 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='01339v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='NA] 3 Jan 2023 DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS JIAN-GUO LIU AND ZIBU LIU Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Oja’s algorithm of principal component analysis (PCA) has been one of the methods utilized in practice to reduce dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In this paper, we focus on the convergence property of the discrete algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' To realize that, we view the algorithm as a stochastic process on the parameter space and semi-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We approximate it by SDEs, and prove large time convergence of the SDEs to ensure its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' This process is completed in three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' First, the discrete algorithm can be viewed as a semigroup: Skϕ = E[ϕ(W(k))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Second, we construct stochastic differential equations (SDEs) on the Stiefel manifold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' the diffusion approximation, to approximate the semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By proving the weak convergence, we verify that the algorithm is ’close to’ the SDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Finally, we use reversibility of the SDEs to prove long time convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Introduction Principal component anlysis (PCA) is a basic tool in dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Due to explo- sion of data, command of efficient PCA algorithms is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In this paper, we focus on the online PCA algorithm proposed by Oja in [14], which is also named as the stochastic gradient ascent (SGA) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Suppose that x ∈ Rn is a mean zero random variable (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Let A := E[xxT ] (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1) be the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Traditional PCA algorithms diagonalize A to derive principal eigenvectors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' the principal components) of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' However, due to limitation of storage and high dimension of data in recent fields such as deep learning, explicit form of the dense matrix A may not be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Therefore, practitioners prefer ’online’ algorithms: it only requires a limited amount of samples of x in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' To solve this problem, Oja proposed the following SGA method in [14]: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) wj(k) = wj(k − 1) + η(k)xT(k)wj(k − 1)[x(k) − (xT(k)wj(k − 1))wj(k − 1) − 2 j−1 � i=1 (xT(k)wi(k − 1))wi(k − 1)], j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' This algorithm iterates the first p principal components wj ∈ Rn, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Here x(k), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' are independent samples of x, η(k), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' are learning rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Algorithm (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) determines a discrete time Markovian process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' W(k) = [w1(k), w2(k), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', wp(k)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The main goal of this paper is to gain a good understanding of this random process (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=') from the view of semigroups, diffusion approximations and SDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' machine learning, dimensionality reduction, online principal component analysis, gradient flow, stochastic differential equations, random matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 1 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU First of all, as η → 0, replacing xxT by A in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2), we derive the corresponding ODE: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ˙q1 = Aq1 − (q1 · Aq1)q1, ˙qj = Aqj − (qj · Aqj)qj − 2 j−1 � i=1 (qi · Aqj)qi, j = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' qi(0) = qi,0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3) Convergence properties including global convergence, stable manifolds and exponential con- vergence were thoroughly investigated in our previous work [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In particular, we proved that for almost every initial value Q0 ∈ O(n), the solution exponentially converges to the eigenbasis (up to a sign).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Moreover, the eigenvectors are aligned in a descending order of the eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' See Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2 in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' As far as we know, this is the first complete result providing global exponential convergence and closed formula for stable manifolds of a PCA flow [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Given convergence of the corresponding ODE, we aim at proving similar result for the discrete algorithm (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We consider this problem in three steps: Viewing the SGA iteration as a semigroup, we construct proper diffusion approximations and prove convergence of diffusion approximations to ensure the performance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' First, we view the SGA method as a semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' It can be reformulated in the following form: W(k + 1) = W(k) + η · G(x(k + 1)xT(k + 1), W(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Here G ∈ Rn×p is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' For arbitrary test function ϕ ∈ C(Rn×p), define Sϕ(W) := Eϕ(W + ηG(xxT, W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='4) Under this notation, if the initial datum of the SGA method is W0, then the Markovian property yields Skϕ(W0) = Eϕ(W(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5) Thus, convergence of the SGA method can also be interpreted as the convergence of the semigroup {Sk}, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='. Second, we construct appropriate diffusion approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Although the SGA method does not preserve W(k) to stay on the Stiefel manifold O(n × p), the desired result, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' the eigenbasis, is in O(n × p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus, we aim at deriving a good diffusion approximation of the semigroup S and the SGA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' It should be an SDE that stays on the Stiefel manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The classical method to derive an SDE on a certain manifold is to project a Stratanovich SDE onto the desired manifold [5]: dW = PTWM(G(A, W)dt + σ ◦ dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Here PTZM is the projection operator onto the tangent space at W on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' If the semigroup is close to the diffusion process, then by proving convergence of the diffusion process in some sense, we can also guarantee the performance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Finally, we prove convergence of the diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The way to prove it is by seeking ’reversibility’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In fact, if the Fokker-Planck equation of the SDE can be recast in the following form: ∂tρ = ∇ · (ρ∇U + ∇ρ) = ∇ · � e−U∇ � ρ eU �� for some potential U, then the diffusion process satisfies detailed-balance condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', the process is reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then, Poincare’s inequality can ensure the exponential convergence DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS 3 of ρ in a certain L2 sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' This proves the convergence of SDEs, which also finishes our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Under this framework of analysis, we will provide our main results and revise previous literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Previous results and unsolves problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' One of the important features of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) which other algorithms do not possess is its semi-decoupling feature: iteration of wj does not depend on wi, i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' This feature facilitates its implementation in neural networks [13], thus researchers focus on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' This feature was also extended to the corresponding ODE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Based on this semi-decoupling property, we also proved all convergence results of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5) in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' However, a satisfying convergence result for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) is still wanting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Since Oja and Karhunen proposed (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) in [14], its convergence behavior has always been a focus in analysis of online PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Oja and Karhunen used stochastic approximation to derive almost sure convergence of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) under an implicit condition on the distribution of x [14, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' This implicit condition requires the iteration to visit a compact set containing the equilibrium for infinitely many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' However, this condition is difficult to verify in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' To improve Oja’s result, more recently, authors in [7] derived weak convergence of the first component of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) to a multidimensional Ornstein-Uhlenbeck process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Following [7], the algorithm conducting full orthonormalization was considered and the weak convergence of all components was derived [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The diffusion approximation of the first component of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) was also considered in our previous work [3] in which both first -order and second-order approximation were derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' As a corollary, the weak convergence of the first component of the SGA method was verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' However, a diffusion approximation of the whole SGA iteration method (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) is still an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The main tool utilized in [3] is the Lax equivalence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' An alternative stochastic analysis approach to prove the convergence is developed by Milstein [11], which was adopted to derive diffusion approximations of the stochastic gradient descent (SGD) method [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' First of all, we investigated properties of the semigroup Sk, k = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=',.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In particular, we proved the stability and the regularity of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' For the stability, we proved that for a fixed terminal time T, if ∥W0∥F ≤ r, then there exist constants C and η0 that depend on r, T and the distribution of x such that ∥W(k)∥F ≤ C holds for all k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', �T η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Here η ∈ (0, η0) is the learning rate in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' See Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We proved the stability because it is necessary for the application of the Lax equivalence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' For the regularity, we prove that Skϕ, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' admit the same order regularity as ϕ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' ∥Skϕ∥Cm(B(0,r)) ≤ C∥ϕ∥Cm(B(0,r′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Here C and r′ are constants that depend on m, r, T and the distribution of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' See Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Second, we constructed the desired diffusion approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We proved that the following family of SDEs ˙W = G(A, W) + ηPTWO(n)F(W) + √ηPTWO(n) ˙Z, W(0) = W0 ∈ O(n) 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU will stay on the Stiefel manifold for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' See Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Here ˙Z = ( ˙Zij)n×n, ˙Zij(W) = Hijkl(W) ◦ ˙Bkl, where H = (Hijkl)n×n×n×n are coefficients and ˙Bkl is the white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' See (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) for detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In fact, the SDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) serves as the first-order diffusion approximation of the SGA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' we proved that under proper regularity conditions of the test function ϕ, there exists a constant C1 = C1(x, T, η) such that sup W∈O(n),kη≤T |Skϕ(W) − u(W, kη)| ≤ C1(x, T, ϕ)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Here u(W, t) is the solution to the Kolmogorov equation determined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13), with the initial value ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' See Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The main idea of the proof comes from the Lax equivalence theorem [6]: stability and consistence is equivalent to convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The consistence is ensured by Taylor’s expansion, see Section 6 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' A natural question is that whether higher- order approximation exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Unfortunately, the answer is no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We proved that the possible second order approximation, which is an SDE, does not stay on the Stiefel manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' See Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' This instability is probably due to the omitted higher-order terms in the SGA algorithm: second and higher-order (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' η) terms were neglected in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) when conducting the Gram-Schmidt orthogonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Finally, for two special cases, we proved the exponential convergence of the SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' As we introduced before, we seek for reversibility to prove the exponential convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' First, we consider the overdamped Langevin equation on the Stiefel manifold: dQ(t) = PTQO(n) ◦ (−∇U(Q)dt + σdW(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The exponential convergence of it is proved in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' If we select the potential U as the weighted Rayleigh quotient (see [10]) and let σ = 0, then the Oja-Brockett flow [2] is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We have to emphasize that the overdamped Langevin equation is not a special case of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) since G in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) is not a gradient of a certain potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Second, for n = 2 of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13), the SDE is rewritten as dW = F1(W)dt + √η · c(W)W ◦ dZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Here F1 is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5) and c(W) is a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' See details in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Exponential conver- gence of this case is proved in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The main approach is to consider the dynamics of the rotational angle of W ∈ O(2), which is a one-dimensional SDE, and the reversibility automatically holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Premier First, we rewrite (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) by matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' For Λ, Q ∈ Rn×n, define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1) Σ(Λ, Q) := n � j=1 j−1 � k=1 EjQTΛQEk − EkQTΛQEj, G(Λ, Q) := ΛQ − QQTΛQ + QΣ(Λ, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) also reads as � A(k) = x(k)xT (k), W(k) = W(k − 1) + ηkG(A(k), W(k − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS 5 In our previous paper [10], we thoroughly investigated corresponding ODE, which can be written as: \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ˙Q = Q n � j=1 j−1 � k=1 (EjQTAQEk − EkQTAQEj), Q(0) = Q0 ∈ O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3) We define F1(Q) := QΣ(A, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='4) Thus one can rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3) as � ˙Q = F1(Q), Q(0) = Q0 ∈ O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5) From now on, we will use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5) in all proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Notations and assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We will follow the convention of notations in our pre- vious paper [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We assume that ν is compact, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', there exists a constant M > 0 such that ∥x∥2 ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='6) In the following sections, we will adopt both the matrix representation and the component- wise representation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5), thus we clarify the notation here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' qi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', n represent the column vectors of Q in order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Q = [q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', qn], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='7) while ˜qi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', n represent the row vectors of Q in order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' QT = [ ˜q1 T, ˜q2 T, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', ˜qn T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='8) For each entry, qi,j, i, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', n represent the entries at ith row, jth column of the matrix Q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' qj = (q1,j, q2,j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', qn,j)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='9) The canonical orthonormal basis in Rn is denoted as ej, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', n, which are written in column vectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In = [e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', en].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='10) Here In is the identity matrix of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' For M, N ∈ Rn×n, ∥M∥F represents the Frobenius norm of M and ⟨M, N⟩F represents the inner product in the Frobenius sense: ∥M∥ = � tr(MMT), ⟨M, N⟩F = tr(MTN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='11) For x = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', xn) ∈ Rn, ∥x∥2 represents the ℓ2 norm of x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' ∥x∥2 = � � � � n � j=1 |xj|2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='12) Suppose that the eigenvalues of A are all single, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' of multiplicity one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Denote them as λ1 > λ2 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' > λn > 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU in descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Without loss of generality, we assume that A is diagonal: A = diag{λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', λn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By default, omitted proofs of Lemmas and other important but complicated computations are available Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Diffusion approximation of the online PCA algorithm In this section, we consider the iteration scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We also assume that the learning rates are constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' ηk = η > 0, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='. Under these assumptions, we derived the diffusion approximation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Our main results imply that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5) is the weak limit of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) as η approaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We also derived families of matrix-valued SDEs (invariant in the Steifel manifold) which serve as first order weak approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In particular, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5) is understood as a special case of these SDEs by taking the time step size η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The matrix-valued discrete time Markov process defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' W(k), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', is time homogeneous because x(k) share the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Following the notations in [3], we denote the expectation under the distribution of this Markov chain starting from W0 as EW0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In our discussion, W0 is assumed to be deterministic though it could be a random variable in general contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Denote the law of W(k) (starting from W0) as µk(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' W0) and the transition probability as µ(V, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then by the Markov property, for any Borel set E ⊂ Rn×n, µk+1(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' W0) = � Rn×n µ(V, E)µk(dV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' W0) = � Rn×n µk(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' U)µ(W0, dU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' For a fixed test function ϕ ∈ L∞(Rn×n), define uk(W0) = EW0 [ϕ(W(k))] = � Rn×n ϕ(V)µk(dV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' W0), k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1) Here W(k) is defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The Markov property yields uk+1(W0) = EW0[EW0[ϕ(W(k + 1))|W(1)]] = EW0[uk(W(1))] = � Rn×n µ(dW1, W0) � Rn×n ϕ(V)µk(dV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' W1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2), we derive that for any W ∈ Rn×n, uk+1(W) = Euk(W + ηG(xxT, W)) := Suk(W), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) hence u0(W) = ϕ(W) and {Sk}k≥0 forms a semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Before discussing the diffusion approximation, we derive some basic properties of the Markov chain and the semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (stability) Fix a real number r > 0 and a terminal time T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Let W(k), k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' be the Markov chain generated by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) with an initial datum W0 satisfying ∥W0∥F ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then there exist constants η(r, M, T) > 0 and C(r, M, T) > 0 which depend on r, T and M in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='6) such that for any 0 < η ≤ η(r, M, T) and k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', �T η � , P � ∥W(k)∥2 F ≤ C(r, M, T) � = 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', W(k) is uniformly bounded for any time discretization with time step size less than η(r, M, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS 7 See Section 6 for the proof of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Based on Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1, we prove that uk(W) possesses the same regularity as the test function ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Admissible sets of test functions are Cm b (Rn×n) := \uf8f1 \uf8f2 \uf8f3f ∈ Cm(Rn×n) ��∥f∥Cm(Rn×n) := � |α|≤m |Dαf|∞ \uf8fc \uf8fd \uf8fe , m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='. (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='4) We denote the open ball in (Rn×n, ∥ · ∥F) centered at M ∈ Rn×n with radius r by B(M, r), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' B(M, r) := {N ∈ Rn×n : ∥N − M∥F < r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (properties of the semigroup) Fix a real number r > 0 and a terminal time T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Let {uk}k≥0, {Sk}k≥0 be the semigroup defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) and M be the upper bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Suppose that ϕ ∈ Cm b (Rn×n) where m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then: (1) (L∞ contraction) S : L∞(Rn×n) → L∞(Rn×n) is a contraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (2) (regularity) there exist constants η(r, m, M, T) > 0 and C(r, m, M, T) > 0 such that for any 0 < η ≤ η(r, m, M, T): ∥uk∥Cm(B(0,r)) ≤ C(r, m, M, T)∥ϕ∥Cm(B(0,eT (2M2+1)/2r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1, for any η ≤ η(r, M, T), there exists C(r, M, T) > 0 such that ∥W∥F ≤ C(r, M, T) holds almost surely for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', [T/η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Notice that G is a polynomial w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' W, so there exists a constant C′(r, M, T) > 0 such that for any W ∈ B(0, r) and indices (i, j) and (i′, j′), ���� ∂(G(x(k)x(k)T , W))i′,j′ ∂wi,j ���� ≤ C′(r, M, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='7) Here (G)i′,j′ represents the entry at the i′th row and j′th column in the matrix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' According to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1, we know that ∥W(k)∥2 F ≤ (1 + (2M2 + 1)η)∥W(k − 1)∥2 F, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', [T/η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus if W ∈ B(0, r), then W + ηG(x(k)x(k)T, W) ∈ B(0, r � 1 + (2M2 + 1)η) ⊂ B � 0, � 1 + (2M2 + 1)η 2 � r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='8) Denote C′′ = (2M2 + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Now we proceed to prove the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (1) Suppose that ψ ∈ L∞(Rn×n), then the L∞ contraction is derived directly by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2): (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='9) ∥Sψ∥L∞(Rn×n) = sup W∈Rn×n |Eψ(W + ηG(xxT, W))| ≤ sup W∈Rn×n |ψ(W)| ≤ ∥ψ∥L∞(Rn×n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (2) We prove the case m = 1 by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The proof for the case of m ≥ 1 is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Because u0 = ϕ, so the conclusion holds for k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Now for k ≥ 0, by the dominant 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU convergence theorem (DCT), we have for any W ∈ B(0, r) and 1 ≤ i, j ≤ n, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='10) ∂uk+1 ∂wi,j (W) = � 1≤i′,j′≤n E � ∂uk ∂wi′,j′ (W + ηG(x(k)x(k)T, W)) · ∂(W + ηG(x(k)x(k)T, W))i′,j′ ∂wi,j � = E � ∂uk ∂wi,j (W + ηG(x(k)x(k)T, W)) � + η � 1≤i′,j′≤n E � ∂uk ∂wi′,j′ (W + ηG(x(k)x(k)T, W)) · ∂(G(x(k)x(k)T , W))i′,j′ ∂wi,j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Remember that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='8) holds, so ����E � ∂uk ∂wi,j (W + ηG(x(k)x(k)T, W)) ����� ≤ ���� ∂uk ∂wi,j ���� L∞(B(0,(1+C′′η)r)) Substituting this into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='10) yields ���� ∂uk+1 ∂wi,j (W) ���� ≤ ���� ∂uk ∂wi,j ���� L∞(B(0,(1+C′′η)r)) + ηC′(r, M, T) � 1≤i′,j′≤n ���� ∂uk ∂wi′,j′ (W + ηG(x(k)x(k)T, W)) ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='7), taking L∞ norm on both sides and summing up over 1 ≤ i, j ≤ n, we derive � 1≤i,j≤n ���� ∂uk+1 ∂wi,j ���� L∞(B(0,r)) ≤ � 1≤i,j≤n ���� ∂uk ∂wi,j ���� L∞(B(0,(1+C′′η)r)) + n2C′(r, M, T)η∥uk∥C1(B(0,(1+C′′η)r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' This inequality yields ∥uk+1∥C1(B(0,r)) ≤ (1 + n2C′(r, M, T)η)∥uk∥C1(B(0,(1+C′′η)r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='11) By induction, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='12) ∥uk∥C1(B(0,r)) ≤ (1 + n2C′(r, M, T)η)k∥u0∥C1(B(0,(1+C′′η)kr)) ≤ (1 + n2C′(r, M, T)η)T/η∥ϕ∥C1(B(0,(1+C′′η)T/ηr)) ≤ en2C′(r,M,T)T∥ϕ∥C1(B(0,eT (2M2+1)/2r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The diffusion approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In this section, we discuss the diffusion approximation of the semigroup (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' SDEs on the Stiefel manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Because the SGA method aims to derive the correct unit eigenbasis, so we expect that our SDE should stay on the Stiefel manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Therefore, we consider the following family of SDEs: ˙W = G(A, W) + ηPTWO(n)F(W) + √ηPTWO(n) ˙Z, W(0) = W0 ∈ O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) Here PTWO(n) is the projection onto the tangent space of O(n) at W, see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' ˙Z is defined as ˙Z(W) = ( ˙Zij)n×n, ˙Zij(W) = Hijkl(W) ◦ ˙Bkl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='14) DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS 9 H(W) = (Hijkl(W))n×n×n×n is the coefficient tensor of the Brownian motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' B = (Bij)n×n is the standard Brownian motion in Rn×n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The notation ’◦’ represents that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) is an SDE in the Stratonovich sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By letting η = 0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13), the SDE degenerates to the ODE ˙W = G(A, W), W(0) = W0 ∈ O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' This is exactly (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus we expect that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) serves as the diffusion approximation of the SGA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We first check that for arbitrary F(W) ∈ Rn×n and H(W) ∈ Rn×n×n×n, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) stays on the Stiefel manifold if W0 ∈ O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' As preparation, we first rewrite the projection operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Let PTWO(n) defined as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5) for some W = (wij)n×n ∈ O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Let P := (Pijkl)n×n×n×n, Pijkl := 1 2(wiswksδjl − wkjwil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='15) Then we have: (i) (projection) for any F = (fij)n×n, (PTWO(n)F)ij = Pijklfkl = 1 2(fij − wilfklwkj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='16) (ii) (symmetry) Pijkl = Pklij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' P is symmetric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (iii) (idempotence) PijklPklrs = Pijrs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' P2 = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Because W ∈ O(n), so wiswks = δik and Pijkl = 1 2(δikδjl − wkjwil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='17) We first prove (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Because PTWO(n)F = 1 2(F − WFTW), we have (PTWO(n)F)ij = 1 2(fij − wilfklwkj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='17), we derive Pijklfkl = 1 2(δikδjl − wkjwil)fkl = 1 2(fij − wilfklwkj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus (PTWO(n)F)ij = Pijklfkl = 1 2(fij − wilfklwkj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (ii) is obvious: Pijkl = 1 2(δikδjl − wkjwil) = 1 2(δkiδlj − wilwkj) = Pklij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' For (iii), we have PijklPklrs = 1 4(δikδjl − wkjwil)(δkrδls − wrlwks) = 1 4(δirδjs − wrjwis − wrjwis + wkjwilwrlwks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Because W ∈ O(n), we have wilwrl = δir, wkjwks = δjs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU Thus PijklPklrs = 1 2(δirδjs − wrjwis) = Pijrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We define Pijkl as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='15) instead of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='17), because in this case, even though W /∈ O(n), the following equality still holds: wijPtjkl + wtjPijkl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='18) This is helpful in proving invariance of Stiefel manifold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Now we are ready to check that the Stiefel manifold is invariant for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Consider (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' If W0 ∈ O(n), then W(t) ∈ O(n) holds for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2, we can rewrite (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) as dwij = gij(A, W)dt + ηPijkl(W)fkl(W)dt + Pijkl(W)Hklrs(W) ◦ dBrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then for 1 ≤ i, t ≤ n, remember that the SDE is in the Stratonovich sense, we have d(wijwtj) = wij ◦ dwtj + wtj ◦ dwij = (wijgtj(A, W) + wtjgij(A, W))dt + (wijPtjkl + wtjPijkl)(fkldt + Hklrs ◦ dBrs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Notice that wijPtjkl + wtjPijkl = 1 2[wij(wtswksδjl − wkjwtl) + wtj(wiswksδjl − wkjwil)] = 1 2(wilwtswks − wijwkjwtl + wiswkswtl − wilwtjwkj) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus d(wijwtj) = (wijgtj(A, W) + wtjgij(A, W))dt, which is an ODE system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We can rewrite this in matrices as d(WWT) dt = WGT(A, W) + G(A, W)WT = WWTA + AWWT − 2WWTAWWT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus X = WWT satisfies the algebraic Riccati equation: dX dt = AX + XA − 2XAX, with initial value X = In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus by results in [18], we know that X(t) = In holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Therefore, W(t) ∈ O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Fokker-Planck equation and Kolmogorov equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In the Itˆo sense, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) reads as ˙W = G(A, W) + ηPTWO(n)F(W) + η 2J(W) + √ηPTWO(n) ˙Y(W), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='19) Here Y is defined as ˙Y(W) = ( ˙Yij)n×n, ˙Yij(W) = Hijkl(W) ˙Bkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='20) The correction term J is then given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='21) K = (Kijkl)n×n×n×n, Kijkl = PijrsHrskl J = (Jij)n×n, Jij(W) := Krsml ∂Kijml ∂wrs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS 11 Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='19) could be rewritten as ˙wij = gij(A, W) + ηPijklfkl + η 2Jij + √ηKijkl ˙ Bkl = gij(A, W) + ηPijklfkl + η 2Jij + √ηPijklHklrs ˙ Brs The corresponding backward Kolmogorov equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='19) is ∂tu(W, t) = Lu(W, t), u(W, 0) = ϕ(W), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='22) where L is an elliptic operator defined as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='23) Lu := � gij(A, W) + ηPijklfkl + η 2Jij � ∂u ∂wij + η 2KijrsKklrs ∂2u ∂wijwkl = � gij(A, W) + ηPijklfkl + η 2Jij � ∂u ∂wij + η 2PijzvHzvrsPklxyHxyrs ∂2u ∂wijwkl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='22) is given by u(W, t) = etLϕ(W) = EW[ϕ(X(t))], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='24) here X(t) ∈ O(n) is the random process (before vectorization) determined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) (or equivalently (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='19)) with starting point X(0) = W ∈ O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' For more details, we refer readers to [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' On the other hand, the solution of the forward Kolmogorov equation (the Fokker-Planck equation) is denoted as ρ(W, t) = etL∗ρ0(W): ∂tρ = L∗ρ, ρ(W, t) = ρ0(W), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='25) where ρ0 is the initial distribution of W and L∗ is defined as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='26) L∗ρ := − ∂ ∂wij �� gij(A, W) + ηPijklfkl + η 2Jij � ρ � + η 2 ∂2 ∂wijwkl (ρKijrsKklrs) = − ∂ ∂wij �� gij(A, W) + ηPijklfkl + η 2Jij � ρ � + η 2 ∂2 ∂wijwkl (ρPijzvHzvrsPklxyHxyrs) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The solution ρ(W, t) = etL∗ρ0(W) is interpreted as the probability distribution of the random process X in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) with the initial distribution ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Notice that {etL}t≥0 and {etL∗}t≥0 form two semigroups, we have the following basic properties: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Consider {etL}t≥0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='23) and {etL∗}t≥0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then: (1) (probability and positivity preserving) if ρ0 ∈ L1(O(n)) and ρ0 ≥ 0, then etL∗ρ0 ≥ 0 and ∥etL∗ρ∥L1(O(n)) = ∥ρ0∥L1(O(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='27) In particular, etL∗ is a contraction from L1(O(n)) to L1(O(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (2) (L∞ contraction) Suppose that ϕ : O(n) → R is a continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' etL : L∞(O(n)) → L∞(O(n)) is a contraction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' ∥etLϕ∥L∞(O(n)) ≤ ∥ϕ∥L∞(O(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='28) 12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The proof of (1) can be found in [3], here we just prove (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Because ϕ is continuous and O(n) is compact, we have ϕ ∈ L∞(O(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Suppose W0 ∈ O(n), let W(t) be the stochastic process generated by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) starting from W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then we have etLϕ(W0) = EW0 [ϕ(W(t))] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='29) By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3, we know that W(t) ∈ O(n) for all t ≥ 0, thus |etLϕ(W0)| ≤ EW0|ϕ(W(t))| ≤ ∥ϕ∥L∞(O(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='30) This holds for any W0 ∈ O(n), thus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='28) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' □ More discussion on contraction and positivity preserving is available in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Now we are ready to verify that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) serves as a weak diffusion approximation of the SGA iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' First-order diffusion approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The method to validate that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) serves as a diffusion approximation originates from the idea of the Lax equivalence theorem [6], which was first adopted in [3] for the same purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (first-order diffusion approximation) Fix time T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Consider {uk(W)}k≥0 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Suppose that ϕ ∈ C4 b (Rn×n) and u(W, t) is the solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='22) with the initial value u(·, 0) = ϕ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then there exist η1(M, T) > 0 and C1(M, T, ϕ) > 0 such that for any η ∈ (0, η1(M, T)], sup W∈O(n),kη≤T |uk(W) − u(W, kη)| ≤ C1(M, T, ϕ)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='31) Here M is the constant in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In the following proof, C is a general constant that varies among equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1, there exist constants η(M, T) and C(M, T) such that for all k = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', [T/η] and η ∈ (0, η(M, T)], ∥uk∥C4(B(0,√n+1)) ≤ C(M, T)∥ϕ∥C4(Rn×n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='32) Then by Taylor’s expansion w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' η, we have that for any W ∈ O(n), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='33) ����uk+1(W) − uk(W) − ηgij(A, W) ∂uk ∂wij ���� ≤ C(M, T)∥uk∥C4(B(0,√n+1))η2 ≤ C(M, T, ϕ)η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Details of this computation can be found in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='4, etL is a contraction from L∞(O(n)) to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Therefore for any W ∈ O(n), there exists η′ ∈ (0, η] such that |eηLuk(W) − uk(W) − ηLuk(W)| ≤ η2 2 |eη′LL2uk(W)| ≤ η2 2 ∥L2uk∥L∞(O(n)) ≤ C(M, T, ∥uk∥C4(B(0,√n+1)))η2 ≤ C(M, T, ϕ)η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Recall the definition of L in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='23), we can rewrite the above estimate as ����eηLuk(W) − uk(W) − ηgij(A, W) ∂uk ∂wij ���� ≤ C(M, T, ϕ)η2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='34) DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS 13 See Section 6 for details of this computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='33) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='34), we have ∥uk+1(W) − eηLuk(W)∥L∞(O(n)) ≤ C(M, T, ϕ)η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='35) Now for k = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', [T/η], define ek := ∥uk(W) − u(W, kη)∥L∞(O(n)), rk := ∥uk+1(W) − eηLuk(W)∥L∞(O(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='36) Thus rk ≤ C(M, T, ϕ)η2 for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', [T/η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Notice that u(W, (k + 1)η) = eηLu(W, kη), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='37) ek+1 = ∥uk+1(W) − u(W, (k + 1)η)∥L∞(O(n)) = ∥uk+1(W) − eηLu(W, kη)∥L∞(O(n)) ≤ ∥uk+1(W) − eηLuk(W)∥L∞(O(n)) + ∥eηL(u(W, kη) − uk(W))∥L∞(O(n)) ≤ rk + ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The last step in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='37) used that eηL is a contraction on L∞(O(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus summing up (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='37) in k yields ek ≤ k−1 � l=1 rl ≤ TC′′(M, T, ϕ) η η2 = C1(M, T, ϕ)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='38) This holds uniformly in k, so the claim is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Unstable second order approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' According to the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2, the key step that ensures first order approximation is the following estimate on the truncation error, which is of second order: ∥Sϕ(W) − eηLϕ(W)∥L∞(O(n)) ≤ Cη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Here L is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' To derive second order approximation, we need to carefully select the drift term F and H in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) such that ∥Sϕ(W) − eηLϕ(W)∥L∞(O(n)) ≤ Cη3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='39) Direct calculation yields Sϕ(W) = E � ϕ(W + ηG(xxT, W)) � = ϕ(W) + ηgij(A, W) ∂ϕ ∂wij + η2 2 E � gij(xxT, W)gkl(xxT, W) � ∂2ϕ ∂wij∂wkl + O(η3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Meanwhile, eηLϕ(W) = ϕ(W) + ηLϕ(W) + 1 2η2L2ϕ(W) + O(η3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='23), we have L2ϕ = L � gij(A, W) ∂ϕ ∂wij + O(η) � = gkl(A, W) ∂ ∂wkl � gij(A, W) ∂ϕ ∂wij � + O(η) = � gkl(A, W)∂gij(A, W) ∂wkl � ∂ϕ ∂wij + gij(A, W)gkl(A, W) ∂2ϕ ∂wij∂wkl + O(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='40) 14 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU Comparing the coefficients of ∂2ϕ ∂wij∂wkl and ∂ϕ ∂wij , we derive (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='41) Pijklfkl = −1 2Jij − 1 2gkl(A, W)∂gij(A, W) ∂wkl , PijuvHuvrsPklxyHxyrs = E[gij(A − xxT , W)gkl(A − xxT , W)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Here F = (fij)n×n and H = (Hijkl)n×n×n×n are coefficients in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Details of the derivation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='41) is summarized in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' To interpret (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='41), one can see that the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' of the second equation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='41) is exactly the covariance tensor of G(xxT, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We denote it as M(W) = (Mijkl(W))n×n×n×n, Mijkl = E � gij(xxT − A, W)gkl(xxT − A, W) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='42) Therefore, we desire suitable H such that PijuvHuvrsPklxyHxyrs = Mijkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='43) This can be realized if W ∈ O(n), which is sufficient for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Consider M in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then there exists a unique N = (Nijkl)n×n×n×n that satisfy (i) (symmetry) Nijkl = Nklij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (ii) (positive semidefinite) for any (mij)n×n, mijNijklmkl ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (iii) (square root of M) NijrsNklrs = Mijkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Moreover, if W ∈ O(n), then PijrsNrskl = Nijkl, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='44) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' PN = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' See Section 6 for the proof of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In the view of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5, we also denote N as N = √ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='45) If we take H = N = √ M, in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13), then PijuvHuvrsPklxyHxyrs = PijuvNuvrsPklxyNxyrs = NijrsNklrs = Mijkl holds if W ∈ O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='43) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Now we take F = 0, H = N = √ M (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='46) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5, we know that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) stays on O(n), so (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) could be rewritten as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='47) dwij = gij(A, W)dt + √ηPijkl(W)Nklrs(W) ◦ dBrs, = gij(A, W)dt + √ηNijrs(W) ◦ dBrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' This SDE seems to serve as the second order approximation for the SGA method: we have gij in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) as the drift term;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' covariance of gij is also reflected in the Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' How- ever, because F in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='46) does not satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='41), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='47) is not a second order approximation, but a first order approximation by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Moreover, there is even no solution to the first equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' This excludes the possibility of deriving a second order approximation on the Stiefel manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS 15 Even with the last try, we require H satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='41) and just replace PTWO(n)F in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) by L = (Lij)n×n, Lij := −1 2Jij − 1 2gkl(A, W)∂gij(A, W) ∂wkl , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='48) then resulted SDE still does not stay on the Stiefel manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Therefore, we have no second order diffusion approximation that stays on the Stiefel manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (unstable second order approximation) Consider (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='41), L in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='48), and J = (Jij)n×n in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then (i) Suppose that H satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then there is no F = (fij)n×n that satisfies Pijklfkl = Lij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' the first equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='41) admits no solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (ii) Consider the solution W(t) to the following SDE: dwij = (gij(A, W) + ηLij) dt + √ηNijrs(W) ◦ dBrs, W(0) = W0 ∈ O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='49) Then there is no time interval [t0, t1] such that W(t) ∈ O(n) for t ∈ [t0, t1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Here N = (Nijkl)n×n×n×n is the square root of the covariance matrix M (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We first prove (i) by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Suppose there is a solution F, then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='18), wijLtj + wtjLij = wijPtjklfkl + wtjPijklfkl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Notice −2(wijLtj + wtjLij) = wijJtj + wtjJlj � �� � I + gkl(A, W) � wij ∂gtj(A, W) ∂wkl + wtj ∂gij(A, W) ∂wkl � � �� � II .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By definition of J, K in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='21), we have I = ∂ ∂wrs ((wijPtjuv + wtjPijuv) HuvmlKrsml) − Ktjml ∂ ∂wrs (wijKrsml) − Kijml ∂ ∂wrs (wtjKrsml).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='18), the first term of I is zero, so we have I = −Ktjml ∂ ∂wrs (wijKrsml) − Kijml ∂ ∂wrs (wtjKrsml) = −2KtjmlKijml − wijPtjuvHuvmlKrsml − wtjPijuvHuvmlKrsml = −2KtjmlKijml = −2E[gtj(A − xxT, W)gij(A − xxT, W)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' For II, we have II = ∂ ∂wkl [(wijgtj(A, W) + wtjgij(A, W))gkl(A, W)] − gtj(A, W) ∂ ∂wkl (wijgkl(A, W)) − gij(A, W) ∂ ∂wkl (wtjgkl(A, W)) = −2gij(A, W)gtj(A, W) + gkl ∂ ∂wkl (gtjwij + gijwtj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 16 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU Thus −2(wijLtj + wtjLij) = I + II = −2E[gtj(A − xxT , W)gij(A − xxT , W)] − 2gij(A, W)gtj(A, W) + gkl ∂ ∂wkl (gtjwij + gijwtj) = −2E[gtj(xxT, W)gij(xxT, W)] + gkl ∂ ∂wkl (gtjwij + gijwtj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' This can not be zero for general R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' x, because the first term depends on the fourth-order momentum while the second term only depends on the second-order momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' This is a contradiction, so wijLtj + wtjLij ̸= 0 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='41) admits no solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' For (ii), we still prove by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Suppose that for some time interval [t0, t1], W(t) ∈ O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then d(WWT) dt = 0, t ∈ (t0, t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5, we know that if W ∈ O(n), then PijrsNrskl = Nijkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' So we can rewrite (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='49) as dwij = (gij(A, W) + ηLij) dt + √ηPijrsNrskl ◦ dBkl However, d(wijwtj) = wijdwtj + wtjdwij = (wijgtj(A, W) + wtjgij(A, W))dt + η(wijLtj + wtjLij)dt + √η(wijPtjrs + wtjPijrs)Nrskl ◦ Bkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Remember that if W ∈ O(n), then G(A, W) = PTWO(n)G(A, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' So gij = Pijklgkl, which yields that for all t ∈ (t0, t1), wijgtj + wtjgij = (wijPtjrs + wtjPijrs)grs = 0 by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Due to the same reason, √η(wijPtjrs + wtjPijrs)Nrskl ◦ Bkl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus d(wijwtj) = η(wijLtj + wtjLij)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' However, (wijLtj + wtjLij) ̸= 0 by (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus d(wijwtj) ̸= 0 for t ∈ (t0, t1), which contradicts with wijwtj = δit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' This is a contradiction, the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' □ In the view of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='6, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='49) fails to stay on the Stiefel manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In this case, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='22) is a degenerate parabolic PDE with unbounded coefficients, which fails to control the diffusion in the normal direction of the Stiefel manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus, utilizing solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='22) to approximate the behavior of the semigroup (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) is meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Reversible diffusion approximation: exponential convergence In Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2, we derived diffusion approximations on the Stiefel manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' A natural question is that whether the SDE is ergodic and converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In fact, reversibility and Poincare’s inequality ensure exponential convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' To see this, the Fokker-Planck operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='26) can be recast as L∗ρ = ∂ ∂wij � ρ � −gij − ηPijklfkl + η 2KijrsKklrs ∂ log ρ ∂wkl + η 2Jij �� = ∂ ∂wij �η 2KijrsKklrsρ �ηJij − 2gij − 2ηPijklfkl ηKijrsKklrs + ∂ log ρ ∂wkl �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' If there exists a function U(W) such that ∂U ∂wkl = ηJij − 2gij − 2ηPijklfkl ηKijrsKklrs , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1) then solutions to ∂tρ = L∗ρ satisfies ∂tρ = ∂ ∂wij �η 2KijrsKklrsρ∂(log ρ + U) ∂wkl � = ∂ ∂wij �η 2e−UKijrsKklrs ∂(eUρ) ∂wkl � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Without loss of generality, we assume that � O(n) e−U(W)dW = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then multiply eUρ − 1 on both sides, and by definition of K in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='21), we derive d dt � O(n) e−U|eUρ − 1|2dW = −η � O(n) e−U � Kijrs ∂ ∂wij (eUρ − 1) � � Kklrs ∂ ∂wkl (eUρ − 1) � dW = −η � O(n) e−U|HT∇W(eUρ − 1)|2dW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Here ∇W is the gradient operator on (O(n), ge) where ge is the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then by Poincare’s inequality, we derive exponential convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Therefore, we desire to carefully select H and F such that the potential condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We provide the following two special cases where reversibility is ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' First, we consider the overdamped Langevin on O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In particular, we select the potential as U(W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' A, N) = tr(NWTAW), where N is a diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then we recover the Oja- Brockett flow with Bronwian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In fact, the Oja-Brockett flow is the gradient flow of U(W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' A, N) = tr(NWTAW) on (O(n), ge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Second, we consider the two-dimensional case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In this case, orthogonal matrices are determined by the rotational angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The SDE of the angle is an SDE on R, which automatically satisfy the potential condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In each case, Poincare’s inequality is verified, so they converge to the invariant measure exponentially fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The overdamped Langevin dynamics on O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Suppose that U(Q) : O(n) → R is a smooth function (which serves as the free energy), consider the overdamped Langevin dynamics on O(n): dQ(t) = PTQO(n) ◦ (−∇U(Q)dt + σdW(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) Here PTQO(n) is the projection onto TQO(n), ∇ represents the derivative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (∇U(Q))ij = ∂u(Q) ∂qij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 18 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU W(t) ∈ Rn×n is the standard Brownian motion and σ is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' ’◦’ means that the above SDE is in the Stratonovich sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In general, σ can be a matrix that depends on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' To illustrate the the Langevin dynamics on O(n), we consider the simplest case here which is sufficient for diffusion approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' If we take U(Q) = −tr(NQTAQ) where N is a diagonal matrix with entries on the diagonal line aligned in the descending order, then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) reads as dQ(t) = (AQN − QNQTAQ)dt + σPTQO(n) ◦ dW(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3) Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3) should be viewed as the disturbed Oja-Brockett flow [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By the conversion rule, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) should be formulated in Ito’s sense as dQ(t) = � −PTQO(n)∇U(Q) − σ2(n − 1) 4 Q � dt + σPTQO(n)dW(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='4) By Ito’s formula, we can derive the Fokker-Planck equation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) which is ∂ρ(Q, t) ∂t = ∇Q · (ρ(Q, t)∇QU(Q)) + σ2 2 ∆Qρ(Q, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5) Here ∇Q·, ∇Q and ∆Q are the divergence, gradient and Laplace-Beltrami operator on (O(n), ge) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' See Section 6 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Compactness of O(n) implies exponential convergence of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Direct calculation yields that the invariant measure of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5) is given by ρeq(Q) := 1 Z e−2U(Q)/σ2, Z := � O(n) e−2U(Q)/σ2dV, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='6) where dV is the volume form on (SO(n), ge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5) can be reformulated as ∂ρ(Q, t) ∂t = σ2 2 ∇Q · � ρeq(Q)∇Q �ρ(Q, t) ρeq(Q) �� := L∗ρ(Q, t), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='7) and we denote the Fokker-Planck operator as L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' This also implies that the invariant measure of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5) is unique since L∗ρ = 0 ⇐⇒ ∇Q · � ρeq(Q)∇Q �ρ(Q, t) ρeq(Q) �� = 0 ⇐⇒ � O(n) ρeq(Q) ����∇Q �ρ(Q, t) ρeq(Q) ����� 2 dV = 0 ⇐⇒ ρ(Q) = cρeq(Q), where c is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' If ρ(Q) is a probability measure on SO(n), then c = 1 and ρ = ρeq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In the above induction we used the positivity of ρeq(Q), which is a consequence of the continuity of U(Q) and the compactness of SO(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Multiplying ρ(Q, t)/ρeq(Q) − 1 on both sides of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5) results in d dt � O(n) ρeq(Q) ���� ρ(Q, t) − ρeq(Q) ρeq(Q) ���� 2 dV = −σ2 � O(n) ρeq(Q) ����∇Q �ρ(Q, t) − ρeq(Q) ρeq(Q) ����� 2 dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='8) DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS 19 If we can prove the Poincare inequality in L2(ρeq(Q)dV), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', there exists a constant C > 0 such that for all f ∈ H1(ρeq(Q)dV) satisfying � SO(n) f(Q)ρeq(Q)dV = 0, we have � O(n) ρeq(Q) |∇Q (f(Q))|2 dV ≥ C � O(n) ρeq(Q)|f(Q)|2dV, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='9) then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='8) yields d dt � O(n) ρeq(Q) ���� ρ(Q, t) − ρeq(Q) ρeq(Q) ���� 2 dV ≤ −C � O(n) ρeq(Q) ���� ρ(Q, t) − ρeq(Q) ρeq(Q) ���� 2 dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By Gronwall’s inequality, this gives exponential convergence of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5) whose initial value is a probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Now we prove the Poincare inequality in L2(ρeqdV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (Poincare’s inequality) Suppose that f ∈ H1(ρeq(Q)dV) satisfying � O(n) f(Q)ρeq(Q)dV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' To prove this, we prove that for any λ > 0, (λ − L∗)−1 : L2 (dV/ρeq(Q)) → L2 (dV/ρeq(Q)) is a compact operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Suppose that {un}n≥1 and {gn}n≥1 are two sequences in L2(dV/ρeq(Q)) such that (λ − L∗)un = gn, n ≥ 1, and {gn}n≥1 is uniformly bounded in L2(dV/ρeq(Q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then multiplying un on both sides yields � O(n) 1 ρeq un · (λ − L∗)undV = � O(n) ρeq ����∇Q � un ρeq ����� 2 dV + λ � SO(n) ρeq ���� un ρeq ���� 2 dV = � O(n) 1 ρeq gnundV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By Cauchy-Schwartz’s inequality, we know that un/ρeq uniformly bounded in H1(ρeqdV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The compact embedding H1(ρeqdV) ֒→֒→ L2(ρeqdV) (see [17]) implies that up to subse- quences, there exists u∗ ∈ L2(ρeqdV), un ρeq → u∗ ρeq in L2(ρeqdV), or equivalently un → u∗ in L2(dV/ρeq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus (λ − L∗)−1 is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus 1/λ ̸= 0 is not an accumulation point of σ((λ − L∗)−1), hence 0 is not an accumulation point of σ(L∗), but the single principal eigenvalue of L∗, whose eigenvectors are c · ρeq where c is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' So for any g ∈ L2(dV/ρeq), we have − � O(n) g ρeq L∗gdV ≥ C � O(n) |g|2 ρeq dV (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='10) for all g satisfying � O(n) gdV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Let g = ρeqf, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' □ 20 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The case of n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' If we ask F, H in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) to satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='46), then the SDE reads as dwij = gij(A, W)dt + √η · Nijkl ◦ dBkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='11) Here M(W) is the covariance matrix of G(xxT , W) defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='11) admits the Stiefel manifold as an invariant set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Utilizing this fact, we can reformulate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='11): Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Suppose n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Consider (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) where H and H satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) (or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='11)) can be reformulated as dW = F1(W)dt + √η · c(W)W ◦ dZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='12) Here F1 is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='4), c(W) is a scalar defined as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) c(W) := � c1(w4 1,1 + w4 1,2) + c2w2 1,1w2 1,2 + c3w1,1w1,2(w1,1w2,2 + w1,2w2,1), c1 := E[x2 1x2 2] − (E[x1x2])2, c2 := E[x4 1 + x4 2 − 4x2 1x2 2] + 2(E[x1x2])2 + 2E[x2 1]E[x2 2] − (E[x2 1])2 − (E[x2 2])2, c3 := 2E[x3 1x2 − x1x3 2] − E[x2 1]E[x1x2] + E[x2 2]E[x1x2] and Z(t) ∈ R2×2 is defined as Z(t) := � 0 −B(t) B(t) 0 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='14) where B(t) is the standard Brownian motion in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' See Section 6 for the proof of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Remember that each element in O(2) can be expressed in either of the following forms: O1(θ) = � cos θ sin θ − sin θ cos θ � , θ ∈ [0, 2π), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='15) O−1(θ) = � cos θ sin θ sin θ − cos θ � , θ ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='16) Because the orbit of W(t) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) is continuous a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' and the determinant is also a continuous function w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' W, so if W0 = O1(θ) for some θ, then W(t) = O1(θ(t)) for some θ(t) ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Without loss of generality, we assume W0 = O1(θ0) for some θ0 ∈ [0, 2π), thus for any t ≥ 0, w1,1(t) = w2,2(t), w1,2(t) + w2,1(t) = 0, w2 1,1(t) + w2 1,2(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='17) To prove the convergence of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='12), we consider the process of θ(t) instead of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We construct the following one dimensional SDE in Ito’s sense: dθ(t) = f(θ)dt + √ηg(θ)dB, θ(t) = θ0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='18) where B(t) is the standard Brownian motion, f, g are defined as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='19) g(θ) = −c(θ) f(θ) = (E[x2 2] − E[x2 1]) cos θ sin θ + η(2c1(θ) − c2(θ)) 2 (cos3 θ sin θ − sin3 θ cos θ) + 3ηc3(θ) 2 cos2 θ sin2 θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS 21 Here c, c1, c2 and c3 are the scalar functions defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) and c(θ) is c(W) where W is replaced by O1(θ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' c(θ) = c �� cos θ sin θ − sin θ cos θ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='20) Then we can use the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='18) to represent the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Consider (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Suppose that the initial value W(0) satisfies W(0) = O1(θ0) (defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='15)) for some θ0 ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Suppose that θ(t) is the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='18), then w1,1(t) = w2,2(t) = cos(θ(t)), w1,2(t) = sin(θ(t)), w2,1(t) = − sin(θ(t)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='21) solves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='12) with the initial value W(0) = O1(θ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' See Section 6 for the proof of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Let T = R/2πZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Now consider (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We denote the invariant measure of this SDE as ρ∞(x) ∈ P(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then ρ∞ is the stationary solution to the Fokker- Planck equation with the periodic boundary condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' ∂ρ(x, t) ∂t = L1ρ(x, t), L1ρ(x, t) := −∂(f(x)ρ(x, t)) ∂x + 1 2 · ∂2(ηg2(x)ρ(x, t)) ∂x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='22) Here f, g are defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='19) A direct calculation yields ρ∞(x) = C exp �� x 0 2f(s) η · g2(s)ds � , C = � exp �� 2π 0 2f(s) η · g2(s)ds ��−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='23) According to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3 (see Section 6), we know that g(x) is strictly positive on T, thus ρ∞(x) > 0 for any x ∈ T, thus the exponential convergence holds by a Poincare’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Define the weighted L2 space L2(T, dx/ρ∞) as L2(T, dx/ρ∞) := � p : � T p2(x) ρ∞(x)dx < ∞ � , ⟨p, q⟩dx/ρ∞ = � T p(x)q(x) ρ∞(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='24) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Suppose that θ(t), t ≥ 0 solves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='18) with the initial value θ0, which is a random variable with density function ρ0 ∈ P(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Let ρ(x, t) ∈ P(T) be the law of θ(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', P(θ(t) ∈ B) = � B ρ(x, t)dx for any Borel set B ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Consider ρ∞(x) ∈ P(T), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' the invariant measure in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then there exists a constant c > 0 only depending on η and momentums of x such that ∥ρ(·, t) − ρ∞∥L2(T,dx/ρ∞) ≤ e−ct∥ρ0 − ρ∞∥L2(T,dx/ρ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='25) See Section 6 for proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' An example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In this section, we explicitly calculated an example here to illustrate our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Suppose that x = (x1, x2)T where x1 and x2 are independent random variables such that they possess density function ρ1 = 1 4 · 1(−2,2), ρ2 = 1 2 · 1(−1,1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' x1 ∼ Uni(−2, 2) and x2 ∼ Uni(−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then the covariance matrix of x is A = � 4/3 0 0 1/3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 22 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU Then by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13), we have c1 = 4 9, c2 = 8 45, c3 = 0, and c = � 4 9(cos4 θ + sin4 θ) + 8 45 cos2 θ sin2 θ = � 4 9 − 32 45 cos2 θ sin2 θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus c ≥ � 4/15 for any θ ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='18) reads as dθ = � − cos θ sin θ + 16η 45 (cos3 θ sin θ − sin3 θ cos θ) � dt + � 4 9 − 32 45 cos2 θ sin2 θ · dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' According to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='23), the invariant measure is ρ∞(x) = exp �1 η � x 0 −45 cos θ sin θ + 16η(cos3 θ sin θ − sin3 θ cos θ) 10 − 16 cos2 θ sin2 θ dθ � , x ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Direct calculation yields ρ∞(x) = exp � −15 √ 6 16η arctan 2 √ 6 sin2 x 5 − 4 sin2 x � � 5 8 sin4 x − 8 sin2 x + 5, x ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='26) The explicit formula of the invariant measure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='26), shows that if η << 1, then the mass of the invariant measure concentrates around x = 0 and x = π, or in terms of the matrix, around −I2 and I2, which gives the right principal component decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The case of p < n All results for the case of p = n can be extended to the case of p < n, by being careful on the size of tensors and rewrite the projection operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' First, all regularity and stability results for the semigroup still hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The proof is exactly the same as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1, by replacing the terminal index n by p for column indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Second, the diffusion approximation is now formulated as ˙W = G(A, W) + ηPTWO(n×p)F(W) + √ηPTWO(n×p) ˙Z, W(0) = W0 ∈ O(n × p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1) Here PTWO(n×p) is the projection onto the tangent space of O(n × p) at W, see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' ˙Z is defined as ˙Z(W) = ( ˙Zij)n×p, ˙Zij(W) = Hijkl(W) ◦ ˙Bkl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) H(W) = (Hijkl(W))n×p×n×p is the coefficient tensor of the Brownian motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' B = (Bij)n×p is the standard Brownian motion in Rn×p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The notation ’◦’ represents that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1) is an SDE in the Stratonovich sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The projection operator onto TWO(n × p) then reads as PTWO(n×p)M = (In − WWT)M + 1 2W(WTM − MT W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3) When p = n, WWT = In so PTWO(n×p) is exactly the projection on O(n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' when p = 1, WTM = MTW are all scalars, then PTWO(n×p)M = PTWSn−1M = (In − WWT)M, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='4) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' PTWO(n×p) degenerates to the projection onto the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS 23 Using the same technique and method as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3, one can prove that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1) stays on O(n × p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' moreover, the diffusion approximation is also of first-order as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' For reversibility, the overdamped Langevin is still reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' All the proof in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1 holds generally on O(n × p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Acknowledgement Jian-Guo Liu was supported in part by the National Science Foundation (NSF) under award DMS-2106988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Appendix 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Riemannian manifolds and the Stiefel manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We denote the tangent space at m on manifold M as TmM, the tangent vector field on M as Γ(TM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The tangent bundle (the disjoint union of the tangent spaces) is denoted as TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (Riemannian manifolds) Suppose that M is a smooth manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' A Rie- mannian manifold (M, g) is a smooth mainfold equipped with an inner product gm on TmM at each m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Moreover, for any tangent vector field ˙x and ˙y, the function ⟨ ˙x(m), ˙y(m)⟩gm : M → R (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1) is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Given a Riemannian metric g on M, the gradient of a smooth function E on M is defined as Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (the gradient on the Riemannian manifold) A tangent vector field ∇gE on M is called the gradient of E w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' the metric g if for every tangent vector field ˙x on M, ⟨E′, ˙x⟩F = ⟨∇gE, ˙x⟩g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) Here E′ is the derivative of E, which is a cotangent vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Now we consider the Stiefel manifold with the Euclidean metric ge, under the global coordinate Q ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We first introduce several important properties of O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' See [4] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (the Stiefel manifold) The Stiefel manifold O(n) is a smooth, compact manifold of dimension n(n − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The tangent space at Q is given by TQO(n) = {QΩ | Ω ∈ Rn×n, Ω + ΩT = 0}, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3) while the normal space at Q is given by TQO(n)⊥ = {QΩ | Ω ∈ Rn×n, Ω = ΩT}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='4) See [4] for proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1, we can prove that for any M ∈ Rn×n, the projection on the tangent spaces and the normal spaces are respectively: PTQO(n)M := 1 2(M − QMT Q), PTQO(n)⊥M := 1 2(M + QMT Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5) Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (Gradient on (O(n), ge)) Suppose that ϕ(Q) : O(n) → R is a restriction of a smooth function (still denoted as ϕ : Rn×n → R) on O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then gradient of ϕ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' ge at point Q is given by ∇geϕ := PTQO(n)(∇ϕ) = 1 2(∇ϕ − Q(∇ϕ)TQ), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='6) 24 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU here ∇ϕ ∈ Rn×n is the gradient of ϕ in Rn×n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (∇ϕ)ij = ∂ϕ ∂qij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We just need to prove that for any Q ∈ O(n) and any tangent vector at Q, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1, a tangent vector at Q can be represented as QΩ where Ω is skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Therefore, for any Ω that is skew-symmetric, we have ⟨∇ϕ, QΩ⟩F = ⟨PTQO(n)(∇ϕ), QΩ⟩F + ⟨PTQO(n)⊥(∇ϕ), QΩ⟩F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='7) Notice that ⟨PTQO(n)⊥(∇ϕ), QΩ⟩F = ⟨Q(QT∇ϕ + (∇ϕ)TQ), QΩ⟩F = ⟨QT∇ϕ + (∇ϕ)TQ, Ω⟩F = 0, since ⟨M, N⟩F = 0 if M is symmetric while N is skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Therefore, ⟨∇ϕ, QΩ⟩F = ⟨PTQO(n)(∇ϕ), QΩ⟩F = ⟨PTQO(n)(∇ϕ), QΩ⟩ge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' So ∇geϕ = PTQO(n)(∇ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' □ Under this global coordinate, the divergence on (O(n), ge) can also be explicitly computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The ij entry of ∇geϕ is given by (∇geϕ)ij = 1 2 � ∂ϕ ∂qij − � k,l qikqlj ∂ϕ ∂qlk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='8) So given a tangent vector field H(Q) = (hij(Q)), the divergence of it is defined by ∇ge · H(Q) := � i,j (∇ge(hij(Q)))ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='9) The Laplace-Beltrami operator is then defined as: ∆geϕ := ∇ge · ∇geϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='10) Explicit expression of the Laplace-Beltrami operator is given by the following lemma: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (Laplace-Beltrami operator) The Laplace-Beltrami operator on (O(n), ge) is given by ∆geϕ = 1 2 �� i,j ∂2ϕ ∂q2 ij − (n − 1) � i,j qij ∂ϕ ∂qij − � i,j,k,l qikqlj ∂2ϕ ∂qij∂qlk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By the expression of the divergence and gradient, we have ∆geϕ = � i,j � ∂hij ∂qij − � k,l qikqlj ∂hij ∂qlk � , hij = � i,j � ∂ϕ ∂qij − � k,l qikqlj ∂ϕ ∂qlk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus ∂hij ∂qij = 1 2 � ∂2ϕ ∂q2 ij − � k,l δklqlj ∂ϕ ∂qlk − � k,l δilqik ∂ϕ ∂qlk − � k,l qikqlj ∂2ϕ ∂qijqlk � = 1 2 � ∂2ϕ ∂q2 ij − � l qlj ∂ϕ ∂qlj − � k qik ∂ϕ ∂qik − � k,l qikqlj ∂2ϕ ∂qijqlk � , DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS 25 and ∂hij ∂qlk = 1 2 � ∂2ϕ ∂qij∂qlk − � k′,l′ δilδkk′ql′j ∂ϕ ∂ql′k′ − � k′,l′ δll′δkjqik′ ∂ϕ ∂ql′k′ − � k′,l′ qik′ql′j ∂2ϕ ∂ql′k′∂qlk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Substituting the above formulas into the Laplacian operator, we can derive (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Proofs of lemmas and omitted calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' According to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2), direct computation yields (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='12) ∥W(k)∥2 F = ∥W(k − 1)∥2 F + 2η · tr(W(k − 1)TA(k)W(k − 1)) − 2η · tr(W(k − 1)TW(k − 1)W(k − 1)TA(k)W(k − 1)) + η2 · tr(W(k − 1)TA(k)2W(k − 1)) − 2η2 · tr(W(k − 1)TA(k)W(k − 1)W(k − 1)TA(k)W(k − 1)) + η2 · tr(W(k − 1)TA(k)W(k − 1)W(k − 1)TW(k − 1)W(k − 1)TA(k)W(k − 1)) − 2η2 · tr(W(k − 1)TA(k)W(k − 1)W(k − 1)TW(k − 1)Σ(A(k), W(k − 1))) + η2 · tr(Σ(A(k), W(k − 1))TW(k − 1)TW(k − 1)Σ(A(k), W(k − 1))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By definition of Σ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1), we have (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13) ∥Σ(A(k), W(k − 1)∥F = �� i̸=j (wi(k − 1)TA(k)wj(k − 1))2 = �� i̸=j (wi(k − 1)Tx(k))2(wj(k − 1)Tx(k))2 ≤ n � i=1 (wi(k − 1)Tx(k))2 ≤ M2 n � i=1 ∥wi(k − 1)∥2 2 = M2∥W(k − 1)∥2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By the following norm inequality: ∥M∥2 ≤ ∥M∥F ≤ √n∥M∥2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='6) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13), we derive the following estimates for each term in the above equality in the almost surely sense: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='14) tr(W(k − 1)TA(k)W(k − 1)) = ∥W(k − 1)Tx(k)∥2 2 ≤ ∥W(k − 1)T∥2 2∥x(k)∥2 2 ≤ M2∥W(k − 1)∥2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='15) tr(W(k − 1)TA(k)2W(k − 1)) = ∥x(k)∥2 2tr(W(k − 1)TA(k)W(k − 1)) ≤ M4∥W(k − 1)∥2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 26 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='16) tr(W(k − 1)TA(k)W(k − 1)W(k − 1)TW(k − 1)W(k − 1)TA(k)W(k − 1)) = tr((W(k − 1)Tx(k))(x(k)TW(k − 1)W(k − 1)TW(k − 1)W(k − 1)TA(k)W(k − 1))) ≤ ∥W(k − 1)Tx(k)∥2∥x(k)TW(k − 1)W(k − 1)TW(k − 1)W(k − 1)TA(k)W(k − 1)∥2 ≤ M2∥W(k − 1)∥6 2∥A(k)∥2 ≤ M4∥W(k − 1)∥6 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='17) |tr(W(k − 1)TA(k)W(k − 1)W(k − 1)TW(k − 1)Σ(A(k), W(k − 1)))| = |tr((W(k − 1)Tx(k))(x(k)TW(k − 1)W(k − 1)TW(k − 1)Σ(A(k), W(k − 1))))| ≤ ∥(W(k − 1)Tx(k)∥2∥x(k)TW(k − 1)W(k − 1)TW(k − 1)Σ(A(k), W(k − 1))∥2 ≤ M2∥W(k − 1)∥4 2∥Σ(A(k), W(k − 1))∥2 ≤ M2∥W(k − 1)∥4 F∥Σ(A(k), W(k − 1))∥F ≤ M4∥W(k − 1)∥6 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='18) tr(Σ(A(k), W(k − 1))TW(k − 1)TW(k − 1)Σ(A(k), W(k − 1))) = ∥W(k − 1)Σ(A(k), W(k − 1))∥2 F ≤ n∥W(k − 1)Σ(A(k), W(k − 1))∥2 2 ≤ n∥W(k − 1)∥2 2∥Σ(A(k), W(k − 1))∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' ≤ nM4∥W(k − 1)∥6 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Substituting (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='14) ∼ (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='18) into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='12) yields ∥W(k)∥2 F ≤ (1 + 2M2η)∥W(k − 1)∥2 F + η2(M4∥W(k − 1)∥2 F + (n + 3)M4∥W(k − 1)∥6 F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='19) Select η(T) as η(T) = 1 M4(1 + (n + 3)r4e2T(2M2+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='20) Then we can prove that for any η ≤ η(T), the Markov chain generated by (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=') which starts from W0 satisfies ∥W(k)∥2 F ≤ r2e2M2+1, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', [T/η], a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='. (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='21) We prove by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Given any η ≤ η(T), if ∥W(k − 1)∥2 F ≤ r2eT(2M2+1), then ∥W(k)∥2 F ≤ (1 + 2M2η)∥W(k − 1)∥2 F + η2(M4∥W(k − 1)∥2 F + (n + 3)M4∥W(k − 1)∥6 F) ≤ (1 + 2M2η)∥W(k − 1)∥2 F + (ηM4(1 + (n + 3)r4e2T(2M2+1))) · η · ∥W(k − 1)∥2 F ≤ (1 + (2M2 + 1)η)∥W(k − 1)∥2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Remember that ∥W0∥2 F ≤ r2, so ∥W0∥2 F ≤ r2e2M2+1, hence for any k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', [T/η], ∥W(k)∥2 F ≤ (1 + (2M2 + 1)η)kr2 ≤ (1 + (2M2 + 1)η)T/ηr2 ≤ r2eT(2M2+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus taking C(T) = r2eT(2M2+1) concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' □ DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS 27 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Because M = (Mijkl) is the covariance matrix of G, it is positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Therefore, the square root of M is well-defined, which satisfies (i),(ii) and (iii) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Denote it as N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' When W ∈ O(n), we have proved that for any symmetric B ∈ Rn×n, G(B, W) ∈ TWO(n), so PTWO(n)G(A − xxT, W) = G(A − xxT, W), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Pijklgkl = gij by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Therefore, by linearity of expectation and symmetry Mijkl = E[gij(A − xxT, W)gkl(A − xxT, W)] = E[Pijrsgrs(A − xxT, W)gxy(A − xxT, W)Pklxy] = PijrsE[grs(A − xxT, W)gxy(A − xxT, W)]Pxykl = PijrsMrsxyPxykl, or in matrix notation, M = PMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus by (iii) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2, we have PijrsMrskl = PijrsPrsuvMuvxyPxykl = PijrsPuvrsMuvxyPxykl = PijuvMuvxyPxykl, or PM = PMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Similarly, MijrsPrskl = PijuvMuvxyPxyrsPrskl = PijuvMuvxyPxykl, or MP = PMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus MijrsPrskl = PijrsMrskl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' So P and M are commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Because P is symmetric, so P is also commutative with the square root of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus NijrsPrskl = PijrsNrskl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus PijrsNrsxyPxyuvNuvkl = PijrsNrsxyNxyuvPuvkl = PijrsMrsuvPuvkl = Mijkl, or PNPN = PNNP = PMP = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' So PN is also the square root of M, by uniqueness, PijrsNrskl = Nijkl, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' PN = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' □ Calculations in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Direct calculation yields (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='22) Sϕ(W) = E � ϕ(W + ηG(xxT, W)) � = ϕ(W) + � gij(A, W) ∂ϕ ∂wij � η + �1 2E � gij(xxT , W)gkl(xxT, W) � ∂2ϕ ∂wij∂wkl � η2 + O(η3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Meanwhile, eηLϕ(W) = ϕ(W) + ηLϕ(W) + 1 2η2L2ϕ(W) + O(η3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 28 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='23), we have L2ϕ = L � gij(A, W) ∂ϕ ∂wij + O(η) � = gkl(A, W) ∂ ∂wkl � gij(A, W) ∂ϕ ∂wij � + O(η) = � gkl(A, W)∂gij(A, W) ∂wkl � ∂ϕ ∂wij + gij(A, W)gkl(A, W) ∂2ϕ ∂wij∂wkl + O(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='23) Thus (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='24) eηLϕ(W) = ϕ(W) + η �� gij(A, W) + ηPijklfkl + η 2Jij � ∂ϕ ∂wij + η 2KijrsKklrs ∂2ϕ ∂wijwkl � + η2 2 �� gkl(A, W)∂gij(A, W) ∂wkl � ∂ϕ ∂wij + gij(A, W)gkl(A, W) ∂2ϕ ∂wij∂wkl � + O(η3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Recast (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='24) in the order of η, we have (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='25) eηLϕ(W) = ϕ(W) + � gij(A, W) ∂ϕ ∂wij � η+ �� Pijklfkl + 1 2Jij + 1 2gkl(A, W)∂gij(A, W) ∂wkl � ∂ϕ ∂wij � η2 + � (KijrsKklrs + gij(A, W)gkl(A, W)) ∂2ϕ ∂wij∂wkl � η2 + O(η3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Comparing the η2 term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='22) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='25), we have ∂ϕ ∂wij : Pijklfkl + 1 2Jij + 1 2gkl(A, W)∂gij(A, W) ∂wkl = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' ∂2ϕ ∂wij∂wkl : KijrsKklrs + gkl(A, W)gij(A, W) = E � gij(xxT, W)gkl(xxT, W) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus F = (fij)n×n and H = (Hijkl)n×n×n×n should satisfy Pijklfkl = −1 2Jij − 1 2gkl(A, W)∂gij(A, W) ∂wkl , PijuvHuvrsPklxyHxyrs = Mijkl = E[gij(A − xxT, W)gkl(A − xxT, W)], which is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='41) □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We first compute H(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3, we know that WWT = I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus G(xxT − A, W) = (xxT − A)W − WWT(xxT − A)W + WΣ(xxT − A, W) = WΣ(xxT − A, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Denote B = xx − A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Since n = 2, we have Σ(B, W) = � 0 −w1 · Bw2 w1 · Bw2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS 29 Denote b = w1 · Bw2, then G(xxT − A, W) = W � 0 −b b 0 � = � w1,2b −w1,1b w2,2b −w2,1b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus vec(G(B, W)) = (w1,2b, −w1,1b, w2,2b, −w2,1b)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Denote u = (w1,2, −w1,1, w2,2, −w2,1)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then the covariance matrix defined in (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=') is (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='26) M(W) = E[vec(G(B, W))vec(G(B, W))T] = E[b2]uuT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Therefore, H(W) = � M(W) = � E[b2] ∥u∥2 uuT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Remember that W ∈ O(2), thus ∥u∥2 = √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' So we have H(W) = � E[b2] √ 2 uuT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='27) Substituting (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='27) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='12), we have dvec(W) = vec(G(A, W))dt + � ηE[b2] √ 2 uuT ◦ dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='28) Moreover, because ∥u∥2 = √ 2 and B(t) is the standard Brownian motion in R4, thus uTdB has the same law with √ 2dB where B(t) is the standard Brownian motion in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='12) can also be reformulated as dvec(W) = vec(G(A, W))dt + � ηE[b2]u ◦ dB(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='29) Moreover, because W(t) ∈ O(2), thus G(A, W) = F(W) where F is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='12) can be directly transformed in the form of matrices: dW = F(W)dt + � ηE[b2] · Q ◦ dZ, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='30) where Z is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Now we compute E[b2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Direct computation yields b = w1 TBw2 = b1,1w1,1w1,2 + b1,2(w1,1w2,2 + w2,1w1,2) + b2,2w2,1w2,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus b2 = b2 1,1w2 1,1w2 1,2 + b2 1,2(w1,1w2,2 + w1,2w2,1)2 + b2 2,2w2 2,1w2 2,2 + 2b1,1b2,2w1,1w1,2w2,1w2,2 + 2b1,1b1,2w1,1w2,2w1,2 + 2b1,1b1,2w1,1w2 1,2w2,1 + 2b1,2b2,2w1,1w2,1w2 2,2 + 2b1,2b2,2w1,2w2 2,1w2,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Meanwhile, we know bi,j = xixj − E[xixj] for 1 ≤ i, j ≤ 2, thus for any i, j, i′, j′ = 1, 2, we have E[bi,jbi′,j′] = E[xixjxi′xj′] − E[xixj]E[xi′xj′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Substituting this into expression of b2, we have E[b2] = var(x2 1)w2 1,1w2 1,2 + var(x1x2)(w2 1,1w2 2,2 + w2 2,1w2 1,2 + 2w1,1w1,2w2,1w2,2) + var(x2 2)w2 2,1w2 2,2 + 2(E[x2 1x2 2] − E[x2 1]E[x2 2])w1,1w1,2w2,1w2,2 + 2(E[x3 1x2] − E[x2 1]E[x1x2])w1,1w1,2(w1,1w2,2 + w1,2w2,1) + 2(E[x1x3 2] − E[x2 2]E[x1x2])w2,1w2,2(w1,1w2,2 + w1,2w2,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 30 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU Using w2 1,1 = w2 2,2, w2 1,2 = w2 2,1, we then derive E[b2] = c1(W)(w4 1,1 + w4 1,2) + c2(W)w2 1,1w2 1,2 + c3(W)w1,1w1,2(w1,1w2,2 + w1,2w2,1), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='31) where c1, c2 and c3 is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus by taking c(W) = � E[b2], we derive (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='12) which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' □ Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' We first write (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='12) in It´o’s sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Because we have assumed that |W(t)| = 1, thus we only need to consider the equation for w1,1 and w1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In the It´o sense, we have dw1,1 = [(F(W))1,1 + h1(W)]dt + √η · c(W)w1,2dB dw1,2 = [(F(W))1,2 + h2(W)]dt − √η · c(W)w1,1dB, where h1 and h2 are defined in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Denote g1 = √ηc(W)w1,2, g2 = −√ηc(W)w1,1, then h1 and h2 are computed as (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='32) h1(W) = 1 2 � g1 ∂g1 ∂w1,1 + g2 ∂g1 ∂w1,2 � = η 2 �1 2w2 1,2 ∂c(W)2 ∂w1,1 − 1 2w1,1w1,2 ∂c(W2) ∂w1,2 − w1,1c(W)2 � h2(W) = 1 2 � g1 ∂g2 ∂w1,1 + g2 ∂g2 ∂w1,2 � = η 2 �1 2w2 1,1 ∂c(W)2 ∂w1,2 − 1 2w1,1w1,2 ∂c(W2) ∂w1,1 − w1,2c(W)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Here we used the chain rule: c ∂c ∂w = 1 2 ∂c2 ∂w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Substituting the above equation into the SDE in It´o’s sense yields (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='33) dw1,1 = � (E[x2 2] − E[x2 1])w1,1w1,2 + η 4w2 1,2 ∂c2 ∂w1,1 − η 4w1,1w1,2 ∂c2 ∂w1,2 − η 2w1,1c(W)2 � dt + √ηc(W)w1,2dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Now consider the process of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Suppose that θ(t) satisfies the following SDE in Ito’s sense: dθ = f(θ)dt + √η · g(θ)dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Then Ito’s isometry yields d cos θ = − sin θ · dθ − ηg2(θ) 2 dt = � − sin θ · f(θ) − η cos θ 2 g2(θ) � dt − √η · sin(θ)g(θ)dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Replacing w1,1 by cos θ, w1,2 by sin θ in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='33) and comparing the coefficients of it with the above equation yields (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='34) g(θ) = −c(θ), f(θ) = (E[x2 2] − E[x2 1]) sin θ cos θ + η · 2c1(θ) − c2(θ) 2 (cos3 θ sin θ − cos θ sin3 θ) + η · 3c3(θ) 2 cos2 θ sin2 θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Here we used (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='31) since c2(W) = E[b2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' This is exactly the SDE in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' □ DIFFUSION APPROXIMATIONS OF OJA’S ONLINE PRINCIPAL COMPONENT ANALYSIS 31 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In the following proof, C is just a general constant that may vary among equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Because ρ(x, t), t ≥ 0 is the law of θ(t), so ρ solves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='22) on T × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus by the periodic boundary condition, d dt � T ρ(x, t)dx = 1 2 � T � −∂(f(x)ρ(x, t)) ∂x + 1 2 · ∂2(ηg2(x)ρ(x, t)) ∂x2 � dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' So � T ρ(x, t)dx = 1, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Now consider the Fokker-Planck operator L∗ 1 which is self-adjoint in L2(T, dx/ρ∞): L∗ 1 : D(L∗ 1) ⊂ L2(T, dx/ρ∞) → L2(T, dx/ρ∞), L∗ 1p := − d dx � ρ∞(x) d dx � p(x) ρ∞(x) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Here D(L∗ 1) = H2(T, dx/µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' A direct calculation yields ⟨L1∗p, q⟩dx/ρ∞ = � T ρ∞(x) d dx � p(x) ρ∞(x) � d dx � q(x) ρ∞(x) � dx, thus L∗ 1 is semi-positive definite, and L∗ 1p = 0 ⇐⇒ p(x) = cρ∞(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' So 0 is the simple principle eigenvalue of L∗ 1 with ρ∞(x) as the eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Moreover, 0 is isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' For any λ > 0, we prove that (λ + L∗ 1)−1 is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Let {gn}∞ n=1 ∈ L2(T, dx/ρ∞) be a bounded sequence, with (λ + L∗)un = gn, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='. To prove that (λ+L∗)−1 is compact, we just need to prove that there exists a subsequence of {un}∞ n=1 which is Cauchy in L2(T, dx/ρ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Because L∗ is semi-positive definite, so (λ + L∗) is bounded, thus {un}∞ n=1 is bounded in L2(T, dx/ρ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By the Cauchy-Schwatz inequality, we have ⟨L∗ 1un, un⟩dx/ρ∞ = ⟨un, gn⟩dx/ρ∞ − λ∥un∥2 L2(T,dx/ρ∞) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='35) Here C is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus ∥un/ρ∞∥2 H1(T,ρ∞dx) = � T ρ∞(x) � d dx un(x) ρ∞(x) �2 dx + � T ρ∞(x) � un(x) ρ∞(x) �2 dx = ∥un∥L2(T,dx/ρ∞) + ⟨L∗ 1un, un⟩dx/ρ∞ ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' So un/ρ∞ is bounded in H1(T, ρ∞dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' By the compact embedding H1(T, ρ∞dx) ⊂⊂ L2(T, ρ∞dx), we know that there exists a subsequence of un (still denoted as un) such that un ρ∞ → u∗ ρ∞ in L2(T, ρ∞dx), or equivalently un → u∗ in L2(T, dx/ρ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' So (λ + L∗)−1 is a compact operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus the spectrum of (λ + L∗)−1 only admits 0 as an accumulation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' So 0 is an isolated point in the spectrum of L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Thus for any p ∈ L2(T, dx/ρ∞) such that � T p(x)dx = 0, we have the following Poincare’s inequality: there exists a constant c > 0 such that � T ρ∞(x) � d dx p(x) ρ∞(x) �2 dx = ⟨L∗ 1p, p⟩L2(T,dx/ρ∞) ≥ c∥p∥2 L2(T,dx/ρ∞) = c � T |p(x)|2 ρ∞(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='36) 32 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' LIU Multiplying ρ(x, t)/ρ∞(x) − 1 on both sides of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='22) and substituting p = ρ(x, t) − ρ∞(x) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='36) yields 1 2 d dt � T (ρ(x, t) − ρ∞(x))2 ρ∞(x) dx = − � T ρ∞(x) � d dx ρ(x, t) ρ∞(x) �2 dx ≤ −c � T |ρ(x, t) − ρ∞(x)|2 ρ∞(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='37) Thus by Gronwall’s inequality, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='25) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' □ References [1] Vincent D Blondel, Alexandre Megretski, and Vincent DD Blondel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Unsolved problems in mathematical systems and control theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Princeton University Press Princeton, NJ, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' [2] Roger W Brockett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Dynamical systems that sort lists, diagonalize matrices, and solve linear program- ming problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Linear Algebra and its applications, 146:79–91, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' [3] Yuanyuan Feng, Lei Li, and Jian-Guo Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Semigroups of stochastic gradient descent and online prin- cipal component analysis: properties and diffusion approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Communications in Mathematical Sciences, 16(3), 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' [4] Uwe Helmke and John B Moore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Optimization and dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Springer Science & Business Media, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' [5] Elton P Hsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Stochastic analysis on manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Number 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' American Mathematical Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=', 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' [6] Peter D Lax and Robert D Richtmyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Survey of the stability of linear finite difference equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Communications on pure and applied mathematics, 9(2):267–293, 1956.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' [7] Chris Junchi Li, Mengdi Wang, Han Liu, and Tong Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Diffusion approximations for online principal component estimation and global convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' [8] Qianxiao Li, Cheng Tai, and Weinan E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Stochastic modified equations and adaptive stochastic gradient algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 2101–2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' PMLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' [9] Xin Liang, Zhen-Chen Guo, Ren-Cang Li, and Wen-Wei Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Nearly optimal stochastic approximation for online principal subspace estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' arXiv preprint arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='06644, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' [10] Jian-Guo Liu and Zibu Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Convergence of oja’s online principal component flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='11308, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' [11] Grigorii Noikhovich Milstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Numerical integration of stochastic differential equations, volume 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Springer Science & Business Media, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' [12] Erkki Oja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Simplified neuron model as a principal component analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Journal of mathematical biology, 15(3):267–273, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' [13] Erkki Oja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Principal components, minor components, and linear neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Neural networks, 5(6):927–935, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' [14] Erkki Oja and Juha Karhunen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' On stochastic approximation of the eigenvectors and eigenvalues of the expectation of a random matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Journal of mathematical analysis and applications, 106(1):69–84, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' [15] Bernt Oksendal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Stochastic differential equations: an introduction with applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Springer Science & Business Media, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' [16] Christian Soize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' The Fokker-Planck equation for stochastic dynamical systems and its explicit steady state solutions, volume 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' World Scientific, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' [17] Michael Eugene Taylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' 1, Basic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Springer, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' [18] Wei-Yong Yan, Uwe Helmke, and John B Moore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Global analysis of oja’s flow for neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' IEEE Transactions on Neural Networks, 5(5):674–683, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content=' Department of Mathematics and Department of Physics, Duke University, Durham, NC Email address: jliu@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='duke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='edu Department of Mathematics, Duke University, Durham, NC Email address: zibu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='liu@duke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf'} diff --git a/Y9AyT4oBgHgl3EQfvvk3/content/2301.00635v1.pdf b/Y9AyT4oBgHgl3EQfvvk3/content/2301.00635v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..bca52170ccf6cf2f26779719d7feb1477252484c --- /dev/null +++ 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(2022) +Preprint 11 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Line Driven Winds from Variable Accretion Discs +Anthony Kirilov∗, Sergei Dyda†, Christopher S. Reynolds +Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK +11 January 2023 +ABSTRACT +We use numerical hydrodynamics simulations to study line driven winds launched from an accreting α−disc. Building +on previous work where the driving radiation field is static, we compute a time-dependent radiation flux from the +local, variable accretion rate of the disc. We find that prior to the establishment of a steady state in the disc, variations +of ∼ 15% in disc luminosity correlate with variations of ∼ 2 − 3 in the mass flux of the wind. After a steady state +is reached, when luminosity variations drop to ∼ 3%, these correlations vanish as the variability in the mass flux is +dominated by the intrinsic variability of the winds. This is especially evident in lower luminosity runs where intrinsic +variability is higher due to a greater prevalence of failed winds. The changing mass flux occurs primarily due to the +formation of clumps and voids near the disc atmosphere that propagate out into the low velocity part of the flow, a +process that can be influenced by local variations in disc intensity. By computing the normalised standard deviation +of the mass outflow, we show that the impact of luminosity variations on mass outflow is more visible at higher +luminosity. However, the absolute change in mass outflow due to luminosity increases is larger for lower luminosity +models due to the luminosity-mass flux scaling relation becoming steeper. We further discuss implications for CVs +and AGN and observational prospects. +Key words: radiation: dynamics - hydrodynamics - stars: winds, outflows - quasars: general - accretion, accretion +discs – novae, cataclysmic variables +1 INTRODUCTION +Many accretion disc systems such as cataclysmic variables +(CVs), X-ray binaries (XRBs) and active galactic nuclei +(AGN) exhibit blue shifted absorption lines, which is inter- +preted as evidence of outflowing gas. In general, outflows may +be accelerated via thermal, radiation or magnetic pressure. +An important question for any system exhibiting outflows is +which of the above mechanisms, or combination thereof, acts +to launch and accelerate this gas? +In the case of CVs and AGN a strong candidate ac- +celeration mechanism is radiation pressure on spectral lines, +so called line driving. Observationally this is particularly in- +teresting because line driving directly couples accretion and +outflow energy. Matter accreting in the disc converts grav- +itational potential energy into photon energy. These pho- +tons irradiate the gas above the disc and drive an outflow +by transferring momentum from the radiation field to the +gas. Drawing intuition from spherically symmetric models of +line driven winds (Castor, Abbot & Klein 1975, hereafter +CAK75), for isothermal winds with a fixed chemical abun- +dance, the strength of the outflow (mass flux and outflow +velocity) should depend strongly on the driving luminosity +and not so much on the density at the base of the outflow. +This is in contrast to thermally driven winds (Parker 1958), +∗antoniikirilov@gmail.com +†sdyda@ast.cam.ac.uk +where for an isothermal wind, the mass flux has a steep de- +pendence on density at the base. +Though only approximately correct in non-spherical ge- +ometries, the correlation between accretion energy and out- +flow properties for line driven winds can be exploited to con- +strain relevant physical processes. In many systems we ob- +serve both the total system luminosity and discern proper- +ties of the outflowing gas by measuring spectra and using +photoionization modeling. Self-consistent treatment of such +a system therefore requires simulating both the physics in the +accretion disc and the outflow. +Originally, line driving, operating in the Sobolev ap- +proximation, was proposed as a driving mechanism for stellar +winds in OB stars (Lucy & Solomon 1970, hereafter LS70 & +CAK75) and was successful in predicting their mass loss rates +and outflow velocities (Pauldrach, Puls & Kudritzki 1986, +Friend & Abbott 1986). Line driving has since been applied +to a variety of systems, in particular those with accretion +discs such as CVs and AGN. +Early work studied line driven disc winds using a va- +riety of simplifying assumptions. Early analytical studies +assumed simple scaling for the strength of the radiation +field (Vitello & Shlosman 1988) or a decoupling of the ra- +dial and polar angle equations of motion (Murray et. al +1995). Later simulations found stationary outflow solutions +(Pereyra, Kallman & Blondin (1997); Pereyra (1997)), but +these were too coarse to properly resolve the critical point. +This transition was resolved using non-uniform meshes near +© 2022 The Authors +arXiv:2301.03632v1 [astro-ph.HE] 9 Jan 2023 + +2 +A. Kirilov, S. Dyda, C. S. Reynolds +the disc midplane (Proga, Stone & Drew 1998; 1999) which +found that as in the CAK picture the strength of the out- +flow was correlated with total system luminosity but the non- +spherical geometry led to these flows being unsteady for low +driving luminosity. Later work in 3D showed that relaxing +the axisymmetry assumption leads to the formation of clumps +(Dyda & Proga 2018a,b hereafter DP18a,b). The geometry of +the radiation field can also affect the strength of the outflow +by altering the flow geometry (Dyda & Proga 2018c). +In this prior work the disc served as a matter reservoir +and winds were driven by a time-independent radiation field. +Some later studies sought to loosen this assumption on the +radiation field. Kurosowa & Proga (2009) used the variable +mass at the inner boundary to estimate the accretion rate +and corresponding disc luminosity and found strong corre- +lations between these and the wind outflow. These models +were extended to include reprocessed and scattered photons, +which were found to further enhance the strength of the out- +flow (Liu et al. 2013; Mosallanezhad et al. 2019). Nomura et +al. (2020) computed disc luminosity by accounting for mass +losses from the wind and found these losses alter the disc UV +continuum responsible for much of the line driving flux. +We build on prior models by allowing for a time- +dependent radiation field, computed self consistently from the +accretion in the disc. We simulate a thin disc where viscous +dissipation is computed using an α-prescription (Shakura & +Sunyaev 1973) and the local disc intensity is computed from +the local accretion rate. The luminous accretion disc drives +an outflow via radiation pressure on spectral lines. We carry +out a series of simulations for different time-averaged disc +luminosities and characterize the properties of the resulting +outflows and their correlation with disc properties. +The structure of our paper is as follows. In Section 2 we +describe our numerical methods, including how the disc in- +tensity is computed from the accretion rate. In Section 3 we +describe the results of varying accretion rate on the time- +averaged and time-dependent outflow properties. We con- +clude in Section 4 where we discuss the implications for CVs +and AGN winds and observational prospects. +2 NUMERICAL METHODS +We performed all numerical simulations with the publicly +available MHD code Athena++ (Stone et al. 2020). The ba- +sic physical setup is a gravitating, central object surrounded +by a thin, axisymmetric, luminous α-disc. The accretion disc +acts as a source of driving radiation, accelerating the gas that +is assumed to be optically thin to the continuum. Our radi- +ation model is described in detail in DP18a,b, but we sketch +the basic setup here. +2.1 Basic Equations +The basic equations for single fluid hydrodynamics driven by +a radiation field are +∂ρ +∂t + ∇ · (ρv) = 0, +(1a) +∂(ρv) +∂t ++ ∇ · +� +ρvv + P + τ +� += −ρ∇Φ + ρFrad, +(1b) +∂E +∂t + ∇ · +� +(E + P)v + (τ · v) +� += −ρv · ∇Φ + ρv · Frad, (1c) +where ρ, v are the fluid density and velocity respectively, P +is a diagonal tensor with components P the gas pressure, τ +is the viscosity tensor and Frad is the radiation force. For +the gravitational potential, we use Φ = −GM/r and E = +1/2ρ|v|2 + E is the total energy where E = P/(γ − 1) is the +internal energy. The isothermal sound speed is a2 = P/ρ +and the adiabatic sound speed c2 +s = γa2. We use a nearly +isothermal equation of state P = kργ with γ = 1.01. The +temperature is then T = (γ − 1)Eµmp/ρkb where µ = 0.6 +is the mean molecular weight and other symbols have their +standard meaning. +We model the viscosity using the Shakura-Sunyaev +α−disc prescription, where the kinematic viscosity is given +by +ν = αν c2 +s +ΩK , +(2) +for dimensionless parameter αν = 0.01 and ΩK = +� +GM/r3 +the Keplerian orbital frequency. +2.2 Radiation Force and Accretion Disc +We assume a time-dependent radiation field, computed from +the local accretion rate of the axisymmetric, thin disc along +the midplane. The frequency integrated intensity of a thin +disc is +I(rd) = 3 +π +GM +r2⋆ +c +σe Γd +�r⋆ +rd +�3 � +1 − +�r⋆ +rd +�1/2� +, +(3) +where +Γd = +˙Maccσe +8πcr⋆ +(4) +is the disc Eddington number, +˙Macc is the accretion rate in +the disc (see for example Pringle 1981), σe is the Thompson +cross section, rd is the radial position on the disc, and r⋆ +the inner radius of the disc and c the speed of light. The +intensity profile (3) assumes a time independent accretion +rate throughout the disc and therefore a constant Eddington +fraction. +In DP18a,b we assumed a fixed +˙Macc but now we com- +pute it within the simulation domain. We break up the disc +into rings and compute a local accretion rate +˙Macc(r, t) = 4π +ˆ π/2 +θd +ρvrr2 sin θdθ, +(5) +where θd is at the surface of the disc as defined by a density +floor ρd = 10−10. We substitute this accretion rate into (3) +to compute the local disc intensity. This is an approximation, +since we use the analytic expression for the local disc intensity +that was derived assuming a constant accretion rate for the +global disc, not just a local patch. An alternative approach +would be to compute the local intensity from the viscous dis- +sipation. We chose our approach to simplify the comparison +between coupled and uncoupled models. The radiation force +is then computed by assuming the gas is optically thin to +MNRAS 000, 1–?? (2022) + +Line Driven Winds from Variable Accretion Discs +3 +this radiation field and every point in the wind experiences a +radiation force +Frad = Frad +e ++ Frad +L , +(6) +which is a sum of the contributions due to electron scattering +Frad +e +and line driving Frad +L . In this continuum, optically thin +approximation, the radiation force due to electron scattering +is +Frad +e += +" � +nσeIdΩ +c +� +, +(7) +where σe is the electron scattering cross section, n is the +normal vector from the radiating surface to the point in the +wind, dΩ is the solid angle and the integration is carried out +over the entire disc. +We treat the radiation due to lines using a modification +of the CAK formulation where the radiation force due to lines +is +Frad +L += +" +M(t) +� +nσeIdΩ +c +� +, +(8) +and M(t) is the so-called force multiplier. We use the Owocki, +Castor & Rybicki (1984) parametrization of the line strength, +where working in the Sobolev approximation, +M(t) = kt−α +�(1 + τmax)1−α − 1 +τ 1−α +max +� +, +(9) +where k and α are constants, τmax = tηmax, ηmax is related +to the maximum force multiplier via Mmax = k(1 − α)ηα +max +and the optical depth parameter +t = σeρvth +|dvl/dl|, +(10) +where vth is the thermal velocity of the gas and dvl/dl is the +velocity gradient along the line of sight. We take k = 0.2, +α = 0.6 and Mmax = 4400. Additional details about our +numerical treatment of the radiation force can be found in +the Appendix of DP18a and DP18b. +2.3 Simulation Setup +The initial conditions consists of a disc in hydrostatic equi- +librium with density profile +ρ = ρ0 +�r⋆ +rd +�3 � +1 − +�r⋆ +rd +� +exp +� +−z2/2h2� +, +(11) +with rd the radial position on the disc, and the scale-height +h = cs/ΩK. The midplane density parameter 8.33 × 10−3 ⩽ +ρ0 ⩽ 1.6 × 10−1 (in g cm−3) is chosen so the accretion rate +at late times produces the same disc luminosity as our un- +coupled disc runs. For reference, the lowest luminosity run +has accretion rate of ∼ 9.8 × 10−9M⊙ yr−1 at late times. +The angular velocity is initialized to its Keplerian value, +ΩK = +� +GM/r3 +d. +The radial computational domain extends over the +range r⋆ ⩽ r ⩽ 16r⋆ with logarithmic spacing between grids +∆ri+1/∆ri = 1.05. The polar angle range is 0 ⩽ θ ⩽ π/2 +and has logarithmic spacing ∆θi+1/∆θi = 0.95 that ensures +that we have sufficient resolution near the disc midplane to +resolve disc accretion and wind acceleration. We use a grid +resolution of nr×nθ=96×96 cells and Nr×Nθ=3×3=9 MPI +meshblocks. +We impose outflow boundary conditions at the inner +and outer radial boundaries, axis boundary conditions along +the θ = 0 axis and reflecting conditions about the θ = π/2 +midplane. +We chose parameters characteristic of a CV system. +The central object has mass and radius M=0.6 M⊙ and +r⋆ = 8.7 × 108 cm, respectively. The orbital period at the +inner radius is then T0 = 18 s. The ratio between the gas +thermal energy and gravitational potential energy, some- +times referred to as the hydrodynamic escape parameter, +HEP = GM/r⋆c2 +s = 8 × 103 at the base of the wind, cor- +responding to a sound speed cs ≈ 3.4 × 106 cm s−1. For +this value of HEP, thermal driving, which requires HEP ≲ 10 +(Stone & Proga 2009; Dyda et al. 2017), is negligible through- +out the domain but it is not too high that we cannot resolve +the disc accretion with our resolution. The density through- +out the domain outside the disc is set to ρ = 10−20 g cm−3. +We impose a density floor of 10−22 g cm−3 and a pressure +floor computed via our nearly isothermal, adiabatic equation +of state. +3 RESULTS +We consider two classes of models for line driven disc winds, +where the disc intensity is computed from the accretion rate +using (3). In uncoupled models the accretion rate is assumed +to be constant whereas coupled models compute the disc in- +tensity from the local, time-dependent accretion rate (5). We +study how the system behaves for four different average lu- +minosities in both the uncoupled and coupled regimes. We +control the luminosity, via the accretion rate, by varying the +disc midplane density parameter, ρ0, in (11). The parame- +ters of the uncoupled runs were chosen to match the average +accretion rate of coupled models at late times. We summa- +rize our list of models in Table 1. The uncoupled models are +identical in setup to those in DP18b and used to benchmark +the coupled models, which are the novel aspect of this work. +Each run proceeds in two phases. First, the disc evolves +for 2000 inner disc orbits T0, with the radiation force turned +off, to reach a quasi-steady accretion rate where variations in +luminosity are low (≲ 15%). This ensures we can still explore +variability while the equation (3), derived for a steady state +disc can be used to approximate the resulting intensity. Over +the next 100 T0 the radiation force is turned on using a linear +ramp up and we study the resulting disc-wind system for +the following 1400 T0. After this time, small variations in +luminosity of ∼ 3% persist as the disc accretion reaches a +steady state . We have verified this for the fiducial run lasting +over 9 000 T0. +In Fig. 1 we plot the time-averaged luminosity of an an- +nulus (in units of the Shakura-Sunyaev disc luminosity, LSS), +as a function of disc radius for uncoupled (black line) and +coupled (orange line) models, U2 and C2, respectively. The +shaded region shows the standard deviation in time of the +luminosity for the coupled run. The variability in the outer +regions of the disc persists into late times, albeit becomes very +small. This convergence of the two classes of models at late +times makes comparison of their resulting outflows easier. +MNRAS 000, 1–?? (2022) + +4 +A. Kirilov, S. Dyda, C. S. Reynolds +Model +Type +ρ0 [g cm−3] +LavgMmax [LEdd] +˙Mavg[M⊙ yr−1] +σL/Lavg +σ ˙ +M/ ˙Mavg +∆ ˙Lmax/ ˙Lavg +∆ ˙Mmax/ ˙Mavg +U1 +Uncoupled +8.33 × 10−3 +1.7 +2.05 × 10−13 +0.06 +0.35 +0% +200% +C1 +Coupled +8.33 × 10−3 +1.7 +3.75 × 10−13 +0.06 +0.36 +15% +200% +U2 +Uncoupled +2.00 × 10−2 +4.1 +3.87 × 10−12 +0.06 +0.18 +0% +50% +C2 +Coupled +2.00 × 10−2 +4.1 +4.74 × 10−12 +0.06 +0.28 +15% +100% +U3 +Uncoupled +6.00 × 10−2 +12 +4.20 × 10−11 +0.06 +0.11 +0% +30% +C3 +Coupled +6.00 × 10−2 +12 +4.98 × 10−11 +0.06 +0.17 +15% +100% +U4 +Uncoupled +1.60 × 10−1 +33 +2.57 × 10−10 +0.06 +0.05 +0% +20% +C4 +Coupled +1.60 × 10−1 +33 +2.94 × 10−10 +0.05 +0.17 +15% +100% +Table 1. Summary of disc wind models, parameters, and results, in cgs units unless otherwise specified. The average disc luminosity +Lavg is for coupled runs, averaged between 3000 and 3500 after variations drop to ∼ 1%. The mass flux +˙M is measured as the outflow +between 0 and 75° at 10r⋆. We have listed the standard deviation σX and maximum variations ∆Xmax for mass flux and disc luminosity +over the time period 2200 − 3600T0. +0 +2 +4 +6 +8 +10 +r [r ] +0.0000 +0.0005 +0.0010 +0.0015 +0.0020 +0.0025 +0.0030 +dL [LSS] +COUPLED +UNCOUPLED +Figure 1. Time averaged luminosity of disc annuli dL as a function +of radius r for 3500 ⩽ t/T0 ⩽ 4000 for coupled (orange line) and +uncoupled (black line) models, C2 and U2. The orange shading +shows the standard deviation over this time interval. We see that +variability at larger r are persistent even into late times though +the inner disc reaches a steady state. +3.1 Global Properties +We find global wind solutions in broad agreement with pre- +vious studies of line driven winds (see Fig. 2). A conical +outflow extending ∼ 45◦ above the disc midplane, with the +fastest parts of the flow at 45◦ and the more radial flow be- +ing slower and denser. Further, the wind has small density +structures, extending out from the disc, formed due to failed +winds. These are winds that launch from the disc but fail to +propagate to the outer boundary and fall back to the disc. +To understand the disc-wind system we search for cor- +relations between the disc luminosity and global properties +of the outflows. In Fig. 3 we plot the wind mass flux as a +function of the disc luminosity. Each color corresponds to a +different wind model with each colored point the average for +a 100 T0 epoch for the coupled model. Ellipses indicate the +one standard deviation in time contours for the correspond- +ing uncoupled model. We see that the mass outflow scales +approximately as +˙M ∝ L2, coming from the slope on Fig. 3. +This is in agreement with previous studies for the region of +phase space we explore, which is close to the turnover point +where the dependence steepens. For very low luminosity the +flow can even halt completely (Drew & Proga 2000). The low- +est luminosity run is not included in this fit since the relation +turns over sharply near LdMmax ≳ 1. +The coupled runs exhibit greater variability, as evi- +denced by some epochs lying outside the uncoupled run con- +tours. In particular, note that the epochs ’0’, ’1’, ’2’, which +correspond to the first 300 T0 of each run (after line driv- +ing is initiated), lie outside the contours. Those early times +harbor the largest luminosity peak of each run, so their po- +sitioning outside of the contours makes sense (see Fig. 4). +We will discuss this period in detail later in the paper. In +relative terms, the higher luminosity runs are less inherently +variable due to less small scale structure and a more smooth +flow. In absolute terms however, the higher luminosity runs +exhibit the largest mass flux and consequently the largest +variability due to changes in luminosity. This is also reflected +in Table 1, where we have calculated the normalised standard +deviation over the first 1400 T0 after line driving is initiated. +The normalised standard deviation, hereafter “NSD”, for the +uncoupled runs monotonically decreases with increasing lu- +minosity, which shows that the higher luminosity runs are +less inherently variable. All coupled runs have a higher NSD +than their corresponding uncoupled run. The difference in +NSD between uncoupled and coupled grows with increasing +luminosity. For the highest luminosity models, the NSD dif- +fers from 0.05 to 0.17, while for the lowest luminosity run, the +difference is negligible, 0.35 and 0.36 respectively. We iden- +tify two sources of variability in the mass flux. Intrinsic peaks +are due to the non-stationary nature of line driven winds and +has been well documented in previous studies of line driven +winds PSD98,99. Luminosity peaks can be directly attributed +to a spike in the disc luminosity. The NSD decreases with in- +creasing luminosity due to a decrease in contributions from +intrinsic peaks. For the low luminosity runs variability is due +to both intrinsic and luminosity peaks, hence the coupled and +uncoupled models have similar NSD. At higher luminosities, +where outflows are more steady, outflow variability is dom- +inated by luminosity peaks, hence the coupled models have +∼ 3 times higher NSD than the uncoupled models. As the +flow is less steady for lower luminosity discs, local variations +MNRAS 000, 1–?? (2022) + +Line Driven Winds from Variable Accretion Discs +5 +16 +15 +14 +13 +12 +11 +[g cm 3] +0.0 +0.2 +0.4 +0.6 +0.8 +Z [cm] +1e10 +C2 +t=2200 +v= 3000 km s 1 +C2 +t=4200 +v= 3000 km s 1 +0.0 +0.2 +0.4 +0.6 +0.8 +X [cm] +1e10 +0.0 +0.2 +0.4 +0.6 +0.8 +Z [cm] +1e10 +U2 +t=2200 +v= 3000 km s 1 +0.0 +0.2 +0.4 +0.6 +0.8 +X [cm] +1e10 +U2 +t=4200 +v= 3000 km s 1 +Figure 2. Yellow line at 50 degrees roughly indicating the separation between the fast and slow parts of the flow. The uncoupled run +(U2) starts off ordered and stable (bottom left) but soon instability develops and the flow becomes turbulent (bottom right). In contrast, +the coupled run is more turbulent at early times due to the varying radiation field (top left). The outflow is initially larger in the coupled +run (C2, top left) than the uncoupled run due to periods of enhanced luminosity due to accretion spikes. This causes a “luminosity peak” +in the flow (see Fig. 4, top right). At late times, the two runs are indistinguishable as luminosity variations drop below 3% and any +luminosity peaks are masked by the intrinsic variability of the wind. +in disc intensity that are small relative to the total variability +of disc intensity can alter the flow. The lowest luminosity runs +exhibit a larger average relative change and relative maxi- +mum change in mass outflow due to a luminosity change, +as expected from the scaling in Fig. 3 and reflected in the +σ ˙ +M/ ˙Mavg and ∆ ˙Mmax/ ˙Mavg values in Table 1. This turnover +in the mass flux luminosity curve is particularly sharp near +LdMmax ≳ 1 where small changes in luminosity can result +in winds failing to launch as the radiation force can barely +overcome gravity. +In Fig. 4 we plot the total disc luminosity and wind +mass flux as a function of time for all models. The lower +panel shows the total disc luminosity for the coupled (colored +solid line) or uncoupled (black dashed line). The upper panel +shows the time dependent outflow mass flux (solid line) and +the time averaged value (dashed line) for the coupled (color +line) or uncoupled (black line) model. We calibrated models +by requiring that the late-time averaged luminosity of cou- +pled and uncoupled models are equal. +The coupled cases show clear evidence of correlations +MNRAS 000, 1–?? (2022) + +6 +A. Kirilov, S. Dyda, C. S. Reynolds +100 +101 +102 +2 +4 +6 +8 +2×101 +4×101 +6×101 8×101 +Ld +max [LEdd] +10 +13 +10 +12 +10 +11 +10 +10 +6×10 +14 +8×10 +14 +2×10 +13 +4×10 +13 +6×10 +13 +8×10 +13 +2×10 +12 +4×10 +12 +6×10 +12 +8×10 +12 +2×10 +11 +4×10 +11 +6×10 +11 +8×10 +11 +2×10 +10 +4×10 +10 +6×10 +10 +M [M +yr +1] +01 +2 +0 +1 +2 +0 +1 +2 +01 +2 +linear fit to average values for each uncoupled run, slope is 2.02 +average of uncoupled run +Figure 3. Outflow mass flux +˙M and disc luminosity multiplied by the maximum of the line driving multiplier, LdM max, for all disc +wind models. Each colored point corresponds to the time-average over a 100 T0 epoch from the coupled runs. Ellipses are defined by +the max/min of the uncoupled runs. The disc evolution (captured in luminosity variation) is the same over all the models but there +are multiple extreme points in the mass flux of the coupled runs (points outside the ellipses that have higher luminosity. At early times +(epochs ’0’, ’1’, ’2’ are indicated on the figure) ∼ 15% variations in luminosity result in mass fluxes which deviate from the uncoupled +models. +between the total disc luminosity and the mass flux. A ∼ 15% +variation in luminosity corresponds to a change in mass flux +by a factor of up to ∼ 3. These numbers depend on the to- +tal disc luminosity, with the lowest luminosity run showing +the largest relative fluctuations. However, as discussed pre- +viously, these increases are more readily masked due to the +inherent variability of the lower luminosity models. There- +fore, the correlations are more apparent at higher luminosity +where the uncoupled models have little intrinsic variability. +The NSD difference between coupled and uncoupled models +is a good proxy for how visible the correlations will be, with +larger differences in NSD for stronger correlations between +disc and outflow variability. Furthermore, we expect the spa- +tial location of this luminosity variation to play a role as well. +Luminosity variations interior to a given fluid streamline are +found to have a stronger effect than variations exterior to it +(Dyda & Proga, 2018). In Section 3.2.2 we examine how lo- +cal, as opposed to global, variations in disc luminosity affect +the outflow. +As the disc tends to a steady state at late times vari- +ations in luminosity decrease to ∼ 3% and the mass flux +varies by ∼ 50% in the C2 model. However, the uncoupled +run has very similar variability, despite the radiation field +being constant in time. At late times, variability correlated +to disc luminosity becomes too small to reliably distinguish +from the inherent variability seen in uncoupled models. +MNRAS 000, 1–?? (2022) + +Line Driven Winds from Variable Accretion Discs +7 +0.0 +2.5 +5.0 +7.5 +M [M +/yr] +1e +13 +C1 +U1 +2500 +3000 +3500 +4000 +4500 +5000 +5500 +6000 +t [T0] +1.6 +1.8 +2.0 +LdMmax [LEdd] +0.2 +0.4 +0.6 +0.8 +1.0 +M [M +/yr] +1e +11 +C2 +U2 +2500 +3000 +3500 +4000 +4500 +5000 +t [T0] +3.8 +4.0 +4.2 +4.4 +4.6 +4.8 +LdMmax [LEdd] +0.2 +0.4 +0.6 +0.8 +1.0 +M [M +/yr] +1e +10 +C3 +U3 +2250 +2500 +2750 +3000 +3250 +3500 +3750 +4000 +t [T0] +11 +12 +13 +14 +15 +LdMmax [LEdd] +2 +3 +4 +5 +6 +M [M +/yr] +1e +10 +C4 +U4 +2200 +2400 +2600 +2800 +3000 +3200 +3400 +3600 +t [T0] +32.5 +35.0 +37.5 +LdMmax [LEdd] +Figure 4. Outflow mass flux (upper panel) and disc luminosity (lower panel) for coupled (solid colored line) and uncoupled (dashed +black line), with dashed lines indicating the temporal average. We show results for models 1 (top left), 2 (top right), 3 (bottom left) and +4 (bottom right). Uncoupled models have constant disc luminosity and the corresponding coupled models luminosity converges to this +value at late times. Mass loss for coupled runs is higher initially due to the luminosity increase in the disc luminosity by ∼ 15%. The +luminosity peaks in the beginning are similar in height to the intrinsic peaks for low luminosity runs. Increases in mass outflow of ∼ 3 +can be masked by intrinsic variability for Model 1. Note the second peak in the coupled run, C3, at 3100. We track its origin to a clump +that was most likely ejected by a local spike in the disc intensity, that did not visibly change the total disc luminosity. This peak occurs +only in the coupled run and greatly exceeds the intrinsic variability at that late time. This illustrates how mass flux is affected by both +local variations in disc intensity and total disc luminosity. +We have seen that for lower luminosity runs, the inher- +ent variability can mask mass flux variations due to luminos- +ity increases (as is seen in Fig. 4 for C1 and C2). There are +differences in the shape of the luminosity and intrinsic peaks. +Intrinsic peaks are short and narrow. In contrast, luminosity +peaks are wider, lasting on timescales of the fluctuations in +luminosity. +Finally, we also note that disc structure is largely un- +affected by the wind. This is to be expected due to the disc +being much more dense than the wind. Lower luminosity sys- +tems, on account of their lower mass winds, are expected +to experience the smallest feedback on the disc. We observe +≲ 1% change in disc luminosity across our suite of simula- +tions, suggesting the feedback on the disc is negligible. While +this applies in a straightforward manner to non-magnetic +CVs, suggesting winds from these discs do not impact the +discs themselves, AGN are more complicated due to their +much higher luminosities, the need for ionization treatment +and strong magnetic fields. +3.2 Outflow Structure and Variability +In this section, we characterize the local outflow structure +and how it relates to global outflow properties. We discuss +how the impact of luminosity variations on outflow structure +result in different types of spikes in mass outflow and contrast +them with spikes due to inherent variability. +In Fig. 5 we plot the time averaged velocity (solid line) +and momentum flux (dashed line) as a function of polar angle +along r = 10r⋆ for models C2, (colored lines) and U2 (black +lines). The shading indicates one standard deviation of tem- +poral variability during the epoch. The wind can be divided +into “fast” and “slow” parts, the boundary between the two, +we define as the angle after which the velocity has dropped +below its minimum value along the rotation axis. +The time dependent behaviour and structure of the flow +are best seen in movies of the simulations.1 +3.2.1 Intrinsic peaks +First, we discuss the intrinsic variability of the wind, present +in both uncoupled and coupled runs. The boundary region +between disc atmosphere and wind is particularly interest- +ing. Close to the central object, there are vertical density +structures, ’fingers’, that can either be very ordered or very +turbulent at different times (see Fig 2). These structures, +despite being close to the central object and more tightly +gravitationally bound, are very important in determining the +1 Those +can +be +seen +at: +https://www.youtube.com/playlist? +list=PLdaxJotZLlw3JYw58T2RqUkkhFP1E380Z +MNRAS 000, 1–?? (2022) + +8 +A. Kirilov, S. Dyda, C. S. Reynolds +0 +5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 + [ ] +0 +2000 +4000 +6000 +8000 +10000 +12000 +14000 +vr [km/s] +25 +20 +15 +10 +5 +vr [g cm +2 s +1] +Figure 5. Time averaged momentum flux (dashed line) and radial +velocity (solid line) as a function of polar angle at 10 r⋆ for 3000 ⩽ +t ⩽ 3450 for C2 (orange) and C4 (blue). The shading indicates +standard deviation in time. In the higher luminosity model, the +slow part of the flow is θ ≳ 35° , while for the lower luminosity +θ ≳ 45°. +flow’s variability. Often clumps form within them that are +later either carried successfully to the outer boundary or fall +back to the disc. In the case of the former, this leads to a peak +in the mass loss rate. Clumps only exit in the “slow” part of +the flow as inhomogeneities are smoothed out by the velocity +shear in the “fast” stream. Hence, the slow stream is respon- +sible for most of the variability in the mass flux. Peaks in the +momentum flux profile correspond to either the passage of +a clump or a more global density fluctuation rather than to +variations in the velocity field, which are insignificant. While +the most variable parts in the velocity field are actually at +the smallest angles, those parts of the flow are orders of mag- +nitude less dense and contribute little to outflow variability. +The flow at moderate values of θ is the least variable and +then, once the above defined slow part starts, variability in- +creases once again. Comparing momentum flux and velocity +profiles as shown in Fig. 5 for epochs both with and without +peaks confirms that variations are due to changes in the slow +parts of the flow. These intrinsic peaks occur for both the +coupled and uncoupled runs and have been appreciated by +the community since early disc wind models with static radi- +ation fields. However, variability of disc intensity can further +couple to produce additional variability, as we explain in the +next section. +3.2.2 Luminosity Peaks +Luminosity peaks refer to increases in the outflow rate cor- +related with increases in the total disc luminosity. In all cou- +pled models, we observe that the initial peak in luminos- +ity (∼ 15%) is correlated with an increase in mass outflow. +The flow is more turbulent close to the very inner regions of +the disc (Fig 2). This turbulence aids clump formation from +early times in the coupled runs. In the uncoupled runs, the +flow eventually becomes turbulent but less so as can be seen +in the movies. At later times, when the magnitude of vari- +ations in luminosity drops below 3%, such luminosity peaks +are less significant than that caused by intrinsic variability. +In the higher luminosity runs, variations in luminosity are +more strongly correlated with peaks in mass flux as intrinsic +variability is weaker (see Fig 4). +0 +5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 + [ ] +0 +2000 +4000 +6000 +8000 +vr [km/s] +25 +20 +15 +10 +5 +vr [g cm +2 s +1] +Figure 6. Time averaged momentum flux (dashed line) and ra- +dial velocity (solid line) as a function of polar angle at 10 r⋆ for +2200 ⩽ t ⩽ 2300 for C3 (green) and U3 (black). The shading in- +dicates standard deviation in time. The coupled run experiences a +luminosity peak during those times due to an increase in disc lumi- +nosity of ∼ 15%. The momentum flux differs by up to 1-2 orders of +magnitude from the uncoupled run at certain angles (45° and 65°) +while only the velocity varies slightly in comparison (∼ 10 − 20%) +and not at those angles. Hence, the dominant variability is in den- +sity fluctuations (through clumps), not velocity. +There are two ways in which an increase in luminosity +causes an increase in mass outflow. If the total disc luminosity +increases, there is a global increase in the mass outflow rate +in both the fast and slow streams. (Fig. 6). This happens +on time scales of the dynamical time for gas moving in the +fast stream. On the other hand, local increases in the disc +intensity can affect the slow stream by accelerating clumps +that would have otherwise failed to launch. Such clumps are +responsible for the narrow structures superimposed on the +broad luminosity peaks (see for example Fig. 4, lower right +panel at t = 2300 and t = 2800). The temporal and spatial +variability of the disc can combine to produce spikes in the +outflow due to clumping. As an example, the mass flux spike +in C3 at t = 3100 (Fig. 4) was found to be produced by two +clumps merging due to an increase in disc intensity directly +below one of the clumps. The merging can also be seen in the +C3 movie. This peak looks exactly like an intrinsic peak but +its magnitude is much higher than expected for late times . +Thus, while overall variations in total luminosity are ∼ 3% at +late times, the increase in the local intensity directly below +the clump was ∼ 10% and sufficient to alter the slow stream +(see Fig. 7). This type of behaviour is of course impossible in +the uncoupled runs. +MNRAS 000, 1–?? (2022) + +Line Driven Winds from Variable Accretion Discs +9 +16 +15 +14 +13 +12 +11 +[g cm 3] +0 +2 +4 +6 +8 +Z [cm] +1e9 +C3 +t=2961 +v= 3000 km s 1 +C3 +t=3002 +v= 3000 km s 1 +0 +2 +4 +6 +8 +X [cm] +1e9 +0 +2 +4 +6 +8 +Z [cm] +1e9 +C3 +t=3033 +v= 3000 km s 1 +0 +2 +4 +6 +8 +X [cm] +1e9 +C3 +t=3090 +v= 3000 km s 1 +Figure 7. The propagation of clump aided by local luminosity increase for model C3. In the top left panel, the clump is forming at the +base of the wind, close to the central object. In the top right panel, the clump is now visible just next to the yellow line, roughly halfway +from the central object to the outer boundary. Shortly before that time, there was a 10% increase in the luminosity directly below the +clump that helped the clump to continue moving towards the outer boundary as opposed to falling back to the disc. The traces from this +event are seen in the low density region that has formed below the clump in the top right panel. In the bottom left panel, the clump +is seen propagating towards the outer boundary, with the low density region below it expanding. In the bottom right panel, the clump +is finally seen crossing the outer boundary. This causes an increase in mass outflow at that time (around 3100) that is not seen in the +uncoupled run (see Fig. 4, bottom left panel). This can be better seen in the movies of the simulation, found on our YouTube playlist, in +particualar, the C3/U3 movies. +4 DISCUSSION AND CONCLUSION +We studied line driven disc winds with a time-dependent ra- +diation field, computed self consistently from an accreting +α-disc. The central object was assumed to be much less lu- +minous than the disc and to not contribute to the radiation +field as expected for a non-magnetic CV. However, we expect +some of our basic methods and results to be applicable to +other line driven wind systems such as AGN. +We see no significant feedback of the wind on the +disc. For the Eddington parameters in this work, the ratio +˙Mout/ ˙Macc ≪ 1, hence little angular momentum and energy +is carried away by the wind, relative to the disc. We studied a +different regime to Nomura et al. (2020), where a significant +fraction of the accreted mass was lost to the wind and had +to be accounted for. Furthermore, we do not study how the +wind can impact the disc continuum as done by Nomura et +al. (2020), which can be a source of feedback even for low +luminosity systems and could be a direction for further work. +Another interesting direction for further work in the case of +CVs is to explore whether winds from the accretion disc can +MNRAS 000, 1–?? (2022) + +10 +A. Kirilov, S. Dyda, C. S. Reynolds +impact the accretion stream from the companion star. +The intrinsic variability of the wind mass loss depends +on disc luminosity. For the lowest luminosity systems, where +LdMmax ≳ 1, the intrinsic variability due to failed winds is +comparable to luminosity peaks when total luminosity fluc- +tuations are ∼ 15% as we see at early times in our simu- +lations. The main distinguishing feature between luminos- +ity peaks and intrinsic peaks is their duration. Luminosity +peaks occur on timescales of variability in the disc luminos- +ity (with the exception of peaks caused by clumps aided by +local disc variability), which can be much shorter than the +viscous timescale of the disc. For a disc that is not in steady +state, luminosity can vary on timescales from the shortest +timescale (orbital) to the longest (viscous) (Pringle 1981). +Indeed, we note luminosity variations on a shorter timescale +than the viscous and much longer than orbital. Since our disc +is almost in steady state, the timescale is closer to the for- +mer. Intrinsic peaks occur on time scales for clumps or voids +to be ejected in the slow stream, that is to say a character- +istic fluid crossing time. By contrast, when the luminosity is +high, LdMmax ≫ 1, variability is dominated by luminosity +peaks, provided the disc has not yet reached a steady state +and luminosity still varies by ≳ 10%. At late times, when +the system approaches a steady state (luminosity variations +≲ 3%), correlations of mass loss with luminosity is lost in all +runs as intrinsic variability dominates for all models. +Variability is dominated by the slow part of the flow. +The variability is primarily due to density inhomogenieties, +clumps and voids, which tend to be sheared away in the fast +stream. The outflow’s high degree of clumpiness and turbu- +lence makes it sensitive to local variations in the disc inten- +sity. Often clumps or voids are formed in inner regions that +propagate to the outer boundary and cause a spike in the +slow part of the flow. Clumps can be aided by local disc in- +tensity spikes of ∼ 10%, which do not appreciably alter the +total disc luminosity, thereby creating a degeneracy between +intrinsic and luminosity peaks. Thus, luminosity peaks in the +mass flux can occur even though there is no apparent increase +in the total disc luminosity (see model C3 at t = 3100 in Fig. +4). Intrinsic variability of momentum flux happens through +the slow parts of the flow primarily due to density fluctua- +tions rather than velocity fluctuations. As can be seen in Fig. +6, the intrinsic variations in flow velocity are much lower and +similar for both high and low luminosity runs and uniform +across the wind. We further note that the ∼ 15% changes in +luminosity cause velocity increases in the low velocity flow. +Those are more easily seen for systems with higher total sys- +tem luminosity. We therefore expect variability due to small +(≲ 15%) variations in luminosity to primarily lead to absorp- +tion troughs becoming deeper as opposed to shifts in velocity +space. +Correlating wind activity with total system luminosity +has thus far proven to be observationally challenging (Bal- +man, Godon & Sion 2014). Observations of V592 Cas (Kafka +et al. 2009) have found no correlations between maximum ve- +locity of absorption and the CV’s brightness. Since the system +is viewed at low inclination (low inclination meaning that we +are observing at low θ), only the highest velocity components +would be observable as per the geometry of our model. There- +fore, we expect that ∼ 15% variations in luminosity to not +have a significant effect on the observed maximum velocity of +absorption as seen in Fig. 6) , which is consistent with their +observations of ≲ 10% luminosity variations. Furthermore, +Kafka et al. (2009) propose that the non-axisymmetry of the +flow could be due to a disc hotspot. In our simulations, we +have seen that variations in disc intensity change the geom- +etry of the flow. Hence, our work shows this is a plausible +explanation, because of the evidence of phase modulation of +the flow, but further study, perhaps using a persistent disc +hot spot, as done by Cranmer & Owocki (1996) in the con- +text of stellar winds is needed. We note that they found a +maximum wind velocity of ∼ 5000 km s−1, consistent with +our most luminous models. As discussed, the higher luminos- +ity models show the most likelihood to have correlated disc +and wind variability since the intrinsic variability of the wind +is suppressed. Wind variability is expected to be in the slow +part of the flow, hence the ideal system would be observed at +high inclination. +For systems where we expect our line of sight to lie +along the fast stream (low inclination systems), the best +prospects are to observe transitions from a low to a high lu- +minosity state or vice versa, the equivalent of a transition be- +tween two models in this work. In our simulations, larger vari- +ations in total driving luminosity can be extrapolated from +the approximate scaling +˙M ∝ L2. For such large variations, +one can expect more dramatic changes to the flow geometry. +Depending on inclination, one might even expect to have pe- +riods where no wind is observed. In particular, if we observe +a low inclination system that undergoes a dramatic drop in +driving luminosity, the wind might become more equatorial +and vice versa. For systems close to the LdMmax ≲ 1 thresh- +old such state changes might also turn on/off the wind. In the +system BZ Cam, estimates for inclination range from 12 to +40 degrees (Honeycut, Kafka & Robertson 2013). This means +that per our model, the disc would be observed at low θ, +much like V592 Cas. However, the luminosity variations in +this system are large, meaning that we need to traverse from +one model to another when the system transitions from a +state of lower to a state of higher luminosity. The momen- +tum flux at low inclination (corresponding to low θ in our +model) would increase dramatically but mostly because of +an increase in density (see Fig. 5), not velocity. This would +still lead to an increase in line equivalent width of absorption. +Indeed, such a correlation is seen in BZ Cam (Honeycut et +al. 2013). The geometry and strength of the flow at lower θ +is altered drastically between models (corresponding to large +luminosity variations). At times of low luminosity, the flow +might even stop completely at low θ as can be deduced from +Fig. 5 (traversing from model C4 to C2 would correspond to +a large change in driving luminosity, which leads to flow be- +tween 20 and 30 degrees to cease almost completely). More +studies of BZ Cam like the one done in Honeycut et al. (2013) +are needed to clearly discern correlations between wind ac- +tivity and luminosity. +A further challenge to observing these correlations is +changes in ionization state. As proposed for BZ Cam by +Greiner et al. (2001), changes in ionizing flux from the binary +companion may explain changes in wind emission lines. Such +a model predicts a periodic variability in ionization state set +by the orbital timescale. Later observations by Honeycut et +al. (2013) found variability on shorter timescales, disfavoring +such a model. Further theoretical work should address this +fundamental question of how variations in driving radiation +MNRAS 000, 1–?? (2022) + +Line Driven Winds from Variable Accretion Discs +11 +(see for example Dyda et al. 2018d) can be distinguished from +variations in ionizing flux from the outflow properties. +The ideal system to observe the effects of lower variabil- +ity (≲ 15%), corresponding to the variability within a model +explored in this paper) is a high inclination system with suf- +ficiently strong driving luminosity (so that the intrinsic vari- +ability is suppressed) and high enough accretion variability +but a nearly constant ionizing flux. As can be seen on Fig. 6, +flow at lower θ is indistinguishable during a luminosity peak +while at high θ, both density and velocity increase during a +luminosity peak (mainly density as can be judged from ρvr +changing by roughly an order of magnitude while velocity in- +creases by less than 25% on Fig. 6). Therefore, looking for +correlations between luminosity and line equivalent width of +absorption in high inclination, high luminosity systems seems +to be the most promising avenue for systems with low varia- +tions in luminosity (≲ 15%). Even for such a system, correla- +tions would be difficult to observe. One reason is that while +mass outflow increases overall during a luminosity peak, that +increase is not present in all parts of the flow (see Fig. 6, +e.g. flow at precisely 60 degrees is less during a luminosity +peak even though flow around it increases significantly), so +one might observe at an ’unlucky’ inclination. A more perti- +nent reason is that intrinsic variability can mask peaks due +to luminosity if driving luminosity is not sufficiently high. +For example, Hartley et al. (2002) study two systems at in- +clinations of around 60 degrees, IX Vel and V3885 Sgr. The +equivalent width of absorption for various lines is presented +at three different times, where the continuum flux varies by +≲ 10%. No correlation is found between flux and wind, which +could be due to the luminosity not being high enough so that +the system is in the regime where variations due to luminosity +dominate intrinsic variability. It is also possible that the in- +clination was particularly unlucky but that is unlikely. Since +IX Vel is one of the brightest CV systems, one can reason- +ably ask the question whether a CV with ’sufficiently’ high +luminosity for these correlations to be visible even exists? A +potential candidate system is ASAS J071404+7004.3 which +is at a similar fortunate inclination of 62 degrees and seems +to possess rapidly changing winds (Inight et al. 2022). +While we have chosen parameters more suited to CVs, +some of our results are applicable to AGN. We expect higher +inclination systems would be better for correlating small +changes in luminosity to wind activity. AGN have much +higher luminosities, so the intrinsic variability of the winds +would be suppressed, making variations due to luminosity +more visible. Low inclination systems (but not completely +face on so that jet effects can be ignored) are better for cor- +relating large variations in luminosity due to the changing ge- +ometry of the flow. However, in the regime of high luminosity +variations, inclination is not as important because even sys- +tems viewed at high inclination (high θ) are expected to have +large fluctuations in density of flow due to changing luminos- +ity (see Fig. 5). In the regime of small luminosity variations +(≲ 15%), AGN might be the only systems where correlations +between luminosity and wind can be observed. We have also +seen that local variations of the accretion rate in higher lumi- +nosity runs can ’conspire’ to produce clumps and thus larger +fluctuations in mass outflow than would be expected from the +total luminosity variations (see Fig. 7). Further work, with +overall much higher total luminosity and including ionization +effects, is needed to understand line driven winds in AGN. +This paper motivates the need for more detailed self- +consistent models of line driven winds from accretion discs +to further understand how local variability can non-trivially +influence mass outflow properties. +ACKNOWLEDGEMENTS +SD and CSR acknowledge the UK Science and Technology +Facilities Council (STFC) for support under the New Appli- +cant grant ST/R000867/1 and the European Research Coun- +cil (ERC) for support under the European Union’s Horizon +2020 research and innovation programme (grant 834203). +DATA AVAILABILITY +The data underlying this article will be shared on reasonable +request to the corresponding authors. +REFERENCES +Balman S., Godon P., Sion E.M., 2014, ApJ, 794, 84 +Castor, J.I., Abbott D.C., Klein R.I., 1975, ApJ, 195, 157 +Clarke C., Carswell B., 2007, Principles of Astrophysical +Fluid Dynamics. Cambridge University Press, Cam- +bridge +Cranmer S. R., Owocki S. P., 1996, ApJ, 462, 469C +Drew J. E., Proga D., 2000, New Astron. Rev., 44, 21 +Dyda S., Proga D., 2018, MNRAS, 475, 3786D +Dyda S., Proga D., 2018, MNRAS, 478, 5006D +Dyda S., Proga D., 2018, MNRAS, 481, 2745D +Dyda S., Proga D., 2018, MNRAS, 481, 5263D +Friend D.B., Abbott D.C., 1986, ApJ, 311, 701 +Gayley K. G., 1995, ApJ, 454, 410 +Greiner +J., +Tovmassian +G., +Orio +G., +Lehmann +H., +Chavushyan V., Rau A., Schwarz R., Casalegno R., +Scholz R.D., 2001, A&A, 376, 1031-1038 +Hartley L. E., Drew J. E., Long K. S., MNRAS, 332, 127 +Honeycut R. K., Kafka S., Robertson J.W., 2013, AJ, 145, +45 +Inight K., G¨ansicke B. T., Blondel D., Boyd D., Ashley R. +P., Knigge C., Long K. S., Marsh T. R., McCleery J., +Scaringi S., Steeghs D., Thorstensen J. R., Vanmunster +T., Wheatley P. J., MNRAS, 510, 3605 +Kafka S., Hoard D. W., Honeycutt R. K., Deliyannis C. P., +2009, AJ, 137, 197 +Kurosawa R., Proga D., 2009, MNRAS, 397, 1791K +Liu C., Yuan F., Ostriker J. P., Gan Z., Yang X., 2013, MN- +RAS, 434, 1721L +Lucy L. B., Solomon P. M., 1970, ApJ, 159, 879 +Mosallanezhad A., Yuan F., Ostriker J. P., Zeraatgari F. Z., +Bu D., 2019, MNRAS, 490, 2567M +Murray N., Chiang J., Grossman S.A., Voit G.M., 1995, ApJ, +451, 498 +Nomura M., Ohsuga K., Done C., 2020, MNRAS, 494, 3616N +Owocki S. P., Castor J. I., Rybicki G. B., 1988, ApJ, 335, 914 +Parker E.N., 1958, PhFl, 1, 171P +Pauldrach A., Puls J., Kudritzki R. P., 1986, A&A, 164, 86 +Pereyra N.A., 1997, PhD thesis, Univ. Maryland +Pereyra N.A., Kallman, T.R., Blondin, J.M., 1997, ApJ, 477, +368 +MNRAS 000, 1–?? (2022) + +12 +A. Kirilov, S. Dyda, C. S. Reynolds +Pringle J.E., 1981, ARA&A, 19, 137 +Proga D., Stone J., Drew J.E., 1998, MNRAS, 295, 595 +Proga D., Stone J., Drew J.E., 1999, MNRAS, 310, 476 +Shakura N.I., Sunyaev R.A., 1973, A&A, 24, 337S +Stone J. M., Tomida K., White C. J., Felker K. G., 2020, +ApJS, 249, 4S +Vitello P.A.J., Shlosman I., 1988, ApJ, 327, 680 +APPENDIX A: DERIVATION OF INITIAL +CONDITION +To improve the convergence time for the simulation, we in- +troduced an appropriate initial condition with axisymmetry, +ρ = ρ(R, z) for our disc. Its derivation is as follows. +Assuming hydrostatic equilibrium, in cylindrical coor- +dinates (R, z), a patch of gas must obey +1 +ρ +dP +dz = − +GM +R2 + z2 cos(θ), +(A1) +where θ is the angle between the ˆR-axis and a radial vector +pointing to from the central object to the patch of gas. If +z/R << 1, +− +GMz +R3(1 + (z/R)2)3/2 ≈ −GMz +R3 . +(A2) +We can write this as, +1 +ρ +dP +dz ≈ −Ω2 +Kz, +(A3) +where ΩK = +� +GM +R3 , the Keplerian angular velocity. The +solution is an exponential in the ˆz-direction, assumming +z/R << 1 and P = c2 +sρ with cs constant along ˆz (verti- +cally isothermal). Hence, we can write our density as +ρ = ρ(R)ρ′ +0 exp +� +−z2/2H2� +, +(A4) +where H = cs/ΩK is the scale height. One can then relate +the density to the surface density via integrating along z from +−∞ to +∞, +Σ = +√ +2πHρ(R). +(A5) +The surface density obeys a diffusion equation, assuming a +Keplerian velocity profile. In the steady state, it can be shown +(Clarke & Carswell, 2007) that +νΣ = +˙M +3π +� +1 − +� +R⋆ +R +� +, +(A6) +where R⋆ is the radius of the central object. This reveals the +origin of the formula for I(rd) with rd ≡ R and r⋆ ≡ R⋆. +Recalling ν = αcsH, and assuming isothermal equation of +state (EoS) throughout (in ˆR as well as ˆz), it also means +that we can find ρ(R), which we choose to write as +ρ(R) = ρ′′ +0 +�R⋆ +R +�3 � +1 − +� +R⋆ +R +� +, +(A7) +where we have absorbed all physical constants into ρ′′ +0. Now +we can write ρ(R, z) as +ρ(R, z) = ρ0 +�R⋆ +R +�3 � +1 − +� +R⋆ +R +� +exp +� +−z2/2H2� +, +(A8) +where we ρ0 = ρ′′ +0ρ′ +0. Since we can choose ρ′ +0, we can choose +ρ0. □ +MNRAS 000, 1–?? (2022) + diff --git a/YNE2T4oBgHgl3EQfEAby/content/tmp_files/load_file.txt b/YNE2T4oBgHgl3EQfEAby/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2e0c903243615e5eedcad597bb099ffcba8d6510 --- /dev/null +++ b/YNE2T4oBgHgl3EQfEAby/content/tmp_files/load_file.txt @@ -0,0 +1,731 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf,len=730 +page_content='MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2022) Preprint 11 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0 Line Driven Winds from Variable Accretion Discs Anthony Kirilov∗, Sergei Dyda†, Christopher S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Reynolds Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK 11 January 2023 ABSTRACT We use numerical hydrodynamics simulations to study line driven winds launched from an accreting α−disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Building on previous work where the driving radiation field is static, we compute a time-dependent radiation flux from the local, variable accretion rate of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We find that prior to the establishment of a steady state in the disc, variations of ∼ 15% in disc luminosity correlate with variations of ∼ 2 − 3 in the mass flux of the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' After a steady state is reached, when luminosity variations drop to ∼ 3%, these correlations vanish as the variability in the mass flux is dominated by the intrinsic variability of the winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This is especially evident in lower luminosity runs where intrinsic variability is higher due to a greater prevalence of failed winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The changing mass flux occurs primarily due to the formation of clumps and voids near the disc atmosphere that propagate out into the low velocity part of the flow, a process that can be influenced by local variations in disc intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' By computing the normalised standard deviation of the mass outflow, we show that the impact of luminosity variations on mass outflow is more visible at higher luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' However, the absolute change in mass outflow due to luminosity increases is larger for lower luminosity models due to the luminosity-mass flux scaling relation becoming steeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We further discuss implications for CVs and AGN and observational prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Key words: radiation: dynamics - hydrodynamics - stars: winds, outflows - quasars: general - accretion, accretion discs – novae, cataclysmic variables 1 INTRODUCTION Many accretion disc systems such as cataclysmic variables (CVs), X-ray binaries (XRBs) and active galactic nuclei (AGN) exhibit blue shifted absorption lines, which is inter- preted as evidence of outflowing gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In general, outflows may be accelerated via thermal, radiation or magnetic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' An important question for any system exhibiting outflows is which of the above mechanisms, or combination thereof, acts to launch and accelerate this gas?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In the case of CVs and AGN a strong candidate ac- celeration mechanism is radiation pressure on spectral lines, so called line driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Observationally this is particularly in- teresting because line driving directly couples accretion and outflow energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Matter accreting in the disc converts grav- itational potential energy into photon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' These pho- tons irradiate the gas above the disc and drive an outflow by transferring momentum from the radiation field to the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Drawing intuition from spherically symmetric models of line driven winds (Castor, Abbot & Klein 1975, hereafter CAK75), for isothermal winds with a fixed chemical abun- dance, the strength of the outflow (mass flux and outflow velocity) should depend strongly on the driving luminosity and not so much on the density at the base of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This is in contrast to thermally driven winds (Parker 1958), ∗antoniikirilov@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='com †sdyda@ast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='uk where for an isothermal wind, the mass flux has a steep de- pendence on density at the base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Though only approximately correct in non-spherical ge- ometries, the correlation between accretion energy and out- flow properties for line driven winds can be exploited to con- strain relevant physical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In many systems we ob- serve both the total system luminosity and discern proper- ties of the outflowing gas by measuring spectra and using photoionization modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Self-consistent treatment of such a system therefore requires simulating both the physics in the accretion disc and the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Originally, line driving, operating in the Sobolev ap- proximation, was proposed as a driving mechanism for stellar winds in OB stars (Lucy & Solomon 1970, hereafter LS70 & CAK75) and was successful in predicting their mass loss rates and outflow velocities (Pauldrach, Puls & Kudritzki 1986, Friend & Abbott 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Line driving has since been applied to a variety of systems, in particular those with accretion discs such as CVs and AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Early work studied line driven disc winds using a va- riety of simplifying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Early analytical studies assumed simple scaling for the strength of the radiation field (Vitello & Shlosman 1988) or a decoupling of the ra- dial and polar angle equations of motion (Murray et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' al 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Later simulations found stationary outflow solutions (Pereyra, Kallman & Blondin (1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Pereyra (1997)), but these were too coarse to properly resolve the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This transition was resolved using non-uniform meshes near © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='03632v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='HE] 9 Jan 2023 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Kirilov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Dyda, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Reynolds the disc midplane (Proga, Stone & Drew 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 1999) which found that as in the CAK picture the strength of the out- flow was correlated with total system luminosity but the non- spherical geometry led to these flows being unsteady for low driving luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Later work in 3D showed that relaxing the axisymmetry assumption leads to the formation of clumps (Dyda & Proga 2018a,b hereafter DP18a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The geometry of the radiation field can also affect the strength of the outflow by altering the flow geometry (Dyda & Proga 2018c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In this prior work the disc served as a matter reservoir and winds were driven by a time-independent radiation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Some later studies sought to loosen this assumption on the radiation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Kurosowa & Proga (2009) used the variable mass at the inner boundary to estimate the accretion rate and corresponding disc luminosity and found strong corre- lations between these and the wind outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' These models were extended to include reprocessed and scattered photons, which were found to further enhance the strength of the out- flow (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Mosallanezhad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Nomura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2020) computed disc luminosity by accounting for mass losses from the wind and found these losses alter the disc UV continuum responsible for much of the line driving flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We build on prior models by allowing for a time- dependent radiation field, computed self consistently from the accretion in the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We simulate a thin disc where viscous dissipation is computed using an α-prescription (Shakura & Sunyaev 1973) and the local disc intensity is computed from the local accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The luminous accretion disc drives an outflow via radiation pressure on spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We carry out a series of simulations for different time-averaged disc luminosities and characterize the properties of the resulting outflows and their correlation with disc properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The structure of our paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In Section 2 we describe our numerical methods, including how the disc in- tensity is computed from the accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In Section 3 we describe the results of varying accretion rate on the time- averaged and time-dependent outflow properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We con- clude in Section 4 where we discuss the implications for CVs and AGN winds and observational prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 2 NUMERICAL METHODS We performed all numerical simulations with the publicly available MHD code Athena++ (Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The ba- sic physical setup is a gravitating, central object surrounded by a thin, axisymmetric, luminous α-disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The accretion disc acts as a source of driving radiation, accelerating the gas that is assumed to be optically thin to the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Our radi- ation model is described in detail in DP18a,b, but we sketch the basic setup here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='1 Basic Equations The basic equations for single fluid hydrodynamics driven by a radiation field are ∂ρ ∂t + ∇ · (ρv) = 0, (1a) ∂(ρv) ∂t + ∇ · � ρvv + P + τ � = −ρ∇Φ + ρFrad, (1b) ∂E ∂t + ∇ · � (E + P)v + (τ · v) � = −ρv · ∇Φ + ρv · Frad, (1c) where ρ, v are the fluid density and velocity respectively, P is a diagonal tensor with components P the gas pressure, τ is the viscosity tensor and Frad is the radiation force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' For the gravitational potential, we use Φ = −GM/r and E = 1/2ρ|v|2 + E is the total energy where E = P/(γ − 1) is the internal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The isothermal sound speed is a2 = P/ρ and the adiabatic sound speed c2 s = γa2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We use a nearly isothermal equation of state P = kργ with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The temperature is then T = (γ − 1)Eµmp/ρkb where µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='6 is the mean molecular weight and other symbols have their standard meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We model the viscosity using the Shakura-Sunyaev α−disc prescription, where the kinematic viscosity is given by ν = αν c2 s ΩK , (2) for dimensionless parameter αν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='01 and ΩK = � GM/r3 the Keplerian orbital frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='2 Radiation Force and Accretion Disc We assume a time-dependent radiation field, computed from the local accretion rate of the axisymmetric, thin disc along the midplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The frequency integrated intensity of a thin disc is I(rd) = 3 π GM r2⋆ c σe Γd �r⋆ rd �3 � 1 − �r⋆ rd �1/2� , (3) where Γd = ˙Maccσe 8πcr⋆ (4) is the disc Eddington number, ˙Macc is the accretion rate in the disc (see for example Pringle 1981), σe is the Thompson cross section, rd is the radial position on the disc, and r⋆ the inner radius of the disc and c the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The intensity profile (3) assumes a time independent accretion rate throughout the disc and therefore a constant Eddington fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In DP18a,b we assumed a fixed ˙Macc but now we com- pute it within the simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We break up the disc into rings and compute a local accretion rate ˙Macc(r, t) = 4π ˆ π/2 θd ρvrr2 sin θdθ, (5) where θd is at the surface of the disc as defined by a density floor ρd = 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We substitute this accretion rate into (3) to compute the local disc intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This is an approximation, since we use the analytic expression for the local disc intensity that was derived assuming a constant accretion rate for the global disc, not just a local patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' An alternative approach would be to compute the local intensity from the viscous dis- sipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We chose our approach to simplify the comparison between coupled and uncoupled models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The radiation force is then computed by assuming the gas is optically thin to MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2022) Line Driven Winds from Variable Accretion Discs 3 this radiation field and every point in the wind experiences a radiation force Frad = Frad e + Frad L , (6) which is a sum of the contributions due to electron scattering Frad e and line driving Frad L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In this continuum, optically thin approximation, the radiation force due to electron scattering is Frad e = " � nσeIdΩ c � , (7) where σe is the electron scattering cross section, n is the normal vector from the radiating surface to the point in the wind, dΩ is the solid angle and the integration is carried out over the entire disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We treat the radiation due to lines using a modification of the CAK formulation where the radiation force due to lines is Frad L = " M(t) � nσeIdΩ c � , (8) and M(t) is the so-called force multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We use the Owocki, Castor & Rybicki (1984) parametrization of the line strength, where working in the Sobolev approximation, M(t) = kt−α �(1 + τmax)1−α − 1 τ 1−α max � , (9) where k and α are constants, τmax = tηmax, ηmax is related to the maximum force multiplier via Mmax = k(1 − α)ηα max and the optical depth parameter t = σeρvth |dvl/dl|, (10) where vth is the thermal velocity of the gas and dvl/dl is the velocity gradient along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We take k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='2, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='6 and Mmax = 4400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Additional details about our numerical treatment of the radiation force can be found in the Appendix of DP18a and DP18b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='3 Simulation Setup The initial conditions consists of a disc in hydrostatic equi- librium with density profile ρ = ρ0 �r⋆ rd �3 � 1 − �r⋆ rd � exp � −z2/2h2� , (11) with rd the radial position on the disc, and the scale-height h = cs/ΩK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The midplane density parameter 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='33 × 10−3 ⩽ ρ0 ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='6 × 10−1 (in g cm−3) is chosen so the accretion rate at late times produces the same disc luminosity as our un- coupled disc runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' For reference, the lowest luminosity run has accretion rate of ∼ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='8 × 10−9M⊙ yr−1 at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The angular velocity is initialized to its Keplerian value, ΩK = � GM/r3 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The radial computational domain extends over the range r⋆ ⩽ r ⩽ 16r⋆ with logarithmic spacing between grids ∆ri+1/∆ri = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The polar angle range is 0 ⩽ θ ⩽ π/2 and has logarithmic spacing ∆θi+1/∆θi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='95 that ensures that we have sufficient resolution near the disc midplane to resolve disc accretion and wind acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We use a grid resolution of nr×nθ=96×96 cells and Nr×Nθ=3×3=9 MPI meshblocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We impose outflow boundary conditions at the inner and outer radial boundaries, axis boundary conditions along the θ = 0 axis and reflecting conditions about the θ = π/2 midplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We chose parameters characteristic of a CV system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The central object has mass and radius M=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='6 M⊙ and r⋆ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='7 × 108 cm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The orbital period at the inner radius is then T0 = 18 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The ratio between the gas thermal energy and gravitational potential energy, some- times referred to as the hydrodynamic escape parameter, HEP = GM/r⋆c2 s = 8 × 103 at the base of the wind, cor- responding to a sound speed cs ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='4 × 106 cm s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' For this value of HEP, thermal driving, which requires HEP ≲ 10 (Stone & Proga 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Dyda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 2017), is negligible through- out the domain but it is not too high that we cannot resolve the disc accretion with our resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The density through- out the domain outside the disc is set to ρ = 10−20 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We impose a density floor of 10−22 g cm−3 and a pressure floor computed via our nearly isothermal, adiabatic equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 3 RESULTS We consider two classes of models for line driven disc winds, where the disc intensity is computed from the accretion rate using (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In uncoupled models the accretion rate is assumed to be constant whereas coupled models compute the disc in- tensity from the local, time-dependent accretion rate (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We study how the system behaves for four different average lu- minosities in both the uncoupled and coupled regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We control the luminosity, via the accretion rate, by varying the disc midplane density parameter, ρ0, in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The parame- ters of the uncoupled runs were chosen to match the average accretion rate of coupled models at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We summa- rize our list of models in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The uncoupled models are identical in setup to those in DP18b and used to benchmark the coupled models, which are the novel aspect of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Each run proceeds in two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' First, the disc evolves for 2000 inner disc orbits T0, with the radiation force turned off, to reach a quasi-steady accretion rate where variations in luminosity are low (≲ 15%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This ensures we can still explore variability while the equation (3), derived for a steady state disc can be used to approximate the resulting intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Over the next 100 T0 the radiation force is turned on using a linear ramp up and we study the resulting disc-wind system for the following 1400 T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' After this time, small variations in luminosity of ∼ 3% persist as the disc accretion reaches a steady state .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We have verified this for the fiducial run lasting over 9 000 T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 1 we plot the time-averaged luminosity of an an- nulus (in units of the Shakura-Sunyaev disc luminosity, LSS), as a function of disc radius for uncoupled (black line) and coupled (orange line) models, U2 and C2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The shaded region shows the standard deviation in time of the luminosity for the coupled run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The variability in the outer regions of the disc persists into late times, albeit becomes very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This convergence of the two classes of models at late times makes comparison of their resulting outflows easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2022) 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Kirilov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Dyda, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Reynolds Model Type ρ0 [g cm−3] LavgMmax [LEdd] ˙Mavg[M⊙ yr−1] σL/Lavg σ ˙ M/ ˙Mavg ∆ ˙Lmax/ ˙Lavg ∆ ˙Mmax/ ˙Mavg U1 Uncoupled 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='33 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='05 × 10−13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='35 0% 200% C1 Coupled 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='33 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='75 × 10−13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='36 15% 200% U2 Uncoupled 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='00 × 10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='87 × 10−12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='18 0% 50% C2 Coupled 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='00 × 10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='74 × 10−12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='28 15% 100% U3 Uncoupled 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='00 × 10−2 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='20 × 10−11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='11 0% 30% C3 Coupled 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='00 × 10−2 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='98 × 10−11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='17 15% 100% U4 Uncoupled 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='60 × 10−1 33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='57 × 10−10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='05 0% 20% C4 Coupled 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='60 × 10−1 33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='94 × 10−10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='17 15% 100% Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Summary of disc wind models, parameters, and results, in cgs units unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The average disc luminosity Lavg is for coupled runs, averaged between 3000 and 3500 after variations drop to ∼ 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The mass flux ˙M is measured as the outflow between 0 and 75° at 10r⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We have listed the standard deviation σX and maximum variations ∆Xmax for mass flux and disc luminosity over the time period 2200 − 3600T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 0 2 4 6 8 10 r [r ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0030 dL [LSS] COUPLED UNCOUPLED Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Time averaged luminosity of disc annuli dL as a function of radius r for 3500 ⩽ t/T0 ⩽ 4000 for coupled (orange line) and uncoupled (black line) models, C2 and U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The orange shading shows the standard deviation over this time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We see that variability at larger r are persistent even into late times though the inner disc reaches a steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='1 Global Properties We find global wind solutions in broad agreement with pre- vious studies of line driven winds (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' A conical outflow extending ∼ 45◦ above the disc midplane, with the fastest parts of the flow at 45◦ and the more radial flow be- ing slower and denser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Further, the wind has small density structures, extending out from the disc, formed due to failed winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' These are winds that launch from the disc but fail to propagate to the outer boundary and fall back to the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' To understand the disc-wind system we search for cor- relations between the disc luminosity and global properties of the outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 3 we plot the wind mass flux as a function of the disc luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Each color corresponds to a different wind model with each colored point the average for a 100 T0 epoch for the coupled model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Ellipses indicate the one standard deviation in time contours for the correspond- ing uncoupled model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We see that the mass outflow scales approximately as ˙M ∝ L2, coming from the slope on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This is in agreement with previous studies for the region of phase space we explore, which is close to the turnover point where the dependence steepens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' For very low luminosity the flow can even halt completely (Drew & Proga 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The low- est luminosity run is not included in this fit since the relation turns over sharply near LdMmax ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The coupled runs exhibit greater variability, as evi- denced by some epochs lying outside the uncoupled run con- tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In particular, note that the epochs ’0’, ’1’, ’2’, which correspond to the first 300 T0 of each run (after line driv- ing is initiated), lie outside the contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Those early times harbor the largest luminosity peak of each run, so their po- sitioning outside of the contours makes sense (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We will discuss this period in detail later in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In relative terms, the higher luminosity runs are less inherently variable due to less small scale structure and a more smooth flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In absolute terms however, the higher luminosity runs exhibit the largest mass flux and consequently the largest variability due to changes in luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This is also reflected in Table 1, where we have calculated the normalised standard deviation over the first 1400 T0 after line driving is initiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The normalised standard deviation, hereafter “NSD”, for the uncoupled runs monotonically decreases with increasing lu- minosity, which shows that the higher luminosity runs are less inherently variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' All coupled runs have a higher NSD than their corresponding uncoupled run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The difference in NSD between uncoupled and coupled grows with increasing luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' For the highest luminosity models, the NSD dif- fers from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='05 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='17, while for the lowest luminosity run, the difference is negligible, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='35 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='36 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We iden- tify two sources of variability in the mass flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Intrinsic peaks are due to the non-stationary nature of line driven winds and has been well documented in previous studies of line driven winds PSD98,99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Luminosity peaks can be directly attributed to a spike in the disc luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The NSD decreases with in- creasing luminosity due to a decrease in contributions from intrinsic peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' For the low luminosity runs variability is due to both intrinsic and luminosity peaks, hence the coupled and uncoupled models have similar NSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' At higher luminosities, where outflows are more steady, outflow variability is dom- inated by luminosity peaks, hence the coupled models have ∼ 3 times higher NSD than the uncoupled models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' As the flow is less steady for lower luminosity discs, local variations MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2022) Line Driven Winds from Variable Accretion Discs 5 16 15 14 13 12 11 [g cm 3] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='8 Z [cm] 1e10 C2 t=2200 v= 3000 km s 1 C2 t=4200 v= 3000 km s 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='8 X [cm] 1e10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='8 Z [cm] 1e10 U2 t=2200 v= 3000 km s 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='8 X [cm] 1e10 U2 t=4200 v= 3000 km s 1 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Yellow line at 50 degrees roughly indicating the separation between the fast and slow parts of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The uncoupled run (U2) starts off ordered and stable (bottom left) but soon instability develops and the flow becomes turbulent (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In contrast, the coupled run is more turbulent at early times due to the varying radiation field (top left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The outflow is initially larger in the coupled run (C2, top left) than the uncoupled run due to periods of enhanced luminosity due to accretion spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This causes a “luminosity peak” in the flow (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 4, top right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' At late times, the two runs are indistinguishable as luminosity variations drop below 3% and any luminosity peaks are masked by the intrinsic variability of the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' in disc intensity that are small relative to the total variability of disc intensity can alter the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The lowest luminosity runs exhibit a larger average relative change and relative maxi- mum change in mass outflow due to a luminosity change, as expected from the scaling in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 3 and reflected in the σ ˙ M/ ˙Mavg and ∆ ˙Mmax/ ˙Mavg values in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This turnover in the mass flux luminosity curve is particularly sharp near LdMmax ≳ 1 where small changes in luminosity can result in winds failing to launch as the radiation force can barely overcome gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 4 we plot the total disc luminosity and wind mass flux as a function of time for all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The lower panel shows the total disc luminosity for the coupled (colored solid line) or uncoupled (black dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The upper panel shows the time dependent outflow mass flux (solid line) and the time averaged value (dashed line) for the coupled (color line) or uncoupled (black line) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We calibrated models by requiring that the late-time averaged luminosity of cou- pled and uncoupled models are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The coupled cases show clear evidence of correlations MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2022) 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Kirilov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Dyda, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Reynolds 100 101 102 2 4 6 8 2×101 4×101 6×101 8×101 Ld max [LEdd] 10 13 10 12 10 11 10 10 6×10 14 8×10 14 2×10 13 4×10 13 6×10 13 8×10 13 2×10 12 4×10 12 6×10 12 8×10 12 2×10 11 4×10 11 6×10 11 8×10 11 2×10 10 4×10 10 6×10 10 M [M yr 1] 01 2 0 1 2 0 1 2 01 2 linear fit to average values for each uncoupled run, slope is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='02 average of uncoupled run Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Outflow mass flux ˙M and disc luminosity multiplied by the maximum of the line driving multiplier, LdM max, for all disc wind models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Each colored point corresponds to the time-average over a 100 T0 epoch from the coupled runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Ellipses are defined by the max/min of the uncoupled runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The disc evolution (captured in luminosity variation) is the same over all the models but there are multiple extreme points in the mass flux of the coupled runs (points outside the ellipses that have higher luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' At early times (epochs ’0’, ’1’, ’2’ are indicated on the figure) ∼ 15% variations in luminosity result in mass fluxes which deviate from the uncoupled models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' between the total disc luminosity and the mass flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' A ∼ 15% variation in luminosity corresponds to a change in mass flux by a factor of up to ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' These numbers depend on the to- tal disc luminosity, with the lowest luminosity run showing the largest relative fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' However, as discussed pre- viously, these increases are more readily masked due to the inherent variability of the lower luminosity models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' There- fore, the correlations are more apparent at higher luminosity where the uncoupled models have little intrinsic variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The NSD difference between coupled and uncoupled models is a good proxy for how visible the correlations will be, with larger differences in NSD for stronger correlations between disc and outflow variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Furthermore, we expect the spa- tial location of this luminosity variation to play a role as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Luminosity variations interior to a given fluid streamline are found to have a stronger effect than variations exterior to it (Dyda & Proga, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='2 we examine how lo- cal, as opposed to global, variations in disc luminosity affect the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' As the disc tends to a steady state at late times vari- ations in luminosity decrease to ∼ 3% and the mass flux varies by ∼ 50% in the C2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' However, the uncoupled run has very similar variability, despite the radiation field being constant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' At late times, variability correlated to disc luminosity becomes too small to reliably distinguish from the inherent variability seen in uncoupled models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2022) Line Driven Winds from Variable Accretion Discs 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='5 M [M /yr] 1e 13 C1 U1 2500 3000 3500 4000 4500 5000 5500 6000 t [T0] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0 LdMmax [LEdd] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0 M [M /yr] 1e 11 C2 U2 2500 3000 3500 4000 4500 5000 t [T0] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='8 LdMmax [LEdd] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0 M [M /yr] 1e 10 C3 U3 2250 2500 2750 3000 3250 3500 3750 4000 t [T0] 11 12 13 14 15 LdMmax [LEdd] 2 3 4 5 6 M [M /yr] 1e 10 C4 U4 2200 2400 2600 2800 3000 3200 3400 3600 t [T0] 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='5 LdMmax [LEdd] Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Outflow mass flux (upper panel) and disc luminosity (lower panel) for coupled (solid colored line) and uncoupled (dashed black line), with dashed lines indicating the temporal average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We show results for models 1 (top left), 2 (top right), 3 (bottom left) and 4 (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Uncoupled models have constant disc luminosity and the corresponding coupled models luminosity converges to this value at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Mass loss for coupled runs is higher initially due to the luminosity increase in the disc luminosity by ∼ 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The luminosity peaks in the beginning are similar in height to the intrinsic peaks for low luminosity runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Increases in mass outflow of ∼ 3 can be masked by intrinsic variability for Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Note the second peak in the coupled run, C3, at 3100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We track its origin to a clump that was most likely ejected by a local spike in the disc intensity, that did not visibly change the total disc luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This peak occurs only in the coupled run and greatly exceeds the intrinsic variability at that late time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This illustrates how mass flux is affected by both local variations in disc intensity and total disc luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We have seen that for lower luminosity runs, the inher- ent variability can mask mass flux variations due to luminos- ity increases (as is seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 4 for C1 and C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' There are differences in the shape of the luminosity and intrinsic peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Intrinsic peaks are short and narrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In contrast, luminosity peaks are wider, lasting on timescales of the fluctuations in luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Finally, we also note that disc structure is largely un- affected by the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This is to be expected due to the disc being much more dense than the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Lower luminosity sys- tems, on account of their lower mass winds, are expected to experience the smallest feedback on the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We observe ≲ 1% change in disc luminosity across our suite of simula- tions, suggesting the feedback on the disc is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' While this applies in a straightforward manner to non-magnetic CVs, suggesting winds from these discs do not impact the discs themselves, AGN are more complicated due to their much higher luminosities, the need for ionization treatment and strong magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='2 Outflow Structure and Variability In this section, we characterize the local outflow structure and how it relates to global outflow properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We discuss how the impact of luminosity variations on outflow structure result in different types of spikes in mass outflow and contrast them with spikes due to inherent variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 5 we plot the time averaged velocity (solid line) and momentum flux (dashed line) as a function of polar angle along r = 10r⋆ for models C2, (colored lines) and U2 (black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The shading indicates one standard deviation of tem- poral variability during the epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The wind can be divided into “fast” and “slow” parts, the boundary between the two, we define as the angle after which the velocity has dropped below its minimum value along the rotation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The time dependent behaviour and structure of the flow are best seen in movies of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='1 Intrinsic peaks First, we discuss the intrinsic variability of the wind, present in both uncoupled and coupled runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The boundary region between disc atmosphere and wind is particularly interest- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Close to the central object, there are vertical density structures, ’fingers’, that can either be very ordered or very turbulent at different times (see Fig 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' These structures, despite being close to the central object and more tightly gravitationally bound, are very important in determining the 1 Those can be seen at: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='com/playlist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' list=PLdaxJotZLlw3JYw58T2RqUkkhFP1E380Z MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2022) 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Kirilov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Dyda, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Reynolds 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 [ ] 0 2000 4000 6000 8000 10000 12000 14000 vr [km/s] 25 20 15 10 5 vr [g cm 2 s 1] Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Time averaged momentum flux (dashed line) and radial velocity (solid line) as a function of polar angle at 10 r⋆ for 3000 ⩽ t ⩽ 3450 for C2 (orange) and C4 (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The shading indicates standard deviation in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In the higher luminosity model, the slow part of the flow is θ ≳ 35° , while for the lower luminosity θ ≳ 45°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' flow’s variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Often clumps form within them that are later either carried successfully to the outer boundary or fall back to the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In the case of the former, this leads to a peak in the mass loss rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Clumps only exit in the “slow” part of the flow as inhomogeneities are smoothed out by the velocity shear in the “fast” stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Hence, the slow stream is respon- sible for most of the variability in the mass flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Peaks in the momentum flux profile correspond to either the passage of a clump or a more global density fluctuation rather than to variations in the velocity field, which are insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' While the most variable parts in the velocity field are actually at the smallest angles, those parts of the flow are orders of mag- nitude less dense and contribute little to outflow variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The flow at moderate values of θ is the least variable and then, once the above defined slow part starts, variability in- creases once again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Comparing momentum flux and velocity profiles as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 5 for epochs both with and without peaks confirms that variations are due to changes in the slow parts of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' These intrinsic peaks occur for both the coupled and uncoupled runs and have been appreciated by the community since early disc wind models with static radi- ation fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' However, variability of disc intensity can further couple to produce additional variability, as we explain in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='2 Luminosity Peaks Luminosity peaks refer to increases in the outflow rate cor- related with increases in the total disc luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In all cou- pled models, we observe that the initial peak in luminos- ity (∼ 15%) is correlated with an increase in mass outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The flow is more turbulent close to the very inner regions of the disc (Fig 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This turbulence aids clump formation from early times in the coupled runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In the uncoupled runs, the flow eventually becomes turbulent but less so as can be seen in the movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' At later times, when the magnitude of vari- ations in luminosity drops below 3%, such luminosity peaks are less significant than that caused by intrinsic variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In the higher luminosity runs, variations in luminosity are more strongly correlated with peaks in mass flux as intrinsic variability is weaker (see Fig 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 [ ] 0 2000 4000 6000 8000 vr [km/s] 25 20 15 10 5 vr [g cm 2 s 1] Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Time averaged momentum flux (dashed line) and ra- dial velocity (solid line) as a function of polar angle at 10 r⋆ for 2200 ⩽ t ⩽ 2300 for C3 (green) and U3 (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The shading in- dicates standard deviation in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The coupled run experiences a luminosity peak during those times due to an increase in disc lumi- nosity of ∼ 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The momentum flux differs by up to 1-2 orders of magnitude from the uncoupled run at certain angles (45° and 65°) while only the velocity varies slightly in comparison (∼ 10 − 20%) and not at those angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Hence, the dominant variability is in den- sity fluctuations (through clumps), not velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' There are two ways in which an increase in luminosity causes an increase in mass outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' If the total disc luminosity increases, there is a global increase in the mass outflow rate in both the fast and slow streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This happens on time scales of the dynamical time for gas moving in the fast stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' On the other hand, local increases in the disc intensity can affect the slow stream by accelerating clumps that would have otherwise failed to launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Such clumps are responsible for the narrow structures superimposed on the broad luminosity peaks (see for example Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 4, lower right panel at t = 2300 and t = 2800).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The temporal and spatial variability of the disc can combine to produce spikes in the outflow due to clumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' As an example, the mass flux spike in C3 at t = 3100 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 4) was found to be produced by two clumps merging due to an increase in disc intensity directly below one of the clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The merging can also be seen in the C3 movie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This peak looks exactly like an intrinsic peak but its magnitude is much higher than expected for late times .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Thus, while overall variations in total luminosity are ∼ 3% at late times, the increase in the local intensity directly below the clump was ∼ 10% and sufficient to alter the slow stream (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This type of behaviour is of course impossible in the uncoupled runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2022) Line Driven Winds from Variable Accretion Discs 9 16 15 14 13 12 11 [g cm 3] 0 2 4 6 8 Z [cm] 1e9 C3 t=2961 v= 3000 km s 1 C3 t=3002 v= 3000 km s 1 0 2 4 6 8 X [cm] 1e9 0 2 4 6 8 Z [cm] 1e9 C3 t=3033 v= 3000 km s 1 0 2 4 6 8 X [cm] 1e9 C3 t=3090 v= 3000 km s 1 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The propagation of clump aided by local luminosity increase for model C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In the top left panel, the clump is forming at the base of the wind, close to the central object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In the top right panel, the clump is now visible just next to the yellow line, roughly halfway from the central object to the outer boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Shortly before that time, there was a 10% increase in the luminosity directly below the clump that helped the clump to continue moving towards the outer boundary as opposed to falling back to the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The traces from this event are seen in the low density region that has formed below the clump in the top right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In the bottom left panel, the clump is seen propagating towards the outer boundary, with the low density region below it expanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In the bottom right panel, the clump is finally seen crossing the outer boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This causes an increase in mass outflow at that time (around 3100) that is not seen in the uncoupled run (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 4, bottom left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This can be better seen in the movies of the simulation, found on our YouTube playlist, in particualar, the C3/U3 movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 4 DISCUSSION AND CONCLUSION We studied line driven disc winds with a time-dependent ra- diation field, computed self consistently from an accreting α-disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The central object was assumed to be much less lu- minous than the disc and to not contribute to the radiation field as expected for a non-magnetic CV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' However, we expect some of our basic methods and results to be applicable to other line driven wind systems such as AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We see no significant feedback of the wind on the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' For the Eddington parameters in this work, the ratio ˙Mout/ ˙Macc ≪ 1, hence little angular momentum and energy is carried away by the wind, relative to the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We studied a different regime to Nomura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2020), where a significant fraction of the accreted mass was lost to the wind and had to be accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Furthermore, we do not study how the wind can impact the disc continuum as done by Nomura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2020), which can be a source of feedback even for low luminosity systems and could be a direction for further work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Another interesting direction for further work in the case of CVs is to explore whether winds from the accretion disc can MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2022) 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Kirilov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Dyda, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Reynolds impact the accretion stream from the companion star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The intrinsic variability of the wind mass loss depends on disc luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' For the lowest luminosity systems, where LdMmax ≳ 1, the intrinsic variability due to failed winds is comparable to luminosity peaks when total luminosity fluc- tuations are ∼ 15% as we see at early times in our simu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The main distinguishing feature between luminos- ity peaks and intrinsic peaks is their duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Luminosity peaks occur on timescales of variability in the disc luminos- ity (with the exception of peaks caused by clumps aided by local disc variability), which can be much shorter than the viscous timescale of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' For a disc that is not in steady state, luminosity can vary on timescales from the shortest timescale (orbital) to the longest (viscous) (Pringle 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Indeed, we note luminosity variations on a shorter timescale than the viscous and much longer than orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Since our disc is almost in steady state, the timescale is closer to the for- mer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Intrinsic peaks occur on time scales for clumps or voids to be ejected in the slow stream, that is to say a character- istic fluid crossing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' By contrast, when the luminosity is high, LdMmax ≫ 1, variability is dominated by luminosity peaks, provided the disc has not yet reached a steady state and luminosity still varies by ≳ 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' At late times, when the system approaches a steady state (luminosity variations ≲ 3%), correlations of mass loss with luminosity is lost in all runs as intrinsic variability dominates for all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Variability is dominated by the slow part of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The variability is primarily due to density inhomogenieties, clumps and voids, which tend to be sheared away in the fast stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The outflow’s high degree of clumpiness and turbu- lence makes it sensitive to local variations in the disc inten- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Often clumps or voids are formed in inner regions that propagate to the outer boundary and cause a spike in the slow part of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Clumps can be aided by local disc in- tensity spikes of ∼ 10%, which do not appreciably alter the total disc luminosity, thereby creating a degeneracy between intrinsic and luminosity peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Thus, luminosity peaks in the mass flux can occur even though there is no apparent increase in the total disc luminosity (see model C3 at t = 3100 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Intrinsic variability of momentum flux happens through the slow parts of the flow primarily due to density fluctua- tions rather than velocity fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 6, the intrinsic variations in flow velocity are much lower and similar for both high and low luminosity runs and uniform across the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We further note that the ∼ 15% changes in luminosity cause velocity increases in the low velocity flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Those are more easily seen for systems with higher total sys- tem luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We therefore expect variability due to small (≲ 15%) variations in luminosity to primarily lead to absorp- tion troughs becoming deeper as opposed to shifts in velocity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Correlating wind activity with total system luminosity has thus far proven to be observationally challenging (Bal- man, Godon & Sion 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Observations of V592 Cas (Kafka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 2009) have found no correlations between maximum ve- locity of absorption and the CV’s brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Since the system is viewed at low inclination (low inclination meaning that we are observing at low θ), only the highest velocity components would be observable as per the geometry of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' There- fore, we expect that ∼ 15% variations in luminosity to not have a significant effect on the observed maximum velocity of absorption as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 6) , which is consistent with their observations of ≲ 10% luminosity variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Furthermore, Kafka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2009) propose that the non-axisymmetry of the flow could be due to a disc hotspot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In our simulations, we have seen that variations in disc intensity change the geom- etry of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Hence, our work shows this is a plausible explanation, because of the evidence of phase modulation of the flow, but further study, perhaps using a persistent disc hot spot, as done by Cranmer & Owocki (1996) in the con- text of stellar winds is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We note that they found a maximum wind velocity of ∼ 5000 km s−1, consistent with our most luminous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' As discussed, the higher luminos- ity models show the most likelihood to have correlated disc and wind variability since the intrinsic variability of the wind is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Wind variability is expected to be in the slow part of the flow, hence the ideal system would be observed at high inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' For systems where we expect our line of sight to lie along the fast stream (low inclination systems), the best prospects are to observe transitions from a low to a high lu- minosity state or vice versa, the equivalent of a transition be- tween two models in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In our simulations, larger vari- ations in total driving luminosity can be extrapolated from the approximate scaling ˙M ∝ L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' For such large variations, one can expect more dramatic changes to the flow geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Depending on inclination, one might even expect to have pe- riods where no wind is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In particular, if we observe a low inclination system that undergoes a dramatic drop in driving luminosity, the wind might become more equatorial and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' For systems close to the LdMmax ≲ 1 thresh- old such state changes might also turn on/off the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In the system BZ Cam, estimates for inclination range from 12 to 40 degrees (Honeycut, Kafka & Robertson 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This means that per our model, the disc would be observed at low θ, much like V592 Cas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' However, the luminosity variations in this system are large, meaning that we need to traverse from one model to another when the system transitions from a state of lower to a state of higher luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The momen- tum flux at low inclination (corresponding to low θ in our model) would increase dramatically but mostly because of an increase in density (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 5), not velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This would still lead to an increase in line equivalent width of absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Indeed, such a correlation is seen in BZ Cam (Honeycut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The geometry and strength of the flow at lower θ is altered drastically between models (corresponding to large luminosity variations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' At times of low luminosity, the flow might even stop completely at low θ as can be deduced from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 5 (traversing from model C4 to C2 would correspond to a large change in driving luminosity, which leads to flow be- tween 20 and 30 degrees to cease almost completely).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' More studies of BZ Cam like the one done in Honeycut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2013) are needed to clearly discern correlations between wind ac- tivity and luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' A further challenge to observing these correlations is changes in ionization state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' As proposed for BZ Cam by Greiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2001), changes in ionizing flux from the binary companion may explain changes in wind emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Such a model predicts a periodic variability in ionization state set by the orbital timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Later observations by Honeycut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2013) found variability on shorter timescales, disfavoring such a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Further theoretical work should address this fundamental question of how variations in driving radiation MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2022) Line Driven Winds from Variable Accretion Discs 11 (see for example Dyda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 2018d) can be distinguished from variations in ionizing flux from the outflow properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The ideal system to observe the effects of lower variabil- ity (≲ 15%), corresponding to the variability within a model explored in this paper) is a high inclination system with suf- ficiently strong driving luminosity (so that the intrinsic vari- ability is suppressed) and high enough accretion variability but a nearly constant ionizing flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' As can be seen on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 6, flow at lower θ is indistinguishable during a luminosity peak while at high θ, both density and velocity increase during a luminosity peak (mainly density as can be judged from ρvr changing by roughly an order of magnitude while velocity in- creases by less than 25% on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Therefore, looking for correlations between luminosity and line equivalent width of absorption in high inclination, high luminosity systems seems to be the most promising avenue for systems with low varia- tions in luminosity (≲ 15%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Even for such a system, correla- tions would be difficult to observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' One reason is that while mass outflow increases overall during a luminosity peak, that increase is not present in all parts of the flow (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 6, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' flow at precisely 60 degrees is less during a luminosity peak even though flow around it increases significantly), so one might observe at an ’unlucky’ inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' A more perti- nent reason is that intrinsic variability can mask peaks due to luminosity if driving luminosity is not sufficiently high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' For example, Hartley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2002) study two systems at in- clinations of around 60 degrees, IX Vel and V3885 Sgr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The equivalent width of absorption for various lines is presented at three different times, where the continuum flux varies by ≲ 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' No correlation is found between flux and wind, which could be due to the luminosity not being high enough so that the system is in the regime where variations due to luminosity dominate intrinsic variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' It is also possible that the in- clination was particularly unlucky but that is unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Since IX Vel is one of the brightest CV systems, one can reason- ably ask the question whether a CV with ’sufficiently’ high luminosity for these correlations to be visible even exists?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' A potential candidate system is ASAS J071404+7004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='3 which is at a similar fortunate inclination of 62 degrees and seems to possess rapidly changing winds (Inight et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' While we have chosen parameters more suited to CVs, some of our results are applicable to AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We expect higher inclination systems would be better for correlating small changes in luminosity to wind activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' AGN have much higher luminosities, so the intrinsic variability of the winds would be suppressed, making variations due to luminosity more visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Low inclination systems (but not completely face on so that jet effects can be ignored) are better for cor- relating large variations in luminosity due to the changing ge- ometry of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' However, in the regime of high luminosity variations, inclination is not as important because even sys- tems viewed at high inclination (high θ) are expected to have large fluctuations in density of flow due to changing luminos- ity (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In the regime of small luminosity variations (≲ 15%), AGN might be the only systems where correlations between luminosity and wind can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' We have also seen that local variations of the accretion rate in higher lumi- nosity runs can ’conspire’ to produce clumps and thus larger fluctuations in mass outflow than would be expected from the total luminosity variations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Further work, with overall much higher total luminosity and including ionization effects, is needed to understand line driven winds in AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This paper motivates the need for more detailed self- consistent models of line driven winds from accretion discs to further understand how local variability can non-trivially influence mass outflow properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' ACKNOWLEDGEMENTS SD and CSR acknowledge the UK Science and Technology Facilities Council (STFC) for support under the New Appli- cant grant ST/R000867/1 and the European Research Coun- cil (ERC) for support under the European Union’s Horizon 2020 research and innovation programme (grant 834203).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' DATA AVAILABILITY The data underlying this article will be shared on reasonable request to the corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' REFERENCES Balman S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Godon P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Sion E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 2014, ApJ, 794, 84 Castor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Abbott D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Klein R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 1975, ApJ, 195, 157 Clarke C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Carswell B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 2007, Principles of Astrophysical Fluid Dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Cambridge University Press, Cam- bridge Cranmer S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Owocki S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 1996, ApJ, 462, 469C Drew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Proga D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 2000, New Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 44, 21 Dyda S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Proga D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 2018, MNRAS, 475, 3786D Dyda S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Proga D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 2018, MNRAS, 478, 5006D Dyda S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Proga D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 2018, MNRAS, 481, 2745D Dyda S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Proga D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 2018, MNRAS, 481, 5263D Friend D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Abbott D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 1986, ApJ, 311, 701 Gayley K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 1995, ApJ, 454, 410 Greiner J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Tovmassian G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Orio G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Lehmann H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Chavushyan V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Rau A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Schwarz R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Casalegno R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Scholz R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 2001, A&A, 376, 1031-1038 Hartley L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Drew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Long K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', MNRAS, 332, 127 Honeycut R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Kafka S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Robertson J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 2013, AJ, 145, 45 Inight K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', G¨ansicke B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Blondel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Boyd D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Ashley R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Knigge C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Long K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Marsh T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', McCleery J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Scaringi S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Steeghs D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Thorstensen J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Vanmunster T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Wheatley P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', MNRAS, 510, 3605 Kafka S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Hoard D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Honeycutt R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Deliyannis C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 2009, AJ, 137, 197 Kurosawa R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Proga D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 2009, MNRAS, 397, 1791K Liu C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Yuan F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Ostriker J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Gan Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Yang X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 2013, MN- RAS, 434, 1721L Lucy L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Solomon P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 1970, ApJ, 159, 879 Mosallanezhad A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Yuan F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Ostriker J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Zeraatgari F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Bu D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 2019, MNRAS, 490, 2567M Murray N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Chiang J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Grossman S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Voit G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 1995, ApJ, 451, 498 Nomura M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Ohsuga K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Done C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 2020, MNRAS, 494, 3616N Owocki S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Castor J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Rybicki G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 1988, ApJ, 335, 914 Parker E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 1958, PhFl, 1, 171P Pauldrach A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Puls J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Kudritzki R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 1986, A&A, 164, 86 Pereyra N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 1997, PhD thesis, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Maryland Pereyra N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Kallman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Blondin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 1997, ApJ, 477, 368 MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2022) 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Kirilov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Dyda, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Reynolds Pringle J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 1981, ARA&A, 19, 137 Proga D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Stone J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Drew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 1998, MNRAS, 295, 595 Proga D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Stone J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Drew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 1999, MNRAS, 310, 476 Shakura N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Sunyaev R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 1973, A&A, 24, 337S Stone J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Tomida K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', White C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Felker K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 2020, ApJS, 249, 4S Vitello P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', Shlosman I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=', 1988, ApJ, 327, 680 APPENDIX A: DERIVATION OF INITIAL CONDITION To improve the convergence time for the simulation, we in- troduced an appropriate initial condition with axisymmetry, ρ = ρ(R, z) for our disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Its derivation is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Assuming hydrostatic equilibrium, in cylindrical coor- dinates (R, z), a patch of gas must obey 1 ρ dP dz = − GM R2 + z2 cos(θ), (A1) where θ is the angle between the ˆR-axis and a radial vector pointing to from the central object to the patch of gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' If z/R << 1, − GMz R3(1 + (z/R)2)3/2 ≈ −GMz R3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (A2) We can write this as, 1 ρ dP dz ≈ −Ω2 Kz, (A3) where ΩK = � GM R3 , the Keplerian angular velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' The solution is an exponential in the ˆz-direction, assumming z/R << 1 and P = c2 sρ with cs constant along ˆz (verti- cally isothermal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Hence, we can write our density as ρ = ρ(R)ρ′ 0 exp � −z2/2H2� , (A4) where H = cs/ΩK is the scale height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' One can then relate the density to the surface density via integrating along z from −∞ to +∞, Σ = √ 2πHρ(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (A5) The surface density obeys a diffusion equation, assuming a Keplerian velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' In the steady state, it can be shown (Clarke & Carswell, 2007) that νΣ = ˙M 3π � 1 − � R⋆ R � , (A6) where R⋆ is the radius of the central object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' This reveals the origin of the formula for I(rd) with rd ≡ R and r⋆ ≡ R⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Recalling ν = αcsH, and assuming isothermal equation of state (EoS) throughout (in ˆR as well as ˆz), it also means that we can find ρ(R), which we choose to write as ρ(R) = ρ′′ 0 �R⋆ R �3 � 1 − � R⋆ R � , (A7) where we have absorbed all physical constants into ρ′′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Now we can write ρ(R, z) as ρ(R, z) = ρ0 �R⋆ R �3 � 1 − � R⋆ R � exp � −z2/2H2� , (A8) where we ρ0 = ρ′′ 0ρ′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' Since we can choose ρ′ 0, we can choose ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' □ MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} +page_content=' (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf'} diff --git a/YdFRT4oBgHgl3EQfOje2/content/tmp_files/2301.13514v1.pdf.txt b/YdFRT4oBgHgl3EQfOje2/content/tmp_files/2301.13514v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b4ac967402cdab95895f7c7c77b68a060b0990d5 --- /dev/null +++ b/YdFRT4oBgHgl3EQfOje2/content/tmp_files/2301.13514v1.pdf.txt @@ -0,0 +1,1571 @@ +Published in Transactions on Machine Learning Research (12/2022) +Fourier Sensitivity and Regularization of Computer Vision +Models +Kiran Krishnamachari +kirank@u.nus.edu +Institute for Infocomm Research, A*STAR, Singapore +School of Computing, National University of Singapore, Singapore +See-Kiong Ng +seekiong@nus.edu.sg +Institute of Data Science, National University of Singapore, Singapore +School of Computing, National University of Singapore, Singapore +Chuan-Sheng Foo +foo_chuan_sheng@i2r.a-star.edu.sg +Institute for Infocomm Research, A*STAR, Singapore +Centre for Frontier AI Research, A*STAR, Singapore +Reviewed on OpenReview: https: // openreview. net/ forum? id= VmTYgjYloM +Abstract +Recent work has empirically shown that deep neural networks latch on to the Fourier statis- +tics of training data and show increased sensitivity to Fourier-basis directions in the input. +Understanding and modifying this Fourier-sensitivity of computer vision models may help +improve their robustness. Hence, in this paper we study the frequency sensitivity character- +istics of deep neural networks using a principled approach. We first propose a basis trick, +proving that unitary transformations of the input-gradient of a function can be used to +compute its gradient in the basis induced by the transformation. Using this result, we pro- +pose a general measure of any differentiable model’s Fourier-sensitivity using the unitary +Fourier-transform of its input-gradient. When applied to deep neural networks, we find that +computer vision models are consistently sensitive to particular frequencies dependent on the +dataset, training method and architecture. Based on this measure, we further propose a +Fourier-regularization framework to modify the Fourier-sensitivities and frequency bias +of models. Using our proposed regularizer-family, we demonstrate that deep neural networks +obtain improved classification accuracy on robustness evaluations. +1 +Introduction +While deep neural networks (DNN) achieve remarkable performance on many challenging image classification +tasks, they can suffer significant drops in performance when evaluated on out-of-distribution (o.o.d.) data. +Intriguingly, this lack of robustness has been partially attributed to the frequency characteristics of data +shifts at test time in relation to the frequency sensitivity characteristics of the model (Yin et al., 2019; Jo +& Bengio, 2017). It is known that distinct spatial frequencies in images contain features at different spatial +scales: low spatial frequencies (LSF) carry global structure and shape information whereas high spatial +frequencies (HSF) carry local information such as edges and borders of objects (Kauffmann et al., 2014). +Moreover, spatial frequencies may also differentially processed in the brain’s visual cortex to learn features +at different scales (Appendix A). We find that when information in frequencies that a model relies on is +corrupted or destroyed, performance can suffer. Hence, understanding the frequency sensitivity of a DNN +can help us characterise and improve them. +DNNs have been demonstrated to be sensitive to Fourier-basis directions in the input (Tsuzuku & Sato, +2019; Yin et al., 2019) both empirically and using theoretical analysis of linear convolutional networks +1 +arXiv:2301.13514v1 [cs.CV] 31 Jan 2023 + +Published in Transactions on Machine Learning Research (12/2022) +(Tsuzuku & Sato, 2019). In fact, the existence of so-called “universal adversarial perturbations” (Moosavi- +Dezfooli et al., 2017), simple semantics-preserving distortions that can degrade models’ accuracy across +inputs and architectures, is attributed to this structural sensitivity. +Yin et al. (2019) also showed that +many natural and digital image corruptions that degrade model performance may also be targeting this +vulnerability. Hence, understanding and modifying Fourier-sensitivity is a promising approach to improve +model robustness. While this problem has been studied empirically, the precise definition and measurement +of a computer vision model’s Fourier-sensitivity still lacks a rigorous approach across studies. In addition, +no principled method has been proposed to study and modify the Fourier-sensitivity of a model. Existing +works have applied heuristic filters on convolution layer parameters (Wang et al., 2020; Saikia et al., 2021) +and input data augmentations (Yin et al., 2019) to modify a model’s frequency sensitivity. +In this work, we first propose a novel basis trick, proving that unitary transformations of a function’s +gradient can be used to compute its gradient in the basis induced by the transformation. Using this result, +we propose a novel and rigorous measure of a DNN’s Fourier-sensitivity using its input-gradient represented +in the Fourier-basis. We demonstrate that DNNs are consistently sensitive to particular frequencies that are +dependent on dataset, training method and architecture. This observation confirms that DNNs tend to rely +on some frequencies more than others, which has implications for robustness when Fourier-statistics change +at test time. Further, using our proposed measure, which is differentiable with respect to model parameters, +we propose a framework of Fourier-regularization to directly modify the Fourier-sensitivities and frequency +bias of a model. We show in extensive empirical evaluations that Fourier-regularization can indeed modify +frequency characteristics of computer vision models, and can improve the generalization performance of +models on o.o.d. datasets where the Fourier-statistics are shifted. In summary, our main contributions are: +1. We propose a basis trick, proving that unitary transformations of the input-gradient of any function +can be used to compute its gradient in the basis induced by the transformation +2. We propose a novel and rigorous measure of a model’s Fourier-sensitivity based on the unitary +Fourier-transform of its input-gradient. We empirically show that Fourier-sensitivity of a model is +dependent on the dataset, training method and architecture +3. We propose a framework of Fourier-regularization to directly induce specific Fourier-sensitivities +in a computer-vision model, which modifies the frequency bias of models and improves generalization +performance on out-of-distribution data where Fourier-statistics are shifted +2 +Related work +2.1 +Frequency perspectives of robustness +Yin et al. (2019); Tsuzuku & Sato (2019) characterised the Fourier characteristics of trained CNNs using +perturbation analysis of their test error under Fourier-basis noise. They showed that a naturally trained +model is most sensitive to all but low frequencies whereas adversarially trained (Madry et al., 2018) models +are sensitive to low-frequency noise. They further showed that these Fourier characteristics relate to model +robustness on corruptions and noise, with models biased towards low frequencies performing better under +high frequency noise and vice versa. Abello et al. (2021) took a different approach by measuring the impact of +removing individual frequency components from the input using filters on accuracy, whereas Ortiz-Jimenez +et al. (2020) computed the margin in input space along basis directions of the discrete cosine transform +(DCT). Wang et al. (2020) made observations about the Fourier characteristics of CNNs in different train- +ing regimes including standard and adversarial training by evaluating accuracy on band-pass filtered data. +Contrary to these empirical approaches, we propose a rigorous measure of a model’s Fourier-sensitivity. +2.2 +Modifying frequency sensitivity of models +Yin et al. (2019) observed that adversarial training (Madry et al., 2018) and Gaussian noise augmentation can +induce a low-frequency sensitivity on some datasets. Wang et al. (2020) proposed smoothing convolution +filter parameters to induce a low-frequency sensitivity in models. We note that such techniques can, in +2 + +Published in Transactions on Machine Learning Research (12/2022) +principle, be undone by subsequent layers of a network. Shi et al. (2022) proposed similar techniques in the +context of deep image priors applied to generative tasks. In addition, data augmentations such as Gaussian +noise do not provide precise control over the Fourier-sensitivity of a model. In this work, we propose a +Fourier-regularization framework to precisely modify the Fourier-sensitivity of any differentiable model. +2.3 +Jacobian regularization +Methods that regularize the input-Jacobian of a model can be broadly classified into two categories: meth- +ods that minimize the norm of the input-Jacobian, and those that regularize its direction or directional +derivatives at the input. Drucker & Le Cun (1991) proposed a method that penalized the norm of the +input-Jacobian to improve generalization; more recently, this has been explored to improve robustness to ad- +versarial perturbations (Ross & Doshi-Velez, 2018; Jakubovitz & Giryes, 2018; Hoffman et al., 2019). Simard +et al. (1992) proposed “Tangent Prop”, which minimized directional derivatives of classifiers in the direction +of local input-transformations (e.g. rotations, translations; called “tangent vectors”) to reduce sensitivity to +such transformations. Czarnecki et al. (2017) proposed Sobolev training of neural networks to improve model +distillation by matching the input-Jacobian of the original model. Regularizing the direction of the input- +Jacobian has also been used to improve adversarial robustness (Chan et al., 2020). In the present work, we +regularize frequency components in the input-gradient to improve performance on out-of-distribution tasks. +As such, we are interested in modifying the input-gradient along certain directions instead of its total norm. +3 +Proposed methods +r=N/6 +r=N/√2 +N +N +r=N/3 +r=N/2 +(cu,cv) +P(u,v) +N +N +P̃ Total = power inside circle +PTotal = power inside entire square +zero-frequency (DC) +P(u,v) +P̃Total = power inside circle +PTotal = power inside entire square +zero-frequency (DC) +(a) +(b) +Figure 1: Power-matrix P of input-gradient. +Preliminaries: +Consider an image classification +task with input x, labels y, and standard cross- +entropy loss function LCE. Let f denote any differ- +entiable model that outputs a scalar loss, F(·) the +unitary discrete Fourier transform (DFT), F−1(·) +its inverse, and F−1∗(·) the adjoint of the inverse- +Fourier transform, and let xf denote the Fourier- +space representation of the input, i.e. xf = F(x). +We denote the input-gradient in the standard basis +as Jf(x), and Jf(xf) as the input-gradient with re- +spect to the input in the Fourier-basis. Let N be +the height of input images (although not necessary, +all images used in this work are square). +DFT notation: The zero-shifted (rearrange DC +component to centre and high frequencies further +from center) 2D-DFT of the input-gradient is denoted F. +Since the input-gradient typically has three +color channels, they are averaged before computing the 2D-DFT. Fourier coefficients in F are complex +numbers with real and imaginary components; F(u, v) = Real(u, v) + i × Imag(u, v), where (u, v) are +indices of coefficients. +The power in a coefficient is its squared amplitude i.e. +P(u, v) = |F(u, v)|2 = +Real(u, v)2+Imag(u, v)2 and the matrix of powers is denoted P (power-matrix). Each coefficient has a radial +distance r(u, v) from the centre of the matrix, r(u, v) = d((u, v), (cu, cv)), where (cu, cv) denotes the center of +P and d(·, ·) is Euclidean distance rounded to the nearest integer. Distinct radial distances of coefficients in +the matrix are the set of integers {1, . . . , N/ +√ +2} and correspond to low to high spatial frequencies, the highest +frequency being limited by the Nyquist-frequency. We denote PT otal as the total power in P, excluding the +zero-frequency coefficient i.e. PT otal = � +r(u,v)>=1 P(u, v). Similarly, we define ˜PT otal as the total power in +P excluding the zero-frequency coefficient and coefficients with radial distance r(u, v) > N/2, i.e. coefficients +outside the largest circle inscribed in P; ˜PT otal = +� +1<=r(u,v)<=N/2 +P(u, v) (see Figure 1 for illustration). We +denote Pk as the power at radial distance k normalized by PT otal, Pk = +1 +PT otal +� +r(u,v)=k +P(u, v) and ˜Pk as the +power at radial distance k normalized by ˜PT otal, ˜Pk = +1 +˜ +PT otal +� +r(u,v)=k +P(u, v). +3 + +Published in Transactions on Machine Learning Research (12/2022) +3.1 +Basis Trick: Unitary transformations of the input-gradient +In this section, we prove that unitary transformations of the input-gradient of a function provide its gradient +in the new basis induced by the transformation. We term this the basis trick and use it to compute the +Fourier-sensitivity of a model using the Fourier-transform of its input-gradient. To illustrate the basis trick, +consider the computation graph in Figure 2 where the input x in the standard basis is mapped to an output via +a function f. We introduce an implicit operation (shown in red) that maps the Fourier-space representation +of the input to the standard basis via the inverse Fourier-transform, i.e. xf +F−1 +−−−→ x. In order to compute the +input-gradient with respect to input in the Fourier-basis, Jf(xf), we must differentiate through this implicit +operation in the forward graph. Since the inverse-Fourier transform is a unitary operator, we have that +F(Jf(x)) = Jf(xf), due to the chain rule (see Corollary 1 below). Hence, even though we do not explicitly +compute the Fourier-space representation of the input, this shows that the Fourier transform of the input- +gradient provides the gradient of the model with respect to the input in Fourier-space. Analogous results can +be obtained for other unitary operators such as the discrete cosine transform (DCT) and discrete wavelet +transform (DWT) (see Proposition 1 below). In addition, this approach can be extended to n-dimensional +input, e.g. time-series or 3D signals, by using the n-dimensional Fourier-transform. We formalize the basis +trick below as a proposition and its corollary when the unitary operator is the Fourier-transform. +Definition 1 (Unitary Operators). A bounded linear operator U : H → H on a Hilbert space H is said to +be unitary if U is bijective and its adjoint U ∗ = U −1. Moreover, if U is unitary, U −1 is also a bounded and +unitary linear operator. +Lemma 1 (Generalized Chain Rule). Let f be a scalar valued function of a vector x, and A be a bijective +linear operator such that x = Axa. Then, A∗(Jf(x)) is the gradient of f with respect to xa i.e. Jf(xa) = +A∗(Jf(x)), where A∗ is the adjoint of A. +Proposition 1 (Basis Trick). Let f be a scalar valued function of a vector x, and A be a bijective linear +operator such that x = Axa. Then, the gradient vector of f w.r.t xa, Jf(xa) = A−1(Jf(x)) iff A is unitary. +Proof. +Since x = Axa, Jf(xa) = A∗(Jf(x)) due to Lemma 1. Since A∗ = A−1 iff A is unitary (Definition +1), we have that Jf(xa) = A−1(Jf(x)) iff A is unitary. +Corollary 1 (Fourier Basis Trick). If A = F−1, the unitary inverse-Fourier operator such that x = F−1xf +with xf being the Fourier-basis representation of x, we have Jf(xf) = F(Jf(x)) where F = (F−1)−1. +∇ +𝗝 (𝑥) +𝗝 ( ) +𝑥𝑓 +𝑥 +Fourier-space +Input-space +Output-space +𝑓(𝑥) +𝑓 +𝑥𝑓 +𝑓 +𝑓 +Figure 2: Fourier-transform of input-gradient is the gradient with respect to input in Fourier-space, i.e., +F(Jf(x)) = Jf(xf). Symbols in red represent the input in Fourier-space and need not explicitly computed. +3.2 +Fourier-sensitivity of computer vision models +In this section, we define the Fourier-sensitivity of any differentiable model using its input-gradient +represented in the Fourier-basis. Fourier-sensitivity is a measure of the relative magnitudes of a model’s input- +gradient with respect to different frequency bands in the input spectrum. As shown in Section 3.1, the input- +gradient of a function with respect to the Fourier-basis can be computed by the unitary Fourier-transform +of Jf(x). To enable interpretation of the complete input-gradient in the Fourier-basis (see Appendix C.5 for +examples), we summarize the information over frequency bands as shown in Figure 3. The Fourier-sensitivity +fSF S(x, y) of a model with respect to an individual input (x, y) is defined as +fSF S(x, y) = [P1, . . . , PN/ +√ +2] +(1) +4 + +F-1FPublished in Transactions on Machine Learning Research (12/2022) +Input +Input-Gradient +DFT Power Matrix +Compute fraction of total power in circular bands +x-axis is radius of circular band +r=15 +r=20 +DC-component +Figure 3: Computing Fourier-sensitivity. The input-gradient of the model is Fourier-transformed to obtain +sensitivities with respect to frequencies. Fourier-sensitivity is then the vector with components being the +proportion of total power in each circular frequency band. +where Pk is the proportion of total power in Fourier coefficients at radial distance k in the power matrix P +of Jf(xf). The overall spatial frequency sensitivity (SFS), or simply Fourier-sensitivity, of a model is defined +as the expectation of fSF S(x, y) over the data distribution p(x, y), i.e. fSF S(·; θ) = E(x,y)∼p[fSF S(x, y)] +(Algorithm 1 in Appendix B.1). +3.3 +Fourier-regularization of computer vision models +In this section, we propose a framework of Fourier-regularization. Fourier-regularization enables control +over the Fourier-sensitivity of a model by modifying the relative magnitude of a model’s sensitivity to +different frequency bands in the input spectrum. Fourier-regularization can modify the natural frequency +sensitivity of neural networks as well as their generalization behavior. Our Fourier-regularizer augments the +usual cross entropy loss: for a single example the new loss is L(x, y) = LCE(x, y) + λSFSLSFS(x, y), where +LSFS is the proposed regularizer and λSFS is a hyperparameter. Our regularizer penalizes the proportion of +power in frequency bands based on the target Fourier-sensitivity. As LSFS is a function of the input-gradient, +optimizing it requires an additional backpropagation step to compute derivatives with respect to parameters, +similar to other gradient-regularization methods. +We now define LSFS for four instances of this regularizer: SFS ∈ {LSF, MSF, HSF, ASF}. Low-spatial- +frequency (lsf) regularization trains a model to be insensitive to medium and high spatial frequencies, +medium-spatial-frequency (msf) regularization trains a model to be insensitive to low and high spatial +frequencies, and high-spatial-frequency (hsf) regularization trains a model to be insensitive to low and +medium spatial frequencies. These are achieved by penalizing the proportion of power, Pk, in the frequencies +we wish the model to be insensitive to. All-spatial-frequency (asf) regularization trains a model to be equally +sensitive to all frequency bands. The motivation behind asf regularization model is to encourage a model to +be sensitive to multiple frequency bands instead of being concentrated in a small frequency range. Hence, the +asf-regularizer loss is defined as the negative entropy of the distribution of power over frequency bands. The +definitions of low, medium and high frequency ranges are based on equally dividing the radius of the largest +circle inscribed in the power-matrix P into three equal parts (Figure 1b). For ASF-regularization, very high +frequency bands, i.e. r(u, v) > N/2 are excluded, which is reflected in the ˜Pk terms. ˜Pk is the proportion +of power in frequency bands within the largest circle inscribed in the power-matrix, P. Concretely, LSFS is +defined for each of these three cases as follows: +LSF +MSF +HSF +ASF +LSFS +� +k>N/6 +Pk +� +kN/3 +Pk +� +k 5 were set to zero. For +medium-pass filtering, Fourier-coefficients with r(u, v) < 5 and r(u, v) > 10 were set to zero. For high-pass +filtering, Fourier-coefficients with r(u, v) < 10 were set to zero. Medium-pass and high-pass filtered images +were contrast-maximised for viewing. +Table 5: Comparing clean accuracy of models standard trained on filtered CIFAR10 images vs Fourier- +regularization (ResNet50). +Method +Accuracy +Low-pass Filtered +86.6 +Medium-pass Filtered +33.8 +High-pass Filtered +15.3 +lsf-regularized +87.1 +msf-regularized +90.6 +hsf-regularized +93.5 +asf-regularized +87.9 +24 + +Published in Transactions on Machine Learning Research (12/2022) +E +Fourier-noise corruptions +E.1 +CIFAR10 +Standard Trained +(a) +(c) +(b) +(d) +LSF-REGULARIZED +MSF-REGULARIZED +ASF-REGULARIZED +Figure 17: (CIFAR10) Heat map of error rates for each Fourier-mode corruption (low-frequencies close to +the center). Each pixel in the heat map is the error of the model when the corresponding Fourier-mode +noise (ϵ = 4) is added to the inputs. The bottom row displays example images containing the corresponding +Fourier-noise. +25 + +1.0 +0.8 +0.6 +0.4 +0.2 +0.0Published in Transactions on Machine Learning Research (12/2022) +E.2 +SVHN +Standard Trained +(a) +(b) +(c) +(d) +LSF-REGULARIZED +MSF-REGULARIZED +ASF-REGULARIZED +Figure 18: (SVHN) Heat map of error rates for each Fourier-mode corruption (low-frequencies close to the +center). Each pixel in the heat map is the error of the model when the corresponding Fourier-mode noise +(ϵ = 4) is added to the inputs. The bottom row displays example images containing the corresponding +Fourier-corruptions. +26 + +1.0 +0.8 +0.6 +0.4 +0.2 +0.0Published in Transactions on Machine Learning Research (12/2022) +E.3 +Accuracy on Fourier-noise distortions +Table 6: Mean accuracy across all Fourier-noise corruptions averaged across 1024 randomly selected test +samples for each corruption. ℓ2 norms of the additive Fourier-noise are ϵ ∈ {3, 4}. +Method +SVHN +CIFAR10 +CIFAR100 +clean +ϵ=3 +ϵ=4 +clean +ϵ=3 +ϵ=4 +clean +ϵ=3 +ϵ=4 +Std. Train +96.4 +81.9 +77.4 +94.9 +40.8 +31.5 +76.2 +22.3 +14.9 +lsf-regularized +95.1 +92.1 +91.0 +87.1 +52.4 +47.5 +62.5 +33.9 +30.0 +msf-regularized +93.1 +77.1 +70.9 +90.6 +62.4 +54.3 +70.7 +42.6 +37.3 +asf-regularized +96.4 +78.3 +71.1 +87.9 +60.8 +48.7 +67.0 +21.6 +14.6 +27 + +Published in Transactions on Machine Learning Research (12/2022) +F +Patch-shuffling images +3x3 +2x2 +original +Figure 19: Patch-shuffling: Images are partitioned into squares whose positions are randomly exchanged. +This operation destroys global structure in the image and is used to evaluate the extent to which a model +relies on global information. +28 + +Published in Transactions on Machine Learning Research (12/2022) +G +Sensitivity to λSFS +Table 7: Clean accuracy on CIFAR10 (ResNet50) at different λSFS values +Method +λSFS = 0 +λSFS = 0.1 +λSFS = 0.2 +λSFS = 0.5 +λSFS = 1 +lsf-regularized +94.8 +93.8 +91.1 +87.1 +84.6 +Figure 20: Fourier-sensitivity of lsf-regularization at different λSFS on CIFAR10 (ResNet50). +29 + diff --git a/YdFRT4oBgHgl3EQfOje2/content/tmp_files/load_file.txt b/YdFRT4oBgHgl3EQfOje2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e3bc009f72666ab823bd72215bd6b371cf358f55 --- /dev/null +++ b/YdFRT4oBgHgl3EQfOje2/content/tmp_files/load_file.txt @@ -0,0 +1,1346 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf,len=1345 +page_content='Published in Transactions on Machine Learning Research (12/2022) Fourier Sensitivity and Regularization of Computer Vision Models Kiran Krishnamachari kirank@u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='edu Institute for Infocomm Research, A*STAR, Singapore School of Computing, National University of Singapore, Singapore See-Kiong Ng seekiong@nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='sg Institute of Data Science, National University of Singapore, Singapore School of Computing, National University of Singapore, Singapore Chuan-Sheng Foo foo_chuan_sheng@i2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='a-star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='sg Institute for Infocomm Research, A*STAR, Singapore Centre for Frontier AI Research, A*STAR, Singapore Reviewed on OpenReview: https: // openreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' net/ forum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' id= VmTYgjYloM Abstract Recent work has empirically shown that deep neural networks latch on to the Fourier statis- tics of training data and show increased sensitivity to Fourier-basis directions in the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Understanding and modifying this Fourier-sensitivity of computer vision models may help improve their robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Hence, in this paper we study the frequency sensitivity character- istics of deep neural networks using a principled approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' We first propose a basis trick, proving that unitary transformations of the input-gradient of a function can be used to compute its gradient in the basis induced by the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Using this result, we pro- pose a general measure of any differentiable model’s Fourier-sensitivity using the unitary Fourier-transform of its input-gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' When applied to deep neural networks, we find that computer vision models are consistently sensitive to particular frequencies dependent on the dataset, training method and architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Based on this measure, we further propose a Fourier-regularization framework to modify the Fourier-sensitivities and frequency bias of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Using our proposed regularizer-family, we demonstrate that deep neural networks obtain improved classification accuracy on robustness evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' 1 Introduction While deep neural networks (DNN) achieve remarkable performance on many challenging image classification tasks, they can suffer significant drops in performance when evaluated on out-of-distribution (o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=') data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Intriguingly, this lack of robustness has been partially attributed to the frequency characteristics of data shifts at test time in relation to the frequency sensitivity characteristics of the model (Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Jo & Bengio, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' It is known that distinct spatial frequencies in images contain features at different spatial scales: low spatial frequencies (LSF) carry global structure and shape information whereas high spatial frequencies (HSF) carry local information such as edges and borders of objects (Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Moreover, spatial frequencies may also differentially processed in the brain’s visual cortex to learn features at different scales (Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' We find that when information in frequencies that a model relies on is corrupted or destroyed, performance can suffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Hence, understanding the frequency sensitivity of a DNN can help us characterise and improve them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' DNNs have been demonstrated to be sensitive to Fourier-basis directions in the input (Tsuzuku & Sato, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=', 2019) both empirically and using theoretical analysis of linear convolutional networks 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='13514v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='CV] 31 Jan 2023 Published in Transactions on Machine Learning Research (12/2022) (Tsuzuku & Sato, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' In fact, the existence of so-called “universal adversarial perturbations” (Moosavi- Dezfooli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=', 2017), simple semantics-preserving distortions that can degrade models’ accuracy across inputs and architectures, is attributed to this structural sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' (2019) also showed that many natural and digital image corruptions that degrade model performance may also be targeting this vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Hence, understanding and modifying Fourier-sensitivity is a promising approach to improve model robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' While this problem has been studied empirically, the precise definition and measurement of a computer vision model’s Fourier-sensitivity still lacks a rigorous approach across studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' In addition, no principled method has been proposed to study and modify the Fourier-sensitivity of a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Existing works have applied heuristic filters on convolution layer parameters (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Saikia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=', 2021) and input data augmentations (Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=', 2019) to modify a model’s frequency sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' In this work, we first propose a novel basis trick, proving that unitary transformations of a function’s gradient can be used to compute its gradient in the basis induced by the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Using this result, we propose a novel and rigorous measure of a DNN’s Fourier-sensitivity using its input-gradient represented in the Fourier-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' We demonstrate that DNNs are consistently sensitive to particular frequencies that are dependent on dataset, training method and architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' This observation confirms that DNNs tend to rely on some frequencies more than others, which has implications for robustness when Fourier-statistics change at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Further, using our proposed measure, which is differentiable with respect to model parameters, we propose a framework of Fourier-regularization to directly modify the Fourier-sensitivities and frequency bias of a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' We show in extensive empirical evaluations that Fourier-regularization can indeed modify frequency characteristics of computer vision models, and can improve the generalization performance of models on o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' datasets where the Fourier-statistics are shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' In summary, our main contributions are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' We propose a basis trick, proving that unitary transformations of the input-gradient of any function can be used to compute its gradient in the basis induced by the transformation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' We propose a novel and rigorous measure of a model’s Fourier-sensitivity based on the unitary Fourier-transform of its input-gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' We empirically show that Fourier-sensitivity of a model is dependent on the dataset, training method and architecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' We propose a framework of Fourier-regularization to directly induce specific Fourier-sensitivities in a computer-vision model, which modifies the frequency bias of models and improves generalization performance on out-of-distribution data where Fourier-statistics are shifted 2 Related work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='1 Frequency perspectives of robustness Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Tsuzuku & Sato (2019) characterised the Fourier characteristics of trained CNNs using perturbation analysis of their test error under Fourier-basis noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' They showed that a naturally trained model is most sensitive to all but low frequencies whereas adversarially trained (Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=', 2018) models are sensitive to low-frequency noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' They further showed that these Fourier characteristics relate to model robustness on corruptions and noise, with models biased towards low frequencies performing better under high frequency noise and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Abello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' (2021) took a different approach by measuring the impact of removing individual frequency components from the input using filters on accuracy, whereas Ortiz-Jimenez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' (2020) computed the margin in input space along basis directions of the discrete cosine transform (DCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' (2020) made observations about the Fourier characteristics of CNNs in different train- ing regimes including standard and adversarial training by evaluating accuracy on band-pass filtered data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Contrary to these empirical approaches, we propose a rigorous measure of a model’s Fourier-sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='2 Modifying frequency sensitivity of models Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' (2019) observed that adversarial training (Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=', 2018) and Gaussian noise augmentation can induce a low-frequency sensitivity on some datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' (2020) proposed smoothing convolution filter parameters to induce a low-frequency sensitivity in models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' We note that such techniques can, in 2 Published in Transactions on Machine Learning Research (12/2022) principle, be undone by subsequent layers of a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' (2022) proposed similar techniques in the context of deep image priors applied to generative tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' In addition, data augmentations such as Gaussian noise do not provide precise control over the Fourier-sensitivity of a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' In this work, we propose a Fourier-regularization framework to precisely modify the Fourier-sensitivity of any differentiable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='3 Jacobian regularization Methods that regularize the input-Jacobian of a model can be broadly classified into two categories: meth- ods that minimize the norm of the input-Jacobian, and those that regularize its direction or directional derivatives at the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Drucker & Le Cun (1991) proposed a method that penalized the norm of the input-Jacobian to improve generalization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' more recently, this has been explored to improve robustness to ad- versarial perturbations (Ross & Doshi-Velez, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Jakubovitz & Giryes, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Hoffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Simard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' (1992) proposed “Tangent Prop”, which minimized directional derivatives of classifiers in the direction of local input-transformations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' rotations, translations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' called “tangent vectors”) to reduce sensitivity to such transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Czarnecki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' (2017) proposed Sobolev training of neural networks to improve model distillation by matching the input-Jacobian of the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Regularizing the direction of the input- Jacobian has also been used to improve adversarial robustness (Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' In the present work, we regularize frequency components in the input-gradient to improve performance on out-of-distribution tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' As such, we are interested in modifying the input-gradient along certain directions instead of its total norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' 3 Proposed methods r=N/6 r=N/√2 N N r=N/3 r=N/2 (cu,cv) P(u,v) N N P̃ Total = power inside circle PTotal = power inside entire square zero-frequency (DC) P(u,v) P̃Total = power inside circle PTotal = power inside entire square zero-frequency (DC) (a) (b) Figure 1: Power-matrix P of input-gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Preliminaries: Consider an image classification task with input x, labels y, and standard cross- entropy loss function LCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Let f denote any differ- entiable model that outputs a scalar loss, F(·) the unitary discrete Fourier transform (DFT), F−1(·) its inverse, and F−1∗(·) the adjoint of the inverse- Fourier transform, and let xf denote the Fourier- space representation of the input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' xf = F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' We denote the input-gradient in the standard basis as Jf(x), and Jf(xf) as the input-gradient with re- spect to the input in the Fourier-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Let N be the height of input images (although not necessary, all images used in this work are square).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' DFT notation: The zero-shifted (rearrange DC component to centre and high frequencies further from center) 2D-DFT of the input-gradient is denoted F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Since the input-gradient typically has three color channels, they are averaged before computing the 2D-DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Fourier coefficients in F are complex numbers with real and imaginary components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' F(u, v) = Real(u, v) + i × Imag(u, v), where (u, v) are indices of coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' The power in a coefficient is its squared amplitude i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' P(u, v) = |F(u, v)|2 = Real(u, v)2+Imag(u, v)2 and the matrix of powers is denoted P (power-matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Each coefficient has a radial distance r(u, v) from the centre of the matrix, r(u, v) = d((u, v), (cu, cv)), where (cu, cv) denotes the center of P and d(·, ·) is Euclidean distance rounded to the nearest integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Distinct radial distances of coefficients in the matrix are the set of integers {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' , N/ √ 2} and correspond to low to high spatial frequencies, the highest frequency being limited by the Nyquist-frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' We denote PT otal as the total power in P, excluding the zero-frequency coefficient i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' PT otal = � r(u,v)>=1 P(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Similarly, we define ˜PT otal as the total power in P excluding the zero-frequency coefficient and coefficients with radial distance r(u, v) > N/2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' coefficients outside the largest circle inscribed in P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' ˜PT otal = � 1<=r(u,v)<=N/2 P(u, v) (see Figure 1 for illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' We denote Pk as the power at radial distance k normalized by PT otal, Pk = 1 PT otal � r(u,v)=k P(u, v) and ˜Pk as the power at radial distance k normalized by ˜PT otal, ˜Pk = 1 ˜ PT otal � r(u,v)=k P(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' 3 Published in Transactions on Machine Learning Research (12/2022) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='1 Basis Trick: Unitary transformations of the input-gradient In this section, we prove that unitary transformations of the input-gradient of a function provide its gradient in the new basis induced by the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' We term this the basis trick and use it to compute the Fourier-sensitivity of a model using the Fourier-transform of its input-gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' To illustrate the basis trick, consider the computation graph in Figure 2 where the input x in the standard basis is mapped to an output via a function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' We introduce an implicit operation (shown in red) that maps the Fourier-space representation of the input to the standard basis via the inverse Fourier-transform, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' xf F−1 −−−→ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' In order to compute the input-gradient with respect to input in the Fourier-basis, Jf(xf), we must differentiate through this implicit operation in the forward graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Since the inverse-Fourier transform is a unitary operator, we have that F(Jf(x)) = Jf(xf), due to the chain rule (see Corollary 1 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Hence, even though we do not explicitly compute the Fourier-space representation of the input, this shows that the Fourier transform of the input- gradient provides the gradient of the model with respect to the input in Fourier-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Analogous results can be obtained for other unitary operators such as the discrete cosine transform (DCT) and discrete wavelet transform (DWT) (see Proposition 1 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' In addition, this approach can be extended to n-dimensional input, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' time-series or 3D signals, by using the n-dimensional Fourier-transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' We formalize the basis trick below as a proposition and its corollary when the unitary operator is the Fourier-transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Definition 1 (Unitary Operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' A bounded linear operator U : H → H on a Hilbert space H is said to be unitary if U is bijective and its adjoint U ∗ = U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Moreover, if U is unitary, U −1 is also a bounded and unitary linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Lemma 1 (Generalized Chain Rule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Let f be a scalar valued function of a vector x, and A be a bijective linear operator such that x = Axa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Then, A∗(Jf(x)) is the gradient of f with respect to xa i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Jf(xa) = A∗(Jf(x)), where A∗ is the adjoint of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Proposition 1 (Basis Trick).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Let f be a scalar valued function of a vector x, and A be a bijective linear operator such that x = Axa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Then, the gradient vector of f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='t xa, Jf(xa) = A−1(Jf(x)) iff A is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Since x = Axa, Jf(xa) = A∗(Jf(x)) due to Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Since A∗ = A−1 iff A is unitary (Definition 1), we have that Jf(xa) = A−1(Jf(x)) iff A is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Corollary 1 (Fourier Basis Trick).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' If A = F−1, the unitary inverse-Fourier operator such that x = F−1xf with xf being the Fourier-basis representation of x, we have Jf(xf) = F(Jf(x)) where F = (F−1)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' ∇ 𝗝 (𝑥) 𝗝 ( ) 𝑥𝑓 𝑥 Fourier-space Input-space Output-space 𝑓(𝑥) 𝑓 𝑥𝑓 𝑓 𝑓 Figure 2: Fourier-transform of input-gradient is the gradient with respect to input in Fourier-space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=', F(Jf(x)) = Jf(xf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Symbols in red represent the input in Fourier-space and need not explicitly computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='2 Fourier-sensitivity of computer vision models In this section, we define the Fourier-sensitivity of any differentiable model using its input-gradient represented in the Fourier-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Fourier-sensitivity is a measure of the relative magnitudes of a model’s input- gradient with respect to different frequency bands in the input spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' As shown in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='1, the input- gradient of a function with respect to the Fourier-basis can be computed by the unitary Fourier-transform of Jf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' To enable interpretation of the complete input-gradient in the Fourier-basis (see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='5 for examples), we summarize the information over frequency bands as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' The Fourier-sensitivity fSF S(x, y) of a model with respect to an individual input (x, y) is defined as fSF S(x, y) = [P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' , PN/ √ 2] (1) 4 F-1FPublished in Transactions on Machine Learning Research (12/2022) Input Input-Gradient DFT Power Matrix Compute fraction of total power in circular bands x-axis is radius of circular band r=15 r=20 DC-component Figure 3: Computing Fourier-sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' The input-gradient of the model is Fourier-transformed to obtain sensitivities with respect to frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Fourier-sensitivity is then the vector with components being the proportion of total power in each circular frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' where Pk is the proportion of total power in Fourier coefficients at radial distance k in the power matrix P of Jf(xf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' The overall spatial frequency sensitivity (SFS), or simply Fourier-sensitivity, of a model is defined as the expectation of fSF S(x, y) over the data distribution p(x, y), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' fSF S(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' θ) = E(x,y)∼p[fSF S(x, y)] (Algorithm 1 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='3 Fourier-regularization of computer vision models In this section, we propose a framework of Fourier-regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Fourier-regularization enables control over the Fourier-sensitivity of a model by modifying the relative magnitude of a model’s sensitivity to different frequency bands in the input spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Fourier-regularization can modify the natural frequency sensitivity of neural networks as well as their generalization behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Our Fourier-regularizer augments the usual cross entropy loss: for a single example the new loss is L(x, y) = LCE(x, y) + λSFSLSFS(x, y), where LSFS is the proposed regularizer and λSFS is a hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Our regularizer penalizes the proportion of power in frequency bands based on the target Fourier-sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' As LSFS is a function of the input-gradient, optimizing it requires an additional backpropagation step to compute derivatives with respect to parameters, similar to other gradient-regularization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' We now define LSFS for four instances of this regularizer: SFS ∈ {LSF, MSF, HSF, ASF}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Low-spatial- frequency (lsf) regularization trains a model to be insensitive to medium and high spatial frequencies, medium-spatial-frequency (msf) regularization trains a model to be insensitive to low and high spatial frequencies, and high-spatial-frequency (hsf) regularization trains a model to be insensitive to low and medium spatial frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' These are achieved by penalizing the proportion of power, Pk, in the frequencies we wish the model to be insensitive to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' All-spatial-frequency (asf) regularization trains a model to be equally sensitive to all frequency bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' The motivation behind asf regularization model is to encourage a model to be sensitive to multiple frequency bands instead of being concentrated in a small frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Hence, the asf-regularizer loss is defined as the negative entropy of the distribution of power over frequency bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' The definitions of low, medium and high frequency ranges are based on equally dividing the radius of the largest circle inscribed in the power-matrix P into three equal parts (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' For ASF-regularization, very high frequency bands, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' r(u, v) > N/2 are excluded, which is reflected in the ˜Pk terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' ˜Pk is the proportion of power in frequency bands within the largest circle inscribed in the power-matrix, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' Concretely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf'} +page_content=' LSFS is defined for each of these three cases as follows: LSF MSF HSF ASF LSFS � k>N/6 Pk � k B∥. Problems related to measurements of the Meissner effect in H3S are also +discussed. The SCPC model, applied on the bulk H3S sample, predicts, that the latter is a high-κ +superconductor with ξ0 ≈ (15 − 20) ˚A , λ0 ≈ (1 − 2) × 103˚A ; κ ≈ (50 − 100), µ0Hc1(0) ≈(18 − 60) +mT, µ0Hc0≈ (0, 6 − 1, 1) T, µ0Hc2(0) ≈ (80 − 140) T. +I. +INTRODUCTION +The first almost room-temperature superconductor +was reported in 2015 in sulphur-hydrides (H3S) with +Tc≈ 203 K under high pressure P ≈ 150 GPa(≈ 1.5 +Mbar), which is based on resistivity measurements [4]. +This finding opens a new frontier in physics and a num- +ber of other HP-hydrides were even predicted before these +were synthesized thereafter. Let us mention some of them +with Tc≥ 200 K - such as LaH10with Tc ≈ 250 K, P≈ 190 +GPa [5]; LaYHx with Tc= 253 K, P≈ 190 GPa [6]. In +all of them Tc goes down by increasing magnetic field, +compatible with the standard theory of superconductiv- +ity. There are also reports on the room temperature su- +perconductivity in the CSH hydride - a superconductor +based on C, S and H, with Tc= 287 K at P≈ 267 GPa +[7]. However, this result was not yet confirmed, neither +experimentally nor theoretically by other groups. +There is also theoretical support for the (almost) room- +temperature superconductivity in HP-hydrides, which +are based on the calculated critical temperature Tc in the +microscopic Migdal-Eliashberg theory for superconduc- +tivity - which is due to the electron-phonon interaction. +Moreover, the theoretical prediction of superconductiv- +ity in HP-hydrides [8] is a rare example in the physics +of superconductivity, that the theory goes ahead of ex- +periments, by predicting Tc(≈ 200K) in H3S. Note, that +in [4] it was assumed that the H2S structure is realized, +while in [9] it is argued, that at high pressures the phase +diagram favors decomposition of H2S into H3S and pure +S. +A main proof for superconductivity is the existence of +the Meissner effect, where the magnetic field H < Hc1 - +applied above Tc, is expelled from the sample at tem- +peratures below Tc - the field cooled (FC) experiment. +However, until now the Meissner effect is not proved ex- +perimentally in HP-hydrides, which caused justified crit- +icism on this subject in [10]-[13]. The failure to measure +the Meissner effect in H3S is due to the following reasons: +(i) Magnetic measurements in small samples are delicate; +(ii) The existence of some extrinsic paramagnetic effects +in samples. In that respect, some contradictory experi- +mental results and their inconsistent theoretical interpre- +tations - given in [2],[4],[14], were among the reasons that +Hirsch and Marsiglio even called into question the exis- +tence of superconductivity in HP-hydrides [10]-[13]. In +the following, we are going to show that some magnetic +measurements in HS can refute this skepticism. +The content of the article is following. In Section II the +SCPC model - firstly introduced in [1] for hard type-II +superconductors with strong columnar pinning centers, +is further elaborated and applied to the magnetic mea- +surements in the H3S superconductor. This model pro- +poses, that the vortex pinning is due to long columnar de- +fects (L ∼ Lv ≫ ξ0) - with the radius of the cross-section +r ∼ ξ0. In that case, the core and electromagnetic pin- +ning contribute almost equally to the elementary pinning +energy Up. This is an optimal situation for pinning in a +superconductor, since there is a maximal gain in the con- +densation and electromagnetic vortex energy . This gives +a maximal pinning force if the superconductivity is fully +suppressed in the columnar defects, i.e. for ∆ = 0 inside +a defect. It seems that ∆ is finite in H3S, but (∆ < ∆0) +and the critical current density is smaller than the max- +imal one, i.e. jco = ηcoljmax +c0 +with ηcol < 1. In Section III +the reduction of the temperature broadening of resistance +arXiv:2301.13153v1 [cond-mat.supr-con] 30 Jan 2023 + +2 +(TBR) in a magnetic field, δtexp +c +(h) (with h = H/Hc2), +in HP-hydrides is studied in the SCPC model and ap- +plied to H3S. The temperature dependence of the irre- +versibility line with Bir ∼ C−1(1 − t)2 is predicted within +the SCPC model also in Section III. In Section IV the +SCPC model is applied in studying the penetration of the +magnetic field (PMF) into the center of the H3S sample. +The PMF measurements are analyzed in the Bean criti- +cal state model, first for a long superconducting cylinder +(the parallel configuration B∥) and then for thin disks +(perpendicular configuration B⊥). It is shown, that the +critical magnetic field Bp at which it penetrates into the +center of the sample is larger for the perpendicular ge- +ometry than for the parallel one, i.e. Bp⊥ > Bp∥. The +comparison of the theory and the experimental results +for PMF in H3S [3] gives a large critical current jc0� 107 +A/cm2. +In Section IV we discuss some experimental controver- +sies related to the Meissner effect and FC experiments in +H3S, which display a large residual paramagnetic magne- +tization. The latter fact does not fit into the classical the- +ory of the Meissner effect. Note, that the paramagnetic +signal can be present in some magnetic superconductors, +superconductors with intrinsic magnetic moments in [15]- +[16] or extrinsic ones [17]. Finally, in Section V the ob- +tained results for TBR and PMF in the SCPC model +for H3S are summarized and discussed. Here, the crucial +difference between pinning forces in HTSC-cuprates and +HP-hydridesis also discussed. Note, in the following we +use the notation ξ0 ≈ ξ(T ≪ Tc) and λ0 ≈ λ(T ≪ Tc) +II. +STRONG VORTEX PINNING BY +COLUMNAR DEFECTS IN H3S +A. +Single vortex pinning +To increase the critical current density it is desirable +to have extended (long columnar) pinning defects - the +correlated disorder. In this case the pinning potential is +correlated over the extended size of the defect L∼ Lv. (Lv +is the vortex length.) This means that the superconduc- +tivity is suppressed in a large volume Vcol= πξ2 +0Lv and +therefore a single vortex prefers sitting on this defect. +Note, that the maximal pinning energy Umax +p += ϵmax +p +Vcol +is reached when the superconductivity is fully suppressed +in the core of defects, i.e. +when ∆ = 0 in dielectric +columnar defects. This might be not the case in H3S, +where the real pinning energy density is ϵp = ηcol · ϵmax +p +with the reducing factor ηcol < 1, because the gap in- +side the columnar defect is finite, ∆ < ∆0. In that case +ηcol = ϵp/ϵmax +p +∼ (1 − ∆2/∆2 +0). As a result, the critical +current density for a single vortex (sitting on such a de- +fect) is given by jc = ηcol · jmax +c +. +Note, that in HTSC-cuprates long columnar pinning +defects are made artificially by irradiating YBa2Cu3O7 +single crystalline samples of small platelets of 1×1×0.02 +mm3 with different doses of 580 MeV 116Sn30+ ions [18]- +[19]. The density of the ion doses are D ≈ 5 × 1010,1.5 × +1011, 2.4×1011ions/cm2, which are equivalent to the cor- +responding vortex densities with the magnetic induc- +tion Bφ ≈ 1, 3, 5 T - Bφ is the matching field. +Since +the ionization energy-loss rate was 2.7 keV/˚A they pro- +duce long tracks with the length L ∼ 20 − 30 µm and +diameter ∼ 50 ˚A. In that case jc(∼ jc0) is of the order +1.5×107A/cm2 at T = 5 K and 106A/cm2at T = 77 K. +These current densities are much larger than those for +point defects, where the pinning is due to oxygen vacan- +cies. In the columnar case the irreversible line Bir(T) (the +line below which the pinning is pronounced) lies higher +than for weak pinning with point defects [18]. +In the following it is argued, that the columnar pin- +ning defects dominate in H3S - see blue colored cylinders +in Fig.2, with the averaged distance dφ = (Φ0/Bφ)1/2 +[20]-[21]. +For an optimal pinning the superconductiv- +ity should be fully destroyed inside the dielectric colum- +nar defect with radius r ≳ ξ [20]-[21]. In this case, both +mechanisms of pinning - the core and the electromagnetic +one, are operative with almost same pinning energy +[21]. The maximal pinning energy is given by Up +scpc≈ +2πξ2Lv(H2 +c /8π) ≡ Lvϵmax +p +with ϵmax +p +≈ Φ2 +0/32π2λ2(T). +Since the elementary pinning force (per unit vortex +length) fp =ϵmax +p +/ξ is balanced in the critical state by the +Lorenz force (per unit length) fL(≡ FL/Lv) = jcΦ0/c, the +critical current density jmax +c +in the SCPC model is given +by [20]-[21] +jmax +c +≈ +cΦ0 +32π2λ2ξ . +(1) +In anisotropic superconductors jmax +c +is given in [21]. (In +SI units jmax +c,SI ≈ Φ0/8πµ0λ2ξ, where λ(T) ≈ λ0(1−t)−1/2 +and ξ(T) ≈ ξ0(1 − t)−1/2 are the Ginzburg-Landau pen- +etration depth and coherence length, respectively). +In +H3S one has λ ≈ (1−2)×103˚A and ξ0 ∼ 20 ˚A [4], which +gives for jmax +c0 +in the range jmax +c0 ≈ (0.8−3)×108A/cm2. In +the following, the critical current density jc0 will be esti- +mated from experiments on the magnetic penetration in +H3S [3], where it is found jc0 ≈ (1, 3 − 1, 5)×107˚A/cm2, +which gives for the reducing factor η ≈ (0.1 − 0, 3). +B. +Crossover field Brb for the single vortex to +vortex bundles pinning +The generic H − T phase diagram for the high tem- +perature superconductors with columnar pinning defects +is shown in Fig. +1 - see [19], and some magnetic +properties of H3S will be discussed in the framework +of this phase diagram. +In II.B the single-vortex pin- +ning is considered, which occurs at small magnetic field +B < Bφ. When the inter-vortex distance av ≈ (Φ0/B)1/2 +is larger than the average distance between columnar de- +fects dφ = (Φ0/Bφ)1/2 and the single vortex pinning en- +ergy is larger than the inter-vortex energy, than the vor- +tices accommodate freely to the pinning sites. However, + +3 +FIG. 1: The generic H − T phase diagram for superconduc- +tors with columnar defects. The doted melting line Bm(T) +of the pure sample is transformed into a Bose-glass transition +line BBG(T). +Also shown are the various pinning regimes +with a single-vortex/single-rod pinning region at low fields +µ0H < Brb(T) < 2BΦ. For T < Tr +dp the fluctuations of a vor- +tex from one defect to another are suppressed, while for +T > Tr +dp the pinning potential is exponentially reduced. At +T > Tdl the individual flux lines are pinned collectively by +an assembly of columnar defects (rods) at high temperatures. +Above the crossover line Brb(T), the largest energy in the +problem is the inter-vortex interaction and pinning involves +vortex bundles - taken from [19]. +when B > BΦ the inter-vortex interaction starts to be +important for pinning and dynamical properties (such +as the vortex creep). In that case, vortex bundles are +pinned. The maximal crossover field Bmax +rb +- see Fig.1, +separates the single vortex regime from the vortex bun- +dle regime and it is obtained by comparing the energy +of the elastic shear deformation (of the order u ∼ dΦ) +with the maximal pinning potential ϵmax +p +per unit length, +i.e. ϵshear = c66(dφ/av)2a2 +v⋍ ϵmax +p +. The shear modulus is +given by c66 ≈ Φ0B/(8πλ)2what gives for Brb +Brb < Bmax +rb +≲ 4ϵmax +p +ϵ0 +Bφ, +(2) +where ϵ0 = Φ2 +0/16π2λ2 [19]. Since ϵmax +p += ϵ0/2 it gives, +that for B < Bbr < Bmax +rb +≈ 2BΦ the columnar defects +outnumber the vortices and the single-vortex pinning pre- +vails. Note, that the real crossover field is Brb < Bmax +rb . +Since, ηcol < 1 this inequality holds for T ≪ Tc, where +ξ(T) ≈ ξ0 and λ(T) ≈ λ0. It is argued bellow, that in +order to explain the TBR experiment in H3S the regime +B > Brb is also realized. +Note, that in cuprates with columnar defects, pinning +properties are highly anisotropic with the maximal criti- +cal current for the magnetic field aligned along the colum- +nar defect (and for jc ⊥ H). The latter is confirmed for +irradiated YBa2Cu3O7 [18]. It seems, that this is not the +case for H3S, where j⊥ +c0 ≈ j∥ +c0 and similar columnar densi- +ties of defects are realized along and perpendicular to the +sample surface, i. e. B⊥ +φ ≃ B∥ +φ [3]. This implies that the +columnar defects in H3S are oriented along, both, per- +pendicular and parallel, axes almost equally - see Fig.2. +C. +Thermal vortex depinning from columnar +defects +For any high temperature superconductor thermal fluc- +tuations of vortex lines are important at higher temper- +atures, because of smoothing of the pinning potential. +This significantly lowers jc(T) at and above some depin- +ning temperature Tr +dp, when the effective pinning po- +tential (per unit vortex length) ϵp(T) = ϵp(0)ϕ(T) be- +comes small, since ϕ(T > Tr +dp) ≪ 1. For the sake of clar- +ity - there are two types of thermal motion of vortex +lines in the presence of pinning centers : (i) Phonon- +like, with small amplitude fluctuations affecting an indi- +vidual pinning potential - intravalley fluctuations, thus +smoothing the pinning potential and reducing jc(T) sig- +nificantly near Tr +dp. +(ii) The second kind of thermal +motion is related to large intervalley thermal fluctua- +tions, which cause jumping of vortices from one to an- +other pinning center (valley). These are mainly respon- +sible for the vortex-creep phenomena - which is not stud- +ied here. +Here, we deal with type (i) thermal effects +in the regime of the single-vortex pinning. Above and at +Tr +dp the amplitude of the thermal fluctuations +� +u2�1/2 +th in- +creases beyond the extent of the vortex core, +� +u2� +th > ξ2. +In that case, the vortex experiences smaller averaged +pinning potential. +The calculation of Tr +dp is sophisti- +cated and based either on: (i) the analogy of the vortex +statistical physics with columnar defects and the quan- +tum 2D-Bose gas placed in a random pinning poten- +tial, or (ii) on the statistical physics of vortices [19]. +It turns out that for r ≳ +√ +2ξ0 the depinning tempera- +ture Tr +dp is approximately given by the self-consistent +equation Tr +dp ≈ r · +� +ϵp(Tdp)ϵ0(Tdp), where ϵp(T) = ηcol· +Φ2 +0/32π2λ2(T) and ϵ0 = Φ0/16π2λ2(T) [19]. +III. +TEMPERATURE BROADENING OF THE +RESISTANCE IN THE MAGNETIC FIELD IN H3S +In Refs. [10]-[13] it was claimed that in order to explain +the temperature broadening of the resistance in magnetic +field (TBR), δtc(h) ≡ (Tc − Tc(h))/Tc, in HP-hydrides +and in the framework of physics of soft superconductors, +it is necessary to invoke an unphysically large critical +current density jc0 > 109A/cm2. In the case of the ques- +tionable CSH superconductor even much larger critical +current is needed, i.e. jc0 > 1011A/cm2- where δtc(h) is + +1 +H +plastic pinning +pinning +1 +1 +1 +Brb +B +BG! +ec' +single vortex pinning +1 +1 +Bm +single rod +many +rods +0 +T +Tc1 +H +plastic pinning +pinning +1 +1 +1 +Brb +B +BG! +ec' +single vortex pinning +1 +1 +Bm +single rod +many +rods +0 +T +Tc4 +FIG. 2: Schematic view of the experiment of the magnetic +flux trapping and penetration in the H3S disk-like sample +[1]: +D = 30µm;P = 5µm for the perpendicular geometry +H0 ⊥ D. According to the SCPC-model, long columnar de- +fects (blue cylinders) strongly pin and trap vortices making +huge magnetization hysteresis and the critical current density +jc ∼ δM . +The non-superconducting 119Sn film (yelow with +d = 20µm; p = 2.6µm) is implemented in the experiment for +the detection of the penetrated magnetic field [3]. +Similar +analyzes holds for the parallel geometry H0 ∥ D. +field independent [7]. In the following it is argued, that +the reduction of δtc(h) in H3S can be explained by invok- +ing the SCPC model - which assumes that H3S is a hard +type − II superconductor with elongated intrinsic colum- +nar pinning defects. The geometry of the experiment is +shown in Fig. 2. +Let us briefly introduce the reader into the subject of +TBR in H3S, which is based on the Tinkham theory for +TBR [22] for the SCPC model [1]. +Namely, in all superconductors dissipationless current +can flow in the vortex state with pinning defects. How- +ever, when the pinning energy is small, especially for +T near Tc, vortices jump easily from one center to an- +other under temperature fluctuations, thus giving rise to +a vortex motion and dissipation of energy - called flux +creep [23]. These jumps are activation-like and propor- +tional to the escape probability (from the pinning cen- +ter) exp(−Up/T) . +Since for point defects Upd +p ∼ ξ3, +then this energy barrier is small in superconductors with +small ξ, what is, for instance, the origin of a pronounced +dissipation in HTSC-cuprates (with oxygen vacancies). +In that sense, the long pinning defects with the energy +U p ∼ Lvξ2 make this barrier much higher, thus sup- +pressing the dissipation effects significantly. In magnetic +fields much higher than the lower critical field, i.e. for +Hc1 ≪ H < Hc2, one has B ≈ µ0H and the vortex dis- +tance is given by a ≈ (Φ0/B)1/2 < λ. In that case bun- +dles of vortices, each with the surface ∼ a2, are pinned +[24] with the pinning energy of the bundle Up∼ Lva2. +The Tinkham TBR theory [22] applied to such prob- +lems is based on the Ambegaokar and Halperin theory +for thermally activated phase motion in Josephson junc- +tions [25]. As the result, it gives for the resistance of the +superconductor (with small transport current) [22] +R/RN ≈ [I0(γ/2)]−2 , γ = Up/T, +(3) +where RN is the resistance of the normal state at Tc and +I0 is the modified Bessel function. +In the SCPC model with columnar pinning centers +with L ≈ Lv and near Tc one obtains for γ +γscpc = βK +�Lv +ξ0 +� (1 − t)2 +h +(2πξ2 +0 +jc0Φ0 +cTc +), +(4) +where +h ≈ H/Hc2(0), +βK +≈ +1 +and +δtc ≡ (1 − t) ≡ 1 − T/Tc += δTc/Tc. +In the case of +point defects one has for γ +γpd ≈ (1 − t)3/2 +h +(2πξ2 +0 +jpd +c0 Φ0 +cTc +). +(5) +Here, the critical current for weak pinning jpd +c0 = ηpd · jc0 +is defined via the weak pining energy Upd +p ≈ ηpdH2 +cξ3 +with ηpd < 1 , while in the SCPC model one has +jc0 = ηcol · jmax +co +- see Eq. (1). The critical current density +for weak pinning centers is usually of the order jpd +c0 ∼ 106 +A/cm2. Note, that γscpc ∼ (1 − t)2/h for long colum- +nar defects, while γpd ∼ (1 − t)3/2/h for point defects. +The latter fact gives rise to the different temperature de- +pendence of the irreversible line Hirr(T) in HTSC (with +oxygen vacancies as point defects) and H3S - see more +below. From these expressions it is seen, that the rela- +tive barrier in the case of long columnar pinning centers +is larger by the factor (Lv/ξ0) ≫ 1, compared with the +one for point-like randomly distributed defects - used in +[10]-[13]. This means that jpd +c0 for point defects is replaced +by much larger quantity (Lv/ξ0)jc0 in the SCPC-model - +where one has (Lv/ξ0)jc0≫jc0. +Let us compare the prediction of the standard theory +for weak pinning (with point-like defects) applied to H3S +- done in [10]-[13]. +In the case when the resistance is +measured at the 10 % level, i.e. R/RN = 0.1, it gives for +I0(γ/2) ≈ 3.2 and γ ≈ 5(= γscpc = γpd in both cases. In +the field B ≈ µ0H ≈ 1T and for µ0Hc2(0) ≈ 100 T one +has h = 0.01. We assume also the following realistic pa- +rameters for the HP-hydrides: ξ0 ≈ 20 ˚A and Tc∼ 200 K, +Φ0 = 2 × 10−7G × cm2. If one takes jpd +c0 =αjc0 ≈ α × 107 +A/cm2 with α < 1, then by using Eq.(5) one obtains +δtpd +c ≡ δT pd +c (h = 0.01) +Tc +≈ 0.1 +α . +(6) +This is a too large value (δtpd +c +> 0.1) - see the red line +in Fig. +3, compared to the experimental one δtexp +c +� +10−2 [2]. Based on Eq.(5) one concludes that in order to +explain the experimental value δtexp +c +(h = 0.01) � 10−2 + +5 +FIG. 3: Broadening of the superconducting transition (TBR) +δtc(h) ≡ ∆T/Tc under external magnetic fields in different +superconducting HP-hydrides derived in Ref. +[10] - points +and solid lines and the experimental values for TBR in H3S +extracted in [2] - blue points. Blue line - the theoretical line +predicted by the SCPC-model δtscpc +c(h) ∼ h1/2 (h = H/Hc2, +Hc2 ≈ 100T) for jc0 ≈ 107A/cm2 and L ≈ 0.5µm. The model +is more suitable for low h < 0.01- see text. +Red line - the +prediction of the standard model for TBR with weak-pinning +δtpd +c (h) ∼ h2/3 and for jc0 ≈ 107A/cm2 is inadequate for H3S +. +in H3S in the framework of the weak pinning theory one +needs a mach larger (effective) critical current jpd +c0 > 3·108 +A/cm2. However, the latter value is far beyond the range +of the weak pinning theory. +However, the experimental value δtexp +c +� 10−2 can +be explained by the SCPC model with the long colum- +nar pinning defects, where the δtscpc +c +(h) depends on the +large factor (L/ξ0)jc0, thus making δtscpc +c +(h) small. In +the case of H3S where γ00 = 2πξ2 +0×jcoΦ0/cTc and jc0 +is in the range jc0 > 107A/cm2 (in CGS units jco> +3×1016 esu/cm2) and for Lv ∼ (0.5 − 1) µm one obtains +δtscpc +c +(h) ≈ +� 5ξ0h +Lvγ00 +�1/2 +≲ 0.01. +(7) +The obtained result for TBR in Eq. (7) in the SCPC- +model for h = 0.01 and jc0 ∼ 107A/cm2 is in satisfactory +agreement with the experimental values shown in Fig. +3 - the blue line. +Since δtscpc +c +≪δtpd +c +this means that +the SCPC model is able to describe TBR in the H3S +superconductor. +Note, that there is TBR also in the zero magnetic filed +(H = 0), i.e. there is an intrinsic TBR, δT0 ̸= 0 , which +should be taken into account in the analyzes of exper- +iments. +This was done in [2] and [10], where one has +δTc(h)≈δTtot(h) − δTc0 and δTtot is the total TBR. The +extracted experimental results for δtc(h)(≡ δTc(h)/Tc) in +H3S are shown in Fig.3 [2] - blue circles, for h within the +range of 0 < h < 0.15. +We stress again, that in order to explain the much +smaller TBR , δtc(h), in H3S (and in other HP-hydrides) +than the standard Tinkham theory predicts for the weak +pinning - see Eq.(5), the authors of [10]-[13] assume an +unrealistically large critical current density jpd∼ (109 − +1011)A/cm2. The latter value is much larger than the ex- +perimental one jexp +c0 � 107A/cm2. The second possibility +is to call into question the existence of superconductiv- +ity in HP-hydrides. This (second) possibility is accepted +in [10]-[13]. +However, in the proposed SCPC model +jc0 in the formula for δtc(h)is in fact replaced by the +much larger quantity (Lv/ξ0)jc0 - see Eq.(4), which for +(Lv/ξ0) ∼ 103 gives realistic values for jc0 � 107 A/cm2. +This analysis confirms our claim, that there is no reason +to call into question the existence of the conventional su- +perconductivity in H3S. +IV. +PENETRATION OF THE MAGNETIC +FIELD IN H3S +In order to confirm the existence of superconductiv- +ity in a sample two kinds of experiments in an external +magnetic field are necessary: (i) At T > Tc the sample +is placed in the magnetic field H < Hc1(or Hc in type- +I superconductors) and than cooled below Tc. - the FC +(field cooled) experiment. If the magnetic field is expelled +from the sample the Meissner effect is realized. Its ex- +istence means a definitive proof for superconductivity in +the sample; (ii) In the zero field cooled (ZFC) experiment +the sample is first cooled to T Tc - when +the sample is non-superconducting and the perpendicu- +lar magnetic field penetrates completely into the 119Sn +film - Fig. 3 in Ref. [3]. Note, that since H3S is a type +II superconductor one expects that if the sample is free +of pinning centers, then for B0 ≫ µ0Hc1 =(18 − 60) mT +the magnetic field should enter into the whole sample in +the form of (almost) homogeneously distributed vortices +- thus showing quantum beats in the 119Sn film. How- +ever, in the perpendicular geometry of the experiment +(H ⊥ D in Fig. 2) at low temperatures and in the field +B0 = 0.68 T [3] the vortices do not show up in the cen- +ter of the 119Sn film up to some temperature, i.e. there +are no quantum beats - Fig. 3 in [3]. This means that +the vortices are strongly pinned in the region outside of +the 119Sn film. Strong pinning of vortices in a supercon- +ductor implies, that the magnetization of the sample is +strongly hysteretic, as revealed experimentally in H3S - +see [2],[4],[14]. +Two +geometries +were +studied +in +[3]: +(A) +The +perpendicular geometry - where the field is perpendicu- +lar to the disk surface (H ⊥ D in Fig.2); (B) The parallel +geometry - the field is parallel to the disk surface (H ∥ D). +As it is seen in Fig. 4A the average field inside the 119Sn +sensor in the perpendicular geometry is approximately +zero for T < 80 K and reaches the value Bext =0.68 T +FIG. 4: +The experimental temperature dependence of the +magnetic field on the sensor 119Sn film (placed) inside H3S +(see Fig. 2) at 153 GPa shown by blue triangles. The exter- +nal field at the reference non-superconducting sample 119Sn +in H2 at 150 GPa is shown by red dots - see [3]. (A) and +(B) are measurements in the perpendicular (H⊥ +ext ⊥ D) and +parallel (H∥ +ext ∥ D) geometry of the external magnetic field, +respectively. Dashed lines are guides to the eye. Taken from +[3]. +at around T≈ 120 K. In the parallel geometry the field +inside the 119Sn film is finite for T > 0 (but smaller than +Bext =0.68 T) starting to increase at T > 80 K saturating +to Bext =0.68 T at T≳ 140 K - Fig. 4B. In the next Sub- +section these experimental results are explained by the +theory which is based on the Bean critical state model +and the SCPC model with high critical current density +jc0 � 107A/cm2. +B. +Penetration of the magnetic field in a long +cylinder with pinning centers +In Section III it is argued, that the experimental re- +sults on the thermal broadening of the resistance (TBR) +in magnetic filed in H3S, δtexp +c +(h), give much smaller +value than those predicted by the Tinkham theory for +standard superconductors with weak pinning, i. +e. +δtexp +c +(h)≪ δtpd +c (h). +However, the SCPC model with + +A +0.8 +119sn in H2 +0.7 +0.6 +(Tesla) +0.5 +0.4 +us +0.3 +P ~ 153 GPa +0.2 +Hext = 0.68 Tesla +0.1 +Hext samp. plane +0.0 +AM AA A +0 +40 +80 +120 +160 +200 +240 +Temperature (K)B +0.8 +119sn in H2 +0.7 +0.6 +0.5 +1I +(Tesla) +0.4 +0.3 +us +P~153 GPa +工 +0.2 +Hext = 0.65 Tesla +0.1 +Hext II samp. plane +0.0 +0 +40 +80 +120 +160 +200 +240 +Temperature (K)7 +the columnar pinning centers and large critical cur- +rents jc(0) > 107A/cm2 explains this property, i. +e. +δtscpc +c +(h) ≈ δtexp +c +(h). Note, that the large value of jc(0) +inevitably causes large magnetization hysteresis, which +is seen experimentally in H3S [2],[4],[14]. +The critical +current density jc(0) is a measure of the strength of pin- +ning forces. Below, it is shown that the large value of +jc(0) is compatible with experiments for the penetration +of the magnetic field into thin H3S film [3]. To maxi- +mally simplify the problem a long cylinder placed in the +external field B0 = µ0H0 is first considered (in absence of +external transport currents), thus escaping demagnetiza- +tion effects. This case is also used in studying the field +penetration in the parallel geometry of H3S, i.e. when +B0 ∥ D. +In an homogeneous system in thermodynamic equilib- +rium - in an ideal type II superconductor (without pin- +ning centers), the vortices are homogeneously distributed +over the bulk sample. In that case the macroscopically +averaged (over the sample) local magnetic induction is +constant, i.e. Beq(r)= const and the macroscopic local +magnetization current density (averaged over the vortex +unit cell) is zero, i.e. +µ0¯jeq(r) = rotBeq=0, as well as +the Lorentz force per unit volume of the vortex lattice +f eqL = ¯jeq ×Beq= 0. However, in the presence of pinning +centers B(r) is inhomogeneous and ¯J(r)̸= 0, thus pro- +ducing finite Lorentz force (per unit volume) on vortices +f L = ¯j × B̸= 0. When the vortices are pinned there is a +force (per unit volume) f p which counteracts the Lorentz +force. In the static case the vortices are not moving and +the condition f L = −fp is fulfilled everywhere in the sam- +ple. So, by increasing the applied field the vortices are +so rearranged that locally the maximal critical current +µ0jc(B) = rotB is achieved. +The simplest but very useful model for the critical state +is the Bean critical state model [26], which assumes that +jc is independent on B. In the case of a long cylinder L≫ +R (no demagnetization effects) one has dB/dr = ±µ0jc +and B(r) is given by +B(r, B0) = B0 − B∗(T)(1 − r +R). +(8) +It is seen from Eq.(8) that for an applied filed +0 ≤ B0 < B∗ the field B(r) penetrates up to the point +rB0 = (1 − B0/B∗)R, where B(rB0, B0) = 0. At B0 = B∗ +the field reaches the center of the cylinder, i.e. +at +rB∗ = 0 one has B(rB∗ = 0, B∗) = 0 - see Fig. 5. In the +experiment [3] - schematically given in Fig.2, one has +D = 30 µm and by assuming that L ≫ D one has 1.8 +T +(averaged over the cylinder) - the virgin (initial) line in bold; +(c) distribution of the current density in the superconducting +cylinder in an axial field: left - in the increasing film from +B0 = 0 to B0 < B∗ the screening current flows between rf and +R where B(rf) = 0; middle - for B0 > B∗ it flows in the whole +sample; right - in decreasing field from B0 +max to B0 the current +is inverted in the external sheath. From [27]. +C. +Penetration of the magnetic field in H3S - +perpendicular geometry +In the perpendicular geometry B0(= µ0H0) ⊥ D of the +thin disk with L ≪ R (i.e. P ≪ D in the experiment of +Ref. [3] - see Fig. 2) there is a geometric barrier when the +sample shape (due to geometric spikes) is different from + +Bz +[a.u.] ↑ +VsB +max +BO = B* +0 +max +max +0 +rf +R +r +(a) +(b)(c)8 +an ellipsoid. It turns out, that in that case the magnetic +field penetrates into the sample in an inhomogeneous way +[28]. The theory [28] predicts that in the perpendicular +geometry the flux is penetrated up the center in the ex- +ternal field B0⊥ = Bp⊥ with +Bp⊥(T) = B∗(T) P +Dln +� +D +P + +� +(1 + D2 +P 2 +� +, +(9) +where B∗(T) = µ0jc(T)D/2. In order to explain the tem- +perature dependence of the field in the 119Sn sensor [3] +- see Fig. 4A, we define an effective field on the 119Sn +sensor +B⊥ +Sn(T) ≡ B0 − Bp⊥(T). +(10) +(Here, T is the absolute temperature.) +By definition, +for B0 ≤ Bp⊥(T) one has B⊥ +Sn = 0, i.e. +the field does +not reach the center of the sample. +When, T ≪ Tc , +jc0 ≈ 1.4 × 107˚A/cm2 and (P/D) ≈ 1/6 - see Fig. 2, one +has Bp⊥ ≈ 1T which gives that +B0(= 0, 68T) 0 and the external field +B0 is large enough to push vortices into the center of the +119Sn film - see Fig. +6B, which is in agreement with +experimental result in Fig. 4B. The fit of ϕ(T) (note, +that jc(T) ∼ ϕ(T)) is analogous to the perpendicular case + +9 +but with slightly different parameters Tkink,∥ ≈ 122K and +Tr +dp,∥ ≈ 142K . Note, that there is a qualitative differ- +ence between B⊥ +Sn(T) and B∥ +Sn(T), since B⊥ +Sn(T) is zero +up to some finite temperature - see Fig. 4 and Fig. 6, +and B∥ +Sn(T) is finite even at T = 0K. This is due to dif- +ferent sample dimensions, i.e. that P ≪ D - see Fig. 2. +These results are in a qualitative and semi-quantitative +agreement with the experimental results in [3]. +It is necessary to point out following items: (a) It +is seen in Fig.5, that in the Bean critical state model +the irreversible magnetization for a long cylinder in +fields B0 > B∗ is constant (field independent). +Since +the measurements of the magnetic moment of the H3S +sample in [3] are given in the CGS unit emu we use +it also here. +After averaging of M = (B − H)/4π over +the sample one has Mir(emu/cm3) ≈(R/30)jc(A/cm2) +and the total magnetic moment µir = ¯ +Mir · Vs. +Note, +that ¯Mir = ( ¯M+ − ¯M−)/2 = ¯M− since in the Bean model +¯M+ = − ¯M holds for B0 > B∗. For the volume of the disk +Vs ≈ 0.8 × P · D2 one obtains the trapped magnetic mo- +ment in the sample to be of the order µ ≈ (0.3−2)×10−5 +emu for B0 > Bp⊥ and for jc0 ≈ (1.4 − 10) × 107A/cm2. +The calculated trapped magnetic moment in the H3S +sample of Ref.[3] is far beyond the SQUID sensitivity +threshold - which is ∼ 10−8 emu [3]. This means that the +measurements of the trapped magnetic flux of vortices +(but without the extrinsic moments) are realizable. (b) In +order to explain the field penetration in the H3S sample +of Ref. [3] it comes out that the perpendicular (H ⊥D) +critical current density j⊥ +c0 must be approximately equal +to the parallel (H ∥ D) one j∥ +c0, i. +e. +j⊥ +c0 ≈ j∥ +c0. +This +means, that in the H3S sample of Ref. [3] the columnar +defects are oriented along both directions, the parallel +and perpendicular one, in a similar way - schematically +shown in Fig. 2. Having in mind the cubic-like struc- +ture of H3S this kind of pinning isotropy is an acceptable +assumption; (c) In order to explain the TBR and PMF +measurements in thin H3S samples in the SCPC model, +it comes out that the bulk H3S sample is a high-κ type II +superconductor with the parameters: ξ0 ≈ (15 − 20) ˚A, +λ0 ≈ (1 − 2) × 103˚A ; κ ≈ (50 − 100), µ0Hc1 ≈ (18 − 60) +mT, µ0Hc0≈ (0.6 − 1.1) T, µ0Hc2 ≈ (80 − 140) T. These +values are very different from those obtained in [2], [14], +where ξ0 ∼ 20 ˚A, λ0 ∼ (1.3 − 2) × 102˚A ; κ ∼ 7 − 10, +µ0Hc0∼ 6 T, µ0Hc1 ∼ 1 T, µ0Hc2 ≈ (80 − 140) T. The +values of the parameters predicted in the SCPC model +are compatible with those obtained in the magnetic mea- +surements and also with the microscopic theory - see the +discussion below. +To this point, Hirsch and Marsiglio have recently real- +ized [29], that their previous interpretation [10]-[11] of the +Troyan’s measurements in H3S [3] in terms of pinning-free +superconductors with unphysically high jc0∼ 1011A/cm2 +may be inadequate. Namely, due to the pronounced mag- +netization hysteresis in [4] they speculated the presence +of pinning forces in H3S - with the critical current density +jc0 ∼ 107A/cm2. However, they did not realize that the +TBR and PMF effects in H3S are due to the strong pin- +ning by the elongated (columnar) defects - as the SCPC +model predicts [1]. +To conclude this Section - the magnetic measurements +of the penetration of the magnetic field B0 = 0.68 T +in the H3S sample [3]can be naturally explained in the +framework of the SCPC model and the Bean critical state +model. This approach also explains naturally the high +critical current density in the H3S samples, which is of the +order jc0 ≈ (1.4 − 1.5) × 107A/cm2. Thereby, the finite- +size effects in the thin H3S disk of Ref. +[3] are taken +into account. This analyzes tell us, that there is no need +to call into question the existence of superconductivity +in H3S (and in other HP-hydrides), as it is claimed in +[10]-[13]. +E. +Meissner effect in H3S +The Meissner effect is an important hallmark of the +superconducting state. +It is realized in the so called +FC (field cooled) experiment, when the sample (ideally +without pinning defects) is in the normal state (T > Tc) +and placed in a magnetic field H0 < Hc1(for type-II su- +perconductors). The latter penetrates into the normal +metallic sample fully, i. +e. +one has B ≈ µ0H0. +How- +ever, if the sample is then cooled down into the su- +perconducting state (T < Tc) the magnetic field will +be expelled from the bulk sample, i. +e. +one has +B = 0 in an ideal non-magnetic superconductor. +The +Meissner effect should not be confused with the ZFC +(zero − field cooled) experiment, where a nonmagnetic +bulk metallic sample is first cooled into the superconduct- +ing state at T < Tc in the zero field (H = 0) and there- +after magnetic field H < Hc1 is turned on. As a result, the +magnetic field (induction) is excluded from the bulk sam- +ple, i.e. B = µH = 0 with µ = µ0(1 + χ) = 0. The ZFC +phenomenon is also called diamagnetic shielding. How- +ever, the ZFC effect would also be realized in a perfect +metal (with ϱ = 0) - if it existed in nature, where it is due +to the classical Lenz law of the electrodynamics. From +this analyzes comes out, that the magnetic susceptibility +of an ideal bulk superconductor is diamagnetic χ = −1 +in the SI system (4πχ = −1 in the CGS) in both types +of experiments. Note, that when the FC experiment is +done in a perfect metal, the magnetic field is not expelled +from the sample at T < Tc, i.e. the magnetic flux stays +frozen in the sample with the same value as in the nor- +mal state, i.e. B ≈ µ0H0. So, for a definite proof of the +Meissner effect in superconductors one should perform +the FC experiment. +In that respect, several inconsistent experimental and +theoretical results related to the magnetization measure- +ments in H3S (and LaH10) were published (in the pe- +riod 2015-2022) in [2],[4],[14]. These results are strongly +criticized in [10]-[13]. We shall not elaborate this criti- +cism in details, but enumerate few, in our opinion, main +points by Hirsch and Marsiglio [10]-[13]. These are: (1) + +10 +FIG. 7: Up - The FC and ZFC magnetization (M) in H3S - +from [2],[4]. Down - the volume susceptibility χV of the weak +ferromagnetic superconductor RuSr2GdCu2O8; a) - ZFC (zfc) +measurements of χVat Hex = 6.5 Oe; doted line is χV in +the non-superconducting state. TM ≈ 137 K is the magnetic +critical temperature; Tms ≈ 30 K is the transition tempera- +ture to the spontaneous vortex state (SVS) which exists at +Tms < T < Tc. Inset: enlarged scale of χV around the super- +conducting transition Tc ≈ 45K ; b) FC measurements of χV +for external fields Hex ≈ (0.5 − 2.5) Oe. Inset: The volume +fraction of the Meissner state f ≈ 40% at Hex = 0, 5Oe - from +[15]. +The Meissner effect is not observed experimentally in +H3S (and other HP-hydrides) [2],[4],[14]). +Namely, in +the FC measurements only some parasitic paramagnetic +susceptibility is observed, i. e. χF C(T < Tc) > 0, and +there are no signs of the flux expulsion. +However, in +the ZFC experiment there is a clear diamagnetic shield- +ing (at T < Tc) but existing on the background of the +parasitic paramagnetic signal. Most probably, the origin +of this paramagnetic signal is not intrinsic, since there +is no reliable reason for strong paramagnetic effects in +H3S. +To this point, there is also a pronounced para- +magnetic contribution in the FC measurements in the +HTSC superconductor RuSr2GdCu2O8 - with Tc≈ 45 K, +where a weak ferromagnetism appears below TM ≈ 137 +K [15]. At T ≥ Tms ≈ 30 K a spontaneous vortex state +(SVS) appears, where Hc1(T) < M and M is the spon- +taneous magnetization. In Fig. 7 it is seen, that in the +FC measurements χv is paramagnetic, even for T < Tc +, i. +e. +χV(T) > 0 . +At Tms ≈ 30 K the suscepti- +bility χV decreases suddenly and the sample is in the +bulk Meissner state at T < Tms with the volume fraction +f =| (χV(0) − χV(Tms)/(1 + χV(Tms)) |≈ 40% at Hex = +0, 5 Oe - see Inset in Fig. 7b. This tells us that pinning +centers are present in the RuSr2GdCu2O8 sample. +Having in mind this analyzes, one expects that in the +FC measurements in H3S a sudden decrease of χV should +be realized below Tc. However, so far, no such results +have been published with clearly realized Meissner ef- +fect in HP-hybrides. The paramagnetic signal in the H3S +sample is most probable due to some extrinsic proper- +ties of the diamond anvil cell (or of the rest of the sulfur +atoms, since H2S is decomposed into H3S and S) and +this fact deserves further studies. (2) Additional critical +remarks, given in [10]-[13], are related to some controver- +sial findings in [14], that in the H3S (and LaH10) sample +the critical fields Hc1(0) and Hc(0) are too large, while +the penetration dept λ0 is too large. If this is true, the +large value of the critical fieldµ0Hc1(0) in [14] would give +very high critical current density in H3S (and LaH10), +i. e. jc0 ∼ 1010 A/cm2. This is much higher value than +the depairing current density jdep ≃ 5 × 108A/cm2, what +is in fact impossible. The large value for Hc(0) (in Ref. +[14]) is incompatible with the microscopic theory of su- +perconductivity, which relates the condensation energy +to Hc(0) by (N(EF)∆2) = H2 +c(0)/8π. Here, N(EF) is the +density of states (per unit volume) at the Fermi surface. +For 2∆ ≈ 3, 5Tc and µ0Hc(0) ∼ 10 T (in [14]) one obtains +that N(EF) ≈30 × NDFT(E), while the density functional +theory gives NDFT(EF) ≈ 0, 2 states/spin × eV×(˚A)3. +Therefore, this highly overestimated value for the density +of states, which are extracted from the magnetic mea- +surements in H3S [2],[4],[14], is an highly unacceptable +value. +Let us discuss the origin of these too large values for +the bulk critical fields µ0Hc1(∼ 1 T) and µ0Hc0(∼ 10 T) +obtained in [14]. +The latter result is based on an in- +adequate experimental definition of Hc1. Namely, it is +determined from the onset field µ0Hp(0)(∼ 0.1T) of the +deviation of M(H) from the linear dependence by assum- +ing that Hc1 = Hp(0)/(1 − N). In the measurements in +Ref. [14] the demagnetization factor is N ≈ 0.96, what +gives too large value for µ0Hc1(T = 0) ≈ 2 T. That this +procedure is not well defined, i. e. Hp(0) is not related +to Hc1, can be seen in Fig. +5b where M(H) curve in +the Bean critical state model, where the saturation field +H∗(P, D) ≫ Hc1(0). +In the perpendicular geometry of +the experiment [3] one has P = 5 µm and D = 30 µm , +N ∼ 0.7 and according to Eq. (9) one has µ0Hp,⊥(0) ≈ 1 +T what is much larger than the real Hc1(T = 0). If we +would apply the same procedure for obtaining Hc1 in the +H3S sample, as it was done in Ref. [14], we would obtain +also an unrealistic value µ0Hc1(T = 0) ≈ 3 T , instead of +the realistic one µ0Hc1(0) ≈ (18 − 60) mT. + +emu) +10 +FC +5 +100um +onset +203K +ZFC +-5 +100 +150 +200 +250 +Temperature(K)fc +2.5 0e +1.50e +0.5. +0.75 0e +0.50e +0.0. +T +ms +0.4 +/ (Xv+1) +0.2 +-0.5. +Axv! +b) +0.0 +2 +4 +681012 +H* (Oe) +-1.0. +0 +255075100125150 +T(K)T +0.2. +zfc; 6.5 Oe +M +0.0. +0.08 +-0.2. +0.07 +0 +T. +0 +0.06. +-0.4- +0 +20 40 +60 80 +0 +50 +100 +150 +T(K)11 +V. +SUMMARY AND DISCUSSION +Recently, the authors o Refs. [10]-[13] raised important +questions on the reliability of magnetic measurements in +high-pressure hydrides (HP-hydrides). Their skepticism +goes so far, that they tend to conclude that supercon- +ductivity does not actually exist in HP-hydrides [10]-[13]. +This attitude is mostly related to the FC magnetic mea- +surements, which are until now unable to prove unam- +biguously the existence of the Meissner effect in small +samples of H3S. On the other side, some ZFC measure- +ments, in small samples of H3S, are experimentally more +reliable, because these are not related to the flux trapping +effects. There are also difficulties to explain some exper- +imental results of magnetic measurements in H3S - done +in [2]-[4],[14], if they are treated by the standard theory +of superconductivity with weak pinning of vortices. The +latter approach is mainly accepted in [10]-[13], thus re- +ducing the possibility for explaining experiments in H3S, +such as: (i) TBR - the temperature broadening of the +resistance in magnetic field, and (ii) PMF - the penetra- +tion of the magnetic field into the center of the sample. +In order to explain these two kinds of experiments - +which can not be explained by the weak pinning the- +ory at all, Ref. [1] introduces the SCPC model - which +holds for superconductors with strong pinning defects. +Moreover, even the quantitative explanation of these two +phenomena (in H3S ) is possible by assuming that the +pinning centers are in the form of long columnar defects, +which are “isotropically” distributed over the sample, i. +e. with the same values of the critical current densities +flowing perpendicular and parallel to the sample surface. +In the framework of the SCPC model it is possible to ex- +plain these two kind of experiments. In the following we +summarize the obtained results: 1. The large reduction +of the temperature broadening of resistance in magnetic +field (TBR), δtscpc +c +(h), is due to the long columnar de- +fects L≈ Lv(ortex)≫ ξ0 with the radius r ∼ ξ0. In such +a case, both the core and the electromagnetic pinning +are operative. +These cause high densities of the criti- +cal current (at T ≪ Tc) - of the order jc0 = (107 − 108) +A/cm2. The SCPC-model predicts, that in H3S the tem- +perature width of TBR is governed by the small param- +eter C =ξ0/Lcol, i.e. δtscpc +c +(h) ∼ C1/2h1/2 with C ∼ 10−3 +and h = H/Hc2. This gives δtscpc +c +(h) � 0.01 for h ≲ 0.01 +and L ∼ 1 µm, which is in satisfactory agreement with +experimental results in H3S [2], as shown in Fig. 3. It +is also seen that the SCPC model fits the experimen- +tal results much better than the model with weak pin- +ning by small defects. The SCPC model also predicts, +that the magnetic irreversible field is governed by 1/C, +i.e. +Hirr ∼ (1 − t)2L/ξ0. +The irreversibility line is not +only significantly increased compared to HTSC-cuprates, +but the temperature dependence, as a measure of the +strength of pinning forces, is given by (1 − t)2 instead +of (1 − t)3/2 - characteristic for materials with point-like +defects. Measurements of the irreversible line Hirr in H3S +are desirable. +These columnar defects cause high density of the criti- +cal current and also a large magnetization hysteresis ∆M +in H3S. +This property opens a possibility for making +powerful high-field superconducting magnets. +For in- +stance, by making (if possible) a long superconducting +cylinder of H3S (with L ≫ R) the magnetic hysteresis in +that case is given by ∆M ∼ jc0 × R, where R is the radius +of the superconducting cylinder. For instance, in H3S for +jc0 > 107A/cm2 and for R ∼ 0.6cm one has µ0M ∼ 100 +T at the temperature T < 50 K. +2. +The magnetic penetration depth (PMF) in H3S +can be naturally explained by the SCPC model, where +the strong columnar pinning of vortices dominates. +The experiment measures PMF in the applied field +B0 = µ0H0 = 0.68 T [3]. If the field is penetrated in the +center of the sample, then the quantum beats should ap- +pear in the 119Sn sensor. It turns out that in the per- +pendicular geometry, when the field is perpendicular to +the sample surface (B0 ⊥ D - see Fig. 2), the magnetic +field does not penetrate to the center, while in the par- +allel case (B0 ∥ D - see Fig. +2) it penetrates partially +even for T ≪ Tc [3]. These experimental results are nat- +urally explained in the SCPC model, where the “isotrop- +ically” distributed strong columnar pinning defects make +a large current density jc0 ≈ (1.3 − 1.5)×107 A/cm2. It +is also predicted, that the depinning temperature Tr +dp - +where the critical current density is strongly weakened +(jc(T) ≪ jc0) and the field B0 is fully penetrated into the +H3S sample, is of the order Tr +dp ∼ (100 − 120)K. +Moreover, in order to explain the TBR and PMF ex- +periments by the SCPC model it comes out that H3S +is a high-κ superconductor with bulk physical parame- +ters: ξ0 ≈ (15 − 20) ˚A, λ0 ≈ (1 − 2) × 103˚A ; κ ≈ +(50−100), µ0Hc1(0) ≈ (18 − 60) mT, µ0Hc0 ≈(0.6 − 1.1) +T, µ0Hc2 ≈ (80−140) T. These values of parameters are +also compatible with the microscopic theory of super- +conductivity. Finally, it is natural to raise the question +- what is the origin of these columnar pinning defects +in H3S? Serious candidates are single edge dislocations +or their bundles, what is a matter of further research. +To conclude - the SCPC model, which assumes the ex- +istence strongly acting columnar pinning centers, is able +to explain important TBR and PMF measurements in a +high-κ H3S superconductor. Therefore, there is no need +for calling into question the existence of superconductiv- +ity in H3S [10]-[13]. +Acknowledgments +The author would like to thank Dirk Rischke and +Radoˇs Gaji´c for permanent support and to Igor Kuli´c +for discussions and support. + +12 +[1] M. L. Kuli´c, arXiv:21.04.12214v1(2021) +[2] M. I. Eremets et al, arXiv:2201.05137 (2022). +[3] I. Troyan et al., Science 351, 133 (2016) +[4] A. Drozdov, M. Eremets, I. Troyan, V. Ksenofontov, S. +Shylin, Nature 525, 73 (2015) +[5] M. Somayazulu, M. Ahart, A. K. Mishra, Z. M. Geballe, +M. Baldini, Y. Meng, V. V. Struzhkin, and R. J. Hemley, +Phys. Rev. Lett., 122:027001, (2019); A. P. Drozdov, P. +P. Kong, V. S. Minkov, S. P. Besedin, M. A. Kuzovnikov, +S. Mozaffari, L. Balicas, F. F. Balakirev, D. E. Graf, V. +B. Prakapenka, E. Greenberg, D. A. Knyazev, M. Tkacz, +and M. I. Eremets, Nature 569, 528, (2019) +[6] D. V. Semenok et al., arXiv:2012.04787 (2020) +[7] E. Snider et al., Nature 586, 373 (2020) +[8] E. Snider et al., Nature 586, 373 (2020)D. Duan, Y. Liu, +F. Tian, D. Li, X. Huang, Z. Zhao, H. Yu, B. Liu, W. +Tian, T. Cui, Sci. Rep. 4, 6968; DOI:10.1038/srep06968 +(2014) +[9] N. Bernstein, C. Stephen Hellberg, M. D. Johannes, I. I. +Mazin, M. J. Mehl, Phys. Rev. B 91, R060511 (2015) +[10] J. E. Hirsch. F. Marsiglio, Phys. Rev. B 103, 134505 +(2021); arXiv: 2101.01701 (2021) +[11] J. E. Hirsch. F. Marsiglio, arXiv: 2103.00701v3 (2021) +[12] J. E. Hirsch. F. Marsiglio, arXiv: 2110.07568v1 (2021); +[13] J. E. Hirsch. F. Marsiglio, arXiv: 2110.07568v5 (2022) +[14] V. S. Minkov et al, +DOI:10.21203/rs.3.rs-936317/v1 +(2021). +[15] C. Bernhard, J. L. Tallon, E. Br¨ucher, R. K. Kremer, +Phys. Rev. B 61, R14 960 (2000) +[16] W. H. Jiao, H. F. Zhai, J. K. Bao, Y. K. Luo, Q. Tao, C. +M. Feng, Z. A. Xu, G. H. Cao, New Journal of Physics +15, 113002 (2013) +[17] Yu. A. Genenko, S. V. Yampolskiib, A. V. Pan, Applied +Physics Letters 84, 3921 (2004) +[18] L. Civale et al., Phys. Rev. Lett. 67, 648 (1991) +[19] G. Blatter et all, Rev. Mod. Phys. 66, 1125 (1994) +[20] G. S. Mkrtchyan, V. V. Schmidt, Sov. Phys. JETP 34, +195 (1972) +[21] M. L. Kuli´c, A. Kr¨amer, K. D. Schotte, Solid State Com- +mun. 82, 541 (1992); A. Kr¨amer, M. L. Kuli´c, Phys. Rev. +B 48, 9673 (1993); A. Kr¨amer , M. L. Kuli´c, Phys. Rev. +B 50, 9484 (1994) +[22] M. Tinkham, Phys. Rev. Lett. 61, 1658 (1988) +[23] P. W. Anderson, Phys. Rev. Lett. 9, 309 (1962) +[24] Y. Yeshurun, A. P. Malozemoff, Phys. Rev. Lett. 60, 2202 +(1988) +[25] V. Ambegaokar, V. I. Halperin, Phys. Rev. Lett. 22, 1364 +(1969) +[26] C. P. Bean, Rev. Mod. Phys. 36, 31 (1964) +[27] P. Mangin, R. Kahn, Superconductivity An Introduction, +Springer (2017) +[28] E. H. Brandt, Phys. Rev. B 58, 6506 (1998) +[29] J. E. Hirisch. F. Marsiglio, arXiv: 2109.10878v2 (2021) + diff --git a/YtFPT4oBgHgl3EQftzUO/content/tmp_files/load_file.txt b/YtFPT4oBgHgl3EQftzUO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7f98b593e91d9bb2d40c577c80f0ebace53e5cbd --- /dev/null +++ b/YtFPT4oBgHgl3EQftzUO/content/tmp_files/load_file.txt @@ -0,0 +1,925 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf,len=924 +page_content='H3S is a high-κ superconductor with columnar pinning defects Miodrag L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Kuli´c Institute for Theoretical Physics, Goethe-University Frankfurt am Main, Germany (Dated: January 31, 2023) Recently, the existence of superconductivity in H3S and other high-pressure hydrides is called into question, because of some flawed magnetic measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' [1]) the SCPC model is proposed, where strong pinning of vortices is due to long columnar defects in H3S - with lengths L of the order of vortex lengths Lv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Two relevant type of experiments in magnetic fields are explained by this model: (1) Reduction (with respect to standard superconductors) of the thermal broadening of resistance (TBR) in magnetic field h = H/Hc2, δtc(h), is governed by the small parameter C =ξ0/L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' δtscpc c (h) ∼ C1/2h1/2 with C ∼ 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' This gives δtscpc c (h) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='01 for h ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='01 and L ∼ 1 µm, which is in satisfactory agreement with measurements in H3S [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The TBR measurements give, that in H3S there is a large shift of the irreversible line Birr(T) towards the Bc2(T) line, with B(H3S) ir ∼ C−1(1 − t)2 instead of Bir ∼ (1 − t)3/2 (t = T/Tc) - in standard superconductors with point defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' (2) In ZFC (zero field cooled) experiments on penetration of the magnetic field (PMF) in H3S [3], the latter reaches the center of a superconducting disk at much larger external fields, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' for B0 ≫ Bc1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The later is due to the pronounced pinning of vortices, but not to the Meissner effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The calculated T- dependence of the penetrated, perpendicular B⊥(T) and parallel B∥(T), magnetic field into the sample is in satisfactory agreement with the experimental results for H3S in [3], where B⊥ > B∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Problems related to measurements of the Meissner effect in H3S are also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The SCPC model, applied on the bulk H3S sample, predicts, that the latter is a high-κ superconductor with ξ0 ≈ (15 − 20) ˚A , λ0 ≈ (1 − 2) × 103˚A ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' κ ≈ (50 − 100), µ0Hc1(0) ≈(18 − 60) mT, µ0Hc0≈ (0, 6 − 1, 1) T, µ0Hc2(0) ≈ (80 − 140) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' INTRODUCTION The first almost room-temperature superconductor was reported in 2015 in sulphur-hydrides (H3S) with Tc≈ 203 K under high pressure P ≈ 150 GPa(≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='5 Mbar), which is based on resistivity measurements [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' This finding opens a new frontier in physics and a num- ber of other HP-hydrides were even predicted before these were synthesized thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Let us mention some of them with Tc≥ 200 K - such as LaH10with Tc ≈ 250 K, P≈ 190 GPa [5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' LaYHx with Tc= 253 K, P≈ 190 GPa [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In all of them Tc goes down by increasing magnetic field, compatible with the standard theory of superconductiv- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' There are also reports on the room temperature su- perconductivity in the CSH hydride - a superconductor based on C, S and H, with Tc= 287 K at P≈ 267 GPa [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' However, this result was not yet confirmed, neither experimentally nor theoretically by other groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' There is also theoretical support for the (almost) room- temperature superconductivity in HP-hydrides, which are based on the calculated critical temperature Tc in the microscopic Migdal-Eliashberg theory for superconduc- tivity - which is due to the electron-phonon interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Moreover, the theoretical prediction of superconductiv- ity in HP-hydrides [8] is a rare example in the physics of superconductivity, that the theory goes ahead of ex- periments, by predicting Tc(≈ 200K) in H3S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Note, that in [4] it was assumed that the H2S structure is realized, while in [9] it is argued, that at high pressures the phase diagram favors decomposition of H2S into H3S and pure S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' A main proof for superconductivity is the existence of the Meissner effect, where the magnetic field H < Hc1 - applied above Tc, is expelled from the sample at tem- peratures below Tc - the field cooled (FC) experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' However, until now the Meissner effect is not proved ex- perimentally in HP-hydrides, which caused justified crit- icism on this subject in [10]-[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The failure to measure the Meissner effect in H3S is due to the following reasons: (i) Magnetic measurements in small samples are delicate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' (ii) The existence of some extrinsic paramagnetic effects in samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In that respect, some contradictory experi- mental results and their inconsistent theoretical interpre- tations - given in [2],[4],[14], were among the reasons that Hirsch and Marsiglio even called into question the exis- tence of superconductivity in HP-hydrides [10]-[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In the following, we are going to show that some magnetic measurements in HS can refute this skepticism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The content of the article is following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In Section II the SCPC model - firstly introduced in [1] for hard type-II superconductors with strong columnar pinning centers, is further elaborated and applied to the magnetic mea- surements in the H3S superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' This model pro- poses, that the vortex pinning is due to long columnar de- fects (L ∼ Lv ≫ ξ0) - with the radius of the cross-section r ∼ ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In that case, the core and electromagnetic pin- ning contribute almost equally to the elementary pinning energy Up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' This is an optimal situation for pinning in a superconductor, since there is a maximal gain in the con- densation and electromagnetic vortex energy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' This gives a maximal pinning force if the superconductivity is fully suppressed in the columnar defects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' for ∆ = 0 inside a defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' It seems that ∆ is finite in H3S, but (∆ < ∆0) and the critical current density is smaller than the max- imal one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' jco = ηcoljmax c0 with ηcol < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In Section III the reduction of the temperature broadening of resistance arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='13153v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='supr-con] 30 Jan 2023 2 (TBR) in a magnetic field, δtexp c (h) (with h = H/Hc2), in HP-hydrides is studied in the SCPC model and ap- plied to H3S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The temperature dependence of the irre- versibility line with Bir ∼ C−1(1 − t)2 is predicted within the SCPC model also in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In Section IV the SCPC model is applied in studying the penetration of the magnetic field (PMF) into the center of the H3S sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The PMF measurements are analyzed in the Bean criti- cal state model, first for a long superconducting cylinder (the parallel configuration B∥) and then for thin disks (perpendicular configuration B⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' It is shown, that the critical magnetic field Bp at which it penetrates into the center of the sample is larger for the perpendicular ge- ometry than for the parallel one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Bp⊥ > Bp∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The comparison of the theory and the experimental results for PMF in H3S [3] gives a large critical current jc0� 107 A/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In Section IV we discuss some experimental controver- sies related to the Meissner effect and FC experiments in H3S, which display a large residual paramagnetic magne- tization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The latter fact does not fit into the classical the- ory of the Meissner effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Note, that the paramagnetic signal can be present in some magnetic superconductors, superconductors with intrinsic magnetic moments in [15]- [16] or extrinsic ones [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Finally, in Section V the ob- tained results for TBR and PMF in the SCPC model for H3S are summarized and discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Here, the crucial difference between pinning forces in HTSC-cuprates and HP-hydridesis also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Note, in the following we use the notation ξ0 ≈ ξ(T ≪ Tc) and λ0 ≈ λ(T ≪ Tc) II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' STRONG VORTEX PINNING BY COLUMNAR DEFECTS IN H3S A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Single vortex pinning To increase the critical current density it is desirable to have extended (long columnar) pinning defects - the correlated disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In this case the pinning potential is correlated over the extended size of the defect L∼ Lv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' (Lv is the vortex length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=') This means that the superconduc- tivity is suppressed in a large volume Vcol= πξ2 0Lv and therefore a single vortex prefers sitting on this defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Note, that the maximal pinning energy Umax p = ϵmax p Vcol is reached when the superconductivity is fully suppressed in the core of defects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' when ∆ = 0 in dielectric columnar defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' This might be not the case in H3S, where the real pinning energy density is ϵp = ηcol · ϵmax p with the reducing factor ηcol < 1, because the gap in- side the columnar defect is finite, ∆ < ∆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In that case ηcol = ϵp/ϵmax p ∼ (1 − ∆2/∆2 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' As a result, the critical current density for a single vortex (sitting on such a de- fect) is given by jc = ηcol · jmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Note, that in HTSC-cuprates long columnar pinning defects are made artificially by irradiating YBa2Cu3O7 single crystalline samples of small platelets of 1×1×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='02 mm3 with different doses of 580 MeV 116Sn30+ ions [18]- [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The density of the ion doses are D ≈ 5 × 1010,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='5 × 1011, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='4×1011ions/cm2, which are equivalent to the cor- responding vortex densities with the magnetic induc- tion Bφ ≈ 1, 3, 5 T - Bφ is the matching field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Since the ionization energy-loss rate was 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='7 keV/˚A they pro- duce long tracks with the length L ∼ 20 − 30 µm and diameter ∼ 50 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In that case jc(∼ jc0) is of the order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='5×107A/cm2 at T = 5 K and 106A/cm2at T = 77 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' These current densities are much larger than those for point defects, where the pinning is due to oxygen vacan- cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In the columnar case the irreversible line Bir(T) (the line below which the pinning is pronounced) lies higher than for weak pinning with point defects [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In the following it is argued, that the columnar pin- ning defects dominate in H3S - see blue colored cylinders in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='2, with the averaged distance dφ = (Φ0/Bφ)1/2 [20]-[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' For an optimal pinning the superconductiv- ity should be fully destroyed inside the dielectric colum- nar defect with radius r ≳ ξ [20]-[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In this case, both mechanisms of pinning - the core and the electromagnetic one, are operative with almost same pinning energy [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The maximal pinning energy is given by Up scpc≈ 2πξ2Lv(H2 c /8π) ≡ Lvϵmax p with ϵmax p ≈ Φ2 0/32π2λ2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Since the elementary pinning force (per unit vortex length) fp =ϵmax p /ξ is balanced in the critical state by the Lorenz force (per unit length) fL(≡ FL/Lv) = jcΦ0/c, the critical current density jmax c in the SCPC model is given by [20]-[21] jmax c ≈ cΦ0 32π2λ2ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' (1) In anisotropic superconductors jmax c is given in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' (In SI units jmax c,SI ≈ Φ0/8πµ0λ2ξ, where λ(T) ≈ λ0(1−t)−1/2 and ξ(T) ≈ ξ0(1 − t)−1/2 are the Ginzburg-Landau pen- etration depth and coherence length, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In H3S one has λ ≈ (1−2)×103˚A and ξ0 ∼ 20 ˚A [4], which gives for jmax c0 in the range jmax c0 ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='8−3)×108A/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In the following, the critical current density jc0 will be esti- mated from experiments on the magnetic penetration in H3S [3], where it is found jc0 ≈ (1, 3 − 1, 5)×107˚A/cm2, which gives for the reducing factor η ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='1 − 0, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Crossover field Brb for the single vortex to vortex bundles pinning The generic H − T phase diagram for the high tem- perature superconductors with columnar pinning defects is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' 1 - see [19], and some magnetic properties of H3S will be discussed in the framework of this phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='B the single-vortex pin- ning is considered, which occurs at small magnetic field B < Bφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' When the inter-vortex distance av ≈ (Φ0/B)1/2 is larger than the average distance between columnar de- fects dφ = (Φ0/Bφ)1/2 and the single vortex pinning en- ergy is larger than the inter-vortex energy, than the vor- tices accommodate freely to the pinning sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' However, 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' 1: The generic H − T phase diagram for superconduc- tors with columnar defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The doted melting line Bm(T) of the pure sample is transformed into a Bose-glass transition line BBG(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Also shown are the various pinning regimes with a single-vortex/single-rod pinning region at low fields µ0H < Brb(T) < 2BΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' For T < Tr dp the fluctuations of a vor- tex from one defect to another are suppressed, while for T > Tr dp the pinning potential is exponentially reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' At T > Tdl the individual flux lines are pinned collectively by an assembly of columnar defects (rods) at high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Above the crossover line Brb(T), the largest energy in the problem is the inter-vortex interaction and pinning involves vortex bundles - taken from [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' when B > BΦ the inter-vortex interaction starts to be important for pinning and dynamical properties (such as the vortex creep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In that case, vortex bundles are pinned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The maximal crossover field Bmax rb see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='1, separates the single vortex regime from the vortex bun- dle regime and it is obtained by comparing the energy of the elastic shear deformation (of the order u ∼ dΦ) with the maximal pinning potential ϵmax p per unit length, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' ϵshear = c66(dφ/av)2a2 v⋍ ϵmax p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The shear modulus is given by c66 ≈ Φ0B/(8πλ)2what gives for Brb Brb < Bmax rb ≲ 4ϵmax p ϵ0 Bφ, (2) where ϵ0 = Φ2 0/16π2λ2 [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Since ϵmax p = ϵ0/2 it gives, that for B < Bbr < Bmax rb ≈ 2BΦ the columnar defects outnumber the vortices and the single-vortex pinning pre- vails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Note, that the real crossover field is Brb < Bmax rb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Since, ηcol < 1 this inequality holds for T ≪ Tc, where ξ(T) ≈ ξ0 and λ(T) ≈ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' It is argued bellow, that in order to explain the TBR experiment in H3S the regime B > Brb is also realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Note, that in cuprates with columnar defects, pinning properties are highly anisotropic with the maximal criti- cal current for the magnetic field aligned along the colum- nar defect (and for jc ⊥ H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The latter is confirmed for irradiated YBa2Cu3O7 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' It seems, that this is not the case for H3S, where j⊥ c0 ≈ j∥ c0 and similar columnar densi- ties of defects are realized along and perpendicular to the sample surface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' B⊥ φ ≃ B∥ φ [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' This implies that the columnar defects in H3S are oriented along, both, per- pendicular and parallel, axes almost equally - see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Thermal vortex depinning from columnar defects For any high temperature superconductor thermal fluc- tuations of vortex lines are important at higher temper- atures, because of smoothing of the pinning potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' This significantly lowers jc(T) at and above some depin- ning temperature Tr dp, when the effective pinning po- tential (per unit vortex length) ϵp(T) = ϵp(0)ϕ(T) be- comes small, since ϕ(T > Tr dp) ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' For the sake of clar- ity - there are two types of thermal motion of vortex lines in the presence of pinning centers : (i) Phonon- like, with small amplitude fluctuations affecting an indi- vidual pinning potential - intravalley fluctuations, thus smoothing the pinning potential and reducing jc(T) sig- nificantly near Tr dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' (ii) The second kind of thermal motion is related to large intervalley thermal fluctua- tions, which cause jumping of vortices from one to an- other pinning center (valley).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' These are mainly respon- sible for the vortex-creep phenomena - which is not stud- ied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Here, we deal with type (i) thermal effects in the regime of the single-vortex pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Above and at Tr dp the amplitude of the thermal fluctuations � u2�1/2 th in- creases beyond the extent of the vortex core, � u2� th > ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In that case, the vortex experiences smaller averaged pinning potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The calculation of Tr dp is sophisti- cated and based either on: (i) the analogy of the vortex statistical physics with columnar defects and the quan- tum 2D-Bose gas placed in a random pinning poten- tial, or (ii) on the statistical physics of vortices [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' It turns out that for r ≳ √ 2ξ0 the depinning tempera- ture Tr dp is approximately given by the self-consistent equation Tr dp ≈ r · � ϵp(Tdp)ϵ0(Tdp), where ϵp(T) = ηcol· Φ2 0/32π2λ2(T) and ϵ0 = Φ0/16π2λ2(T) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' TEMPERATURE BROADENING OF THE RESISTANCE IN THE MAGNETIC FIELD IN H3S In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' [10]-[13] it was claimed that in order to explain the temperature broadening of the resistance in magnetic field (TBR), δtc(h) ≡ (Tc − Tc(h))/Tc, in HP-hydrides and in the framework of physics of soft superconductors, it is necessary to invoke an unphysically large critical current density jc0 > 109A/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In the case of the ques- tionable CSH superconductor even much larger critical current is needed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' jc0 > 1011A/cm2- where δtc(h) is 1 H plastic pinning pinning 1 1 1 Brb B BG!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=" ec' single vortex pinning 1 1 Bm single rod many rods 0 T Tc1 H plastic pinning pinning 1 1 1 Brb B BG!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=" ec' single vortex pinning 1 1 Bm single rod many rods 0 T Tc4 FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' 2: Schematic view of the experiment of the magnetic flux trapping and penetration in the H3S disk-like sample [1]: D = 30µm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='P = 5µm for the perpendicular geometry H0 ⊥ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' According to the SCPC-model, long columnar de- fects (blue cylinders) strongly pin and trap vortices making huge magnetization hysteresis and the critical current density jc ∼ δM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The non-superconducting 119Sn film (yelow with d = 20µm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='6µm) is implemented in the experiment for the detection of the penetrated magnetic field [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Similar analyzes holds for the parallel geometry H0 ∥ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' field independent [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In the following it is argued, that the reduction of δtc(h) in H3S can be explained by invok- ing the SCPC model - which assumes that H3S is a hard type − II superconductor with elongated intrinsic colum- nar pinning defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The geometry of the experiment is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Let us briefly introduce the reader into the subject of TBR in H3S, which is based on the Tinkham theory for TBR [22] for the SCPC model [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Namely, in all superconductors dissipationless current can flow in the vortex state with pinning defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' How- ever, when the pinning energy is small, especially for T near Tc, vortices jump easily from one center to an- other under temperature fluctuations, thus giving rise to a vortex motion and dissipation of energy - called flux creep [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' These jumps are activation-like and propor- tional to the escape probability (from the pinning cen- ter) exp(−Up/T) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Since for point defects Upd p ∼ ξ3, then this energy barrier is small in superconductors with small ξ, what is, for instance, the origin of a pronounced dissipation in HTSC-cuprates (with oxygen vacancies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In that sense, the long pinning defects with the energy U p ∼ Lvξ2 make this barrier much higher, thus sup- pressing the dissipation effects significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In magnetic fields much higher than the lower critical field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' for Hc1 ≪ H < Hc2, one has B ≈ µ0H and the vortex dis- tance is given by a ≈ (Φ0/B)1/2 < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In that case bun- dles of vortices, each with the surface ∼ a2, are pinned [24] with the pinning energy of the bundle Up∼ Lva2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The Tinkham TBR theory [22] applied to such prob- lems is based on the Ambegaokar and Halperin theory for thermally activated phase motion in Josephson junc- tions [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' As the result, it gives for the resistance of the superconductor (with small transport current) [22] R/RN ≈ [I0(γ/2)]−2 , γ = Up/T, (3) where RN is the resistance of the normal state at Tc and I0 is the modified Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In the SCPC model with columnar pinning centers with L ≈ Lv and near Tc one obtains for γ γscpc = βK �Lv ξ0 � (1 − t)2 h (2πξ2 0 jc0Φ0 cTc ), (4) where h ≈ H/Hc2(0), βK ≈ 1 and δtc ≡ (1 − t) ≡ 1 − T/Tc = δTc/Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In the case of point defects one has for γ γpd ≈ (1 − t)3/2 h (2πξ2 0 jpd c0 Φ0 cTc ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' (5) Here, the critical current for weak pinning jpd c0 = ηpd · jc0 is defined via the weak pining energy Upd p ≈ ηpdH2 cξ3 with ηpd < 1 , while in the SCPC model one has jc0 = ηcol · jmax co see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The critical current density for weak pinning centers is usually of the order jpd c0 ∼ 106 A/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Note, that γscpc ∼ (1 − t)2/h for long colum- nar defects, while γpd ∼ (1 − t)3/2/h for point defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The latter fact gives rise to the different temperature de- pendence of the irreversible line Hirr(T) in HTSC (with oxygen vacancies as point defects) and H3S - see more below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' From these expressions it is seen, that the rela- tive barrier in the case of long columnar pinning centers is larger by the factor (Lv/ξ0) ≫ 1, compared with the one for point-like randomly distributed defects - used in [10]-[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' This means that jpd c0 for point defects is replaced by much larger quantity (Lv/ξ0)jc0 in the SCPC-model - where one has (Lv/ξ0)jc0≫jc0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Let us compare the prediction of the standard theory for weak pinning (with point-like defects) applied to H3S done in [10]-[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In the case when the resistance is measured at the 10 % level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' R/RN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='1, it gives for I0(γ/2) ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='2 and γ ≈ 5(= γscpc = γpd in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In the field B ≈ µ0H ≈ 1T and for µ0Hc2(0) ≈ 100 T one has h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' We assume also the following realistic pa- rameters for the HP-hydrides: ξ0 ≈ 20 ˚A and Tc∼ 200 K, Φ0 = 2 × 10−7G × cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' If one takes jpd c0 =αjc0 ≈ α × 107 A/cm2 with α < 1, then by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' (5) one obtains δtpd c ≡ δT pd c (h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='01) Tc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='1 α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' (6) This is a too large value (δtpd c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='1) - see the red line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' 3, compared to the experimental one δtexp c � 10−2 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' (5) one concludes that in order to explain the experimental value δtexp c (h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='01) � 10−2 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' 3: Broadening of the superconducting transition (TBR) δtc(h) ≡ ∆T/Tc under external magnetic fields in different superconducting HP-hydrides derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' [10] - points and solid lines and the experimental values for TBR in H3S extracted in [2] - blue points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Blue line - the theoretical line predicted by the SCPC-model δtscpc c(h) ∼ h1/2 (h = H/Hc2, Hc2 ≈ 100T) for jc0 ≈ 107A/cm2 and L ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='5µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The model is more suitable for low h < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='01- see text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Red line - the prediction of the standard model for TBR with weak-pinning δtpd c (h) ∼ h2/3 and for jc0 ≈ 107A/cm2 is inadequate for H3S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' in H3S in the framework of the weak pinning theory one needs a mach larger (effective) critical current jpd c0 > 3·108 A/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' However, the latter value is far beyond the range of the weak pinning theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' However, the experimental value δtexp c � 10−2 can be explained by the SCPC model with the long colum- nar pinning defects, where the δtscpc c (h) depends on the large factor (L/ξ0)jc0, thus making δtscpc c (h) small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' In the case of H3S where γ00 = 2πξ2 0×jcoΦ0/cTc and jc0 is in the range jc0 > 107A/cm2 (in CGS units jco> 3×1016 esu/cm2) and for Lv ∼ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='5 − 1) µm one obtains δtscpc c (h) ≈ � 5ξ0h Lvγ00 �1/2 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' (7) The obtained result for TBR in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' (7) in the SCPC- model for h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='01 and jc0 ∼ 107A/cm2 is in satisfactory agreement with the experimental values shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' 3 - the blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Since δtscpc c ≪δtpd c this means that the SCPC model is able to describe TBR in the H3S superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Note, that there is TBR also in the zero magnetic filed (H = 0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' there is an intrinsic TBR, δT0 ̸= 0 , which should be taken into account in the analyzes of exper- iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' This was done in [2] and [10], where one has δTc(h)≈δTtot(h) − δTc0 and δTtot is the total TBR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The extracted experimental results for δtc(h)(≡ δTc(h)/Tc) in H3S are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='3 [2] - blue circles, for h within the range of 0 < h < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' We stress again, that in order to explain the much smaller TBR , δtc(h), in H3S (and in other HP-hydrides) than the standard Tinkham theory predicts for the weak pinning - see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' (5), the authors of [10]-[13] assume an unrealistically large critical current density jpd∼ (109 − 1011)A/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The latter value is much larger than the ex- perimental one jexp c0 � 107A/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' The second possibility is to call into question the existence of superconductiv- ity in HP-hydrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' This (second) possibility is accepted in [10]-[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' However, in the proposed SCPC model jc0 in the formula for δtc(h)is in fact replaced by the much larger quantity (Lv/ξ0)jc0 - see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' (4), which for (Lv/ξ0) ∼ 103 gives realistic values for jc0 � 107 A/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' This analysis confirms our claim, that there is no reason to call into question the existence of the conventional su- perconductivity in H3S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' PENETRATION OF THE MAGNETIC FIELD IN H3S In order to confirm the existence of superconductiv- ity in a sample two kinds of experiments in an external magnetic field are necessary: (i) At T > Tc the sample is placed in the magnetic field H < Hc1(or Hc in type- I superconductors) and than cooled below Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' - the FC (field cooled) experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' If the magnetic field is expelled from the sample the Meissner effect is realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' Its ex- istence means a definitive proof for superconductivity in the sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf'} +page_content=' (ii) In the zero field cooled (ZFC) experiment the sample is first cooled to T